Algebra is Useful: 5000 Practice Problems to Show how Algebraic Identities can be used to Simplify Arithmetic Calculations 4900000000, 4900280004, 2500000000, 2500100001, 4900140001, 6400000000, 6400480009, 3600000000
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Algebra is Useful: 5000 Practice Problems to Show how Algebraic Identities can be used to Simplify Arithmetic Calculations Norman Sanko
What This Book Is About Many students who encounter algebra for the first time when they enter middle school think that it is only about letters and operations and identities and factors and equations. They are partly right. Elementary and intermediate algebra are definitely about all those things, but their applications in other fields - physics, economics, biology, business modeling, and so on - are vast, and growing rapidly. This book focuses on the application of elementary algebraic identities to simplify arithmetic and numerical calculations. In the age of the calculator and of calculator apps, tedious computations done by hand are almost a thing of the past - but not entirely. As part of the steps required to solve a long problem in mathematics or in allied fields, long numerical calculations are often encountered. It is tedious to pull out the calculator every single time; indeed, some tests do not allow the usage of calculators, even in this day and age. In such situations, it is very useful to be able to use algebraic identities to simplify complex computations. And who can forget the fact that you look like a genius when you reduce what looks like a difficult task even for a calculator, to something you can do in your head? This book focuses on the three basic square identities that students spend the first few months of an introductory algebra course mastering. As you go through the theory, through the solved examples, through the Challenge Section, and through the detailed drills, you will find that it is flexible, and that the difficulty level rises slowly. You will be able to level up till your expertise in the subject matches that which you require in your chosen field of
study; you can spend ten hours or a thousand hours on the exercises in this book, according to your requirement. Future books in this series will explore the applications of other identities from intermediate algebra, including the cubic identities. I am confident that the tools you learn in this book will stand you in good stead throughout your education, and throughout your career. Best of luck!
Table of Contents What This Book Is About The Identity (a + b)² = (a² + b² + 2ab): Basic Theory The Identity (a + b)² = (a² + b² + 2ab): Practice Problems Solutions: The Identity (a + b)² = (a² + b² + 2ab): Practice Problems The Identity (a - b)² = (a² + b² - 2ab): Basic Theory The Identity (a - b)² = (a² + b² - 2ab): Practice Problems Solutions: The Identity (a - b)² = (a² + b² - 2ab): Practice Problems The Identity (a + b)(a - b) = (a² - b²): Basic Theory The Identity (a + b)(a - b) = (a² - b²): Practice Problems Solutions: The Identity (a + b)(a - b) = (a² - b²): Practice Problems The Identity (a + b)² = (a² + b² + 2ab): Challenges Solutions: The Identity (a + b)² = (a² + b² + 2ab): Challenges The Identity (a - b)² = (a² + b² - 2ab): Challenges Solutions: The Identity (a - b)² = (a² + b² - 2ab): Challenges The Identity (a + b)(a - b) = (a² - b²): Challenges Solutions: The Identity (a + b)(a - b) = (a² - b²): Challenges Detailed Drills Solutions: Detailed Drills
The Identity (a + b)² = (a² + b² + 2ab): Basic Theory The algebraic identity: (a + b)² = (a² + b² + 2ab), Where a and b can take on any numerical values, is the first identity that is taught in most middle school algebra classrooms. It is also one of the most useful. This identity is usually used in simplifying long algebraic expressions, but it can also be very useful in making arithmetic calculations more simple. Let’s find out how! Using this algebraic identity to help with arithmetic calculations For small values of a and b, the identity does not give us any useful shortcuts in calculation. Assume a = 5, and b = 2, in the identity above. The Left Hand Side of the identity = (a + b)² = (5 + 2)² = 7² = 49. The Right Hand Side = (a² + b² + 2ab) = (5² + 2² + 2 x 5 x 2) = 25 + 4 + 20 = 49. Both calculations are relatively equal in complexity - and they, of course, give us the same result. However, for certain values of a and b, this identity can drastically reduce the amount of computational work we have to do. Here is an example: Let us imagine that we were asked to find the value of 10012 (without using a calculator app). This can be done by multiplying 1001 by 1001, by hand - but this is quite cumbersome, and errors might happen. Here is another way: In the identity (a + b)² = (a² + b² + 2ab), we put a = 1000, and b = 1. The Left Hand Side of this equation becomes (1000 + 1)² = (1001)², which is exactly what we want to find.
The Right Hand Side of this equation is (a² + b² + 2ab) = (1000² + 1² + 2 x 1000 x 1) = (1000000 + 1 + 2000) = 1002001. You will have noticed that this method was very simple, because we were able to avoid difficult calculations. It is also more likely to be error-free, because all we have to worry about is addition, which is much easier to do by hand than a long multiplication. The same identity can help us simplify many other cumbersome arithmetic calculations. In the next chapter, you will see a few different calculations for which you can take the help of our trusty identity. Detailed solutions are also provided, so that you can arm yourself with all the examples you need, before you move to the Challenge Section!
The Identity (a + b)² = (a² + b² + 2ab): Practice Problems Detailed solutions can be found in the next chapter.
1. Please find the value of 1002², by using algebraic identities to simplify computation. -2. What is the value of 557² + 443² + (557 x 886)? Please use algebraic identities to make the computation simple. -3. What is the value of [4408² - 2244² - 2164²] / [4488]? -4. What is the square root of [983² + 182² + 357812]? --
Solutions: The Identity (a + b)² = (a² + b² + 2ab): Practice Problems
1. Please find the value of 1002², by using algebraic identities to simplify computation. -Solution: Since 1002 is only a little higher than 1000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 1000 and b = 2. After the substitution, we get 1002² = (1000 + 2)² = 1000² + 2² + (2 x 1000 x 2) = 1000000 + 4 + 4000 = 1004004. ---2. What is the value of 557² + 443² + (557 x 886)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 886 = 2 x 443. Using this fact, we can rewrite the given expression as 557² + 443² + (2 x 557 x 443). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 557² + 443² + (557 x 886) = 557² + 443² + (2 x 557 x 443) = (557 + 443)² = 1000² = 1000000. ---3. What is the value of [4408² - 2244² - 2164²] / [4488]? -Solution: First, we look at the three numbers under the square signs in the numerator: 4408, 2244, and 2164. We note that 4408 = 2244 + 2164. We also note, in the denominator, that 4488 = 2 x 2244. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 2244, and b = 2164, to get [(2244 + 2164)² - 2244² - 2164²] / 4488 = [4408² - 2244² - 2164²] / 4488 = 2164. The Left Hand Side of this equation is exactly the expression in our question, so
the value of that expression is equal to 2164. ---4. What is the square root of [983² + 182² + 357812]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 983, and b = 182, we see that 2ab = 357812, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [983² + 182² + 357812] = [983 + 182]². Therefore, the square root of the expression given is equal to [983 + 182] = 1165. ----
The Identity (a - b)² = (a² + b² - 2ab): Basic Theory In the previous identity, if we replace b with -b, we get the closely related: (a - b)² = (a² + b² - 2ab), Where, again, a and b can take on any numerical values. We can also use this identity to aid us in arithmetic calculations. Finding the value of 992 If we had to find the value of the square of 99 by hand, we would do a multiplication that was fairly complex, and involves carrying numbers. There is an easier way. In the identity (a - b)² = (a² + b² - 2ab), we put a = 100, and b = 1. This gives us: 992 = (100 - 1)² = (100² + 1² - 2 x 100 x 1) = 10000 + 1 - 200 = 9801. As you can see, this is very quick, and more likely to help us avoid errors. In the next chapter, you will look at multiple practice problems that extend this concept, with detailed solutions available for each.
The Identity (a - b)² = (a² + b² - 2ab): Practice Problems Detailed solutions can be found in the next chapter.
1. Please find the value of 9998², by using algebraic identities to simplify computation. -2. What is the value of 2953² + 1953² - (2953 x 3906)? Please use algebraic identities to make the computation simple. -3. What is the value of [4437² + 1630² - 2807²] / [8874]? -4. What is the positive square root of [179² + 50² - 17900]? --
Solutions: The Identity (a - b)² = (a² + b² - 2ab): Practice Problems
1. Please find the value of 9998², by using algebraic identities to simplify computation. -Solution: Since 9998 is only a little lower than 10000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 10000 and b = 2. After the substitution, we get 9998² = (10000 - 2)² = 10000² + 2² - (2 x 10000 x 2) = 100000000 + 4 - 40000 = 99960004. ---2. What is the value of 2953² + 1953² - (2953 x 3906)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3906 = 2 x 1953. Using this fact, we can rewrite the given expression as 2953² + 1953² - (2 x 2953 x 1953). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² - 2ab), we get 2953² + 1953² - (2953 x 3906) = 2953² + 1953² - (2 x 2953 x 1953) = (2953 1953)² = 1000² = 1000000. ---3. What is the value of [4437² + 1630² - 2807²] / [8874]? -Solution: First, we look at the three numbers under the square signs in the numerator: 4437, 1630, and 2807. We note that 2807 = 4437 - 1630. We also note, in the denominator, that 8874 = 2 x 4437. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 4437, and b = 1630, to get [(4437² + 1630² - (4437 - 1630)²] / 8874 = [4437² + 1630² - 2807²] / 8874 = 1630. The Left Hand Side of this equation is exactly the expression in our question, so
the value of that expression is equal to 1630. ---4. What is the positive square root of [179² + 50² - 17900]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 179, and b = 50, we see that 2ab = 17900, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [179² + 50² - 17900] = [179 - 50]². Therefore, the positive square root of the expression given is equal to [179 50] = 129. ----
The Identity (a + b)(a - b) = (a² - b²): Basic Theory To find the value of (a + b)(a - b), we can just multiply out the terms: (a + b)(a - b) = a (a - b) + b (a - b) = a² - ab + ab - b² = a² - b². How to use this identity to make certain calculations easier If we had to find the value of 101 x 99, we could either multiply it out, or use the identity. Putting a = 100, and b = 1, in the given identity, we get: (a + b)(a - b) = (a² - b²) (100 + 1)(100 - 1) = (100² - 1²) 101 x 99 = 10000 - 1 = 9999 We have achieved our result with minimal effort. In the next chapter, we will look at solved problems.
The Identity (a + b)(a - b) = (a² - b²): Practice Problems Detailed solutions can be found in the next chapter.
1. Evaluate the expression (805² - 195²), by using algebraic identities to simplify computation. -2. Evaluate the expression (10001² - 9999²), by using algebraic identities to simplify computation. -3. Find the value of (1001 x 999), by using algebraic identities to simplify the calculation. -4. What is the value of the positive square root of [(630 + 279)(630 - 279) + 77841]? --
Solutions: The Identity (a + b)(a - b) = (a² - b²): Practice Problems
1. Evaluate the expression (805² - 195²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 805 and b = 195 in the difference-of-squares identity. We then get (805² - 195²) = (805 + 195)(805 - 195). The Right Hand Side of this equation can be simplified to 1000 x 610 = 610000. ---2. Evaluate the expression (10001² - 9999²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 10001 and b = 9999 in the difference-of-squares identity. We then get (10001² - 9999²) = (10001 + 9999)(10001 - 9999). The Right Hand Side of this equation simplifies to 20000 x 2 = 40000. ---3. Find the value of (1001 x 999), by using algebraic identities to simplify the calculation. -Solution: We observe that both 1001 and 999 are close in value to 1000. We can rewrite the expression (1001 x 999) as (1000 + 1)(1000 - 1). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (1000 + 1)(1000 - 1) = 1000² - 1². This is relatively easy to compute; we have 1000² - 1² = 1000000 - 1 = 999999. ----
4. What is the value of the positive square root of [(630 + 279)(630 - 279) + 77841]? -Solution: First, we note that the number 77841 looks approximately like the square of 279. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(630 + 279)(630 - 279) + 77841] = [(630 + 279)(630 - 279) + 279²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(630 + 279)(630 - 279) + 279²] = [630² - 279² + 279²]. The second and third terms cancel, so the expression is just equal to 630². Clearly, the square root of the expression is equal to 630. ----
The Identity (a + b)² = (a² + b² + 2ab): Challenges Detailed solutions can be found in the next chapter.
1. Please find the value of 503², by using algebraic identities to simplify computation. -2. What is the value of 0.48² + 6.52² + (0.48 x 13.04)? Please use algebraic identities to make the computation simple. -3. What is the value of [184.33² - 101.47² - 82.86²] / [202.94]? -4. What is the square root of [33.3² + 61.3² + 4082.58]? --
Solutions: The Identity (a + b)² = (a² + b² + 2ab): Challenges Many more challenging questions on this topic can be found later in this book.
1. Please find the value of 503², by using algebraic identities to simplify computation. -Solution: Since 503 is only a little higher than 500, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 500 and b = 3. After the substitution, we get 503² = (500 + 3)² = 500² + 3² + (2 x 500 x 3) = 250000 + 9 + 3000 = 253009. ---2. What is the value of 0.48² + 6.52² + (0.48 x 13.04)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 13.04 = 2 x 6.52. Using this fact, we can rewrite the given expression as 0.48² + 6.52² + (2 x 0.48 x 6.52). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.48² + 6.52² + (0.48 x 13.04) = 0.48² + 6.52² + (2 x 0.48 x 6.52) = (0.48 + 6.52)² = 7² = 49. ---3. What is the value of [184.33² - 101.47² - 82.86²] / [202.94]? -Solution: First, we look at the three numbers under the square signs in the numerator: 184.33, 101.47, and 82.86. We note that 184.33 = 101.47 + 82.86. We also note, in the denominator, that 202.94 = 2 x 101.47. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 101.47, and b = 82.86, to get [(101.47 + 82.86)² - 101.47² - 82.86²] / 202.94 = [184.33² - 101.47² - 82.86²] / 202.94
= 82.86. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 82.86. ---4. What is the square root of [33.3² + 61.3² + 4082.58]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 33.3, and b = 61.3, we see that 2ab = 4082.58, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [33.3² + 61.3² + 4082.58] = [33.3 + 61.3]². Therefore, the square root of the expression given is equal to [33.3 + 61.3] = 94.6. ----
The Identity (a - b)² = (a² + b² - 2ab): Challenges Detailed solutions can be found in the next chapter.
1. Please find the value of 698², by using algebraic identities to simplify computation. -2. What is the value of 2.295² + 1.695² - (2.295 x 3.39)? Please use algebraic identities to make the computation simple. -3. What is the value of [567.2² + 156.6² - 410.6²] / [1134.4]? -4. What is the positive square root of [1722² + 90² - 309960]? --
Solutions: The Identity (a - b)² = (a² + b² - 2ab): Challenges Many more challenging questions on this topic can be found later in this book.
1. Please find the value of 698², by using algebraic identities to simplify computation. -Solution: Since 698 is only a little lower than 700, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 2. After the substitution, we get 698² = (700 - 2)² = 700² + 2² - (2 x 700 x 2) = 490000 + 4 - 2800 = 487204. ---2. What is the value of 2.295² + 1.695² - (2.295 x 3.39)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.39 = 2 x 1.695. Using this fact, we can rewrite the given expression as 2.295² + 1.695² - (2 x 2.295 x 1.695). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.295² + 1.695² - (2.295 x 3.39) = 2.295² + 1.695² - (2 x 2.295 x 1.695) = (2.295 - 1.695)² = 0.6² = 0.36. ---3. What is the value of [567.2² + 156.6² - 410.6²] / [1134.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 567.2, 156.6, and 410.6. We note that 410.6 = 567.2 - 156.6. We also note, in the denominator, that 1134.4 = 2 x 567.2. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 567.2, and b = 156.6, to get [(567.2² + 156.6² - (567.2 - 156.6)²] / 1134.4 = [567.2² + 156.6² - 410.6²] / 1134.4 =
156.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 156.6. ---4. What is the positive square root of [1722² + 90² - 309960]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1722, and b = 90, we see that 2ab = 309960, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1722² + 90² - 309960] = [1722 - 90]². Therefore, the positive square root of the expression given is equal to [1722 90] = 1632. ----
The Identity (a + b)(a - b) = (a² - b²): Challenges Detailed solutions can be found in the next chapter.
1. Evaluate the expression (7826² - 2174²), by using algebraic identities to simplify computation. -2. Evaluate the expression (8004² - 7996²), by using algebraic identities to simplify computation. -3. Find the value of (4008 x 3992), by using algebraic identities to simplify the calculation. -4. What is the value of the positive square root of [(810 + 134)(810 - 134) + 17956]? --
Solutions: The Identity (a + b)(a - b) = (a² - b²): Challenges Many more challenging questions on this topic can be found later in this book.
1. Evaluate the expression (7826² - 2174²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7826 and b = 2174 in the difference-of-squares identity. We then get (7826² - 2174²) = (7826 + 2174)(7826 - 2174). The Right Hand Side of this equation can be simplified to 10000 x 5652 = 56520000. ---2. Evaluate the expression (8004² - 7996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8004 and b = 7996 in the difference-of-squares identity. We then get (8004² - 7996²) = (8004 + 7996)(8004 - 7996). The Right Hand Side of this equation can be simplified to 16000 x 8 = 128000. ---3. Find the value of (4008 x 3992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 4008 and 3992 are close in value to 4000. We can rewrite the expression (4008 x 3992) as (4000 + 8)(4000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (4000 + 8)(4000 - 8) = 4000² - 8². This is relatively easy to compute; we have 4000² - 8² = 16000000 - 64 = 15999936.
---4. What is the value of the positive square root of [(810 + 134)(810 - 134) + 17956]? -Solution: First, we note that the number 17956 looks approximately like the square of 134. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(810 + 134)(810 - 134) + 17956] = [(810 + 134)(810 - 134) + 134²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(810 + 134)(810 - 134) + 134²] = [810² - 134² + 134²]. The second and third terms cancel, so the expression is just equal to 810². Clearly, the square root of the expression is equal to 810. ----
Detailed Drills
1. Please find the value of 4001², by using algebraic identities to simplify computation. -2. What is the value of 0.575² + 0.425² + (0.575 x 0.85)? Please use algebraic identities to make the computation simple. -3. What is the value of [222.6² - 113.84² - 108.76²] / [227.68]? -4. What is the square root of [42.8² + 38.5² + 3295.6]? -5. Please find the value of 70002², by using algebraic identities to simplify computation. -6. What is the value of 0.447² + 7.553² + (0.447 x 15.106)? Please use algebraic identities to make the computation simple. -7. What is the value of [127.08² - 52.72² - 74.36²] / [105.44]? -8. What is the square root of [55.3² + 79.6² + 8803.76]? -9. Please find the value of 7001², by using algebraic identities to simplify computation. -10. What is the value of 0.461² + 8.539² + (0.461 x 17.078)? Please use algebraic identities to make the computation simple. --
11. What is the value of [170.52² - 113.14² - 57.38²] / [226.28]? -12. What is the square root of [80.7² + 93² + 15010.2]? -13. Please find the value of 203², by using algebraic identities to simplify computation. -14. What is the value of 0.533² + 2.467² + (0.533 x 4.934)? Please use algebraic identities to make the computation simple. -15. What is the value of [141.25² - 74.79² - 66.46²] / [149.58]? -16. What is the square root of [75.7² + 65.6² + 9931.84]? -17. Please find the value of 30001², by using algebraic identities to simplify computation. -18. What is the value of 0.628² + 9.372² + (0.628 x 18.744)? Please use algebraic identities to make the computation simple. -19. What is the value of [250.09² - 129.05² - 121.04²] / [258.1]? -20. What is the square root of [88.3² + 69.2² + 12220.72]? -21. Please find the value of 602², by using algebraic identities to simplify computation. -22. What is the value of 0.514² + 4.486² + (0.514 x 8.972)? Please use algebraic identities to make the computation simple. -23. What is the value of [154.15² - 59.18² - 94.97²] / [118.36]? --
24. What is the square root of [82.7² + 13.2² + 2183.28]? -25. Please find the value of 50001², by using algebraic identities to simplify computation. -26. What is the value of 0.695² + 0.305² + (0.695 x 0.61)? Please use algebraic identities to make the computation simple. -27. What is the value of [123.06² - 52.33² - 70.73²] / [104.66]? -28. What is the square root of [21.8² + 46.3² + 2018.68]? -29. Please find the value of 602², by using algebraic identities to simplify computation. -30. What is the value of 0.631² + 5.369² + (0.631 x 10.738)? Please use algebraic identities to make the computation simple. -31. What is the value of [185.14² - 96.09² - 89.05²] / [192.18]? -32. What is the square root of [30.3² + 82.4² + 4993.44]? -33. Please find the value of 30001², by using algebraic identities to simplify computation. -34. What is the value of 0.44² + 8.56² + (0.44 x 17.12)? Please use algebraic identities to make the computation simple. -35. What is the value of [146.85² - 71.13² - 75.72²] / [142.26]? -36. What is the square root of [35.7² + 85² + 6069.]?
-37. Please find the value of 70001², by using algebraic identities to simplify computation. -38. What is the value of 0.483² + 5.517² + (0.483 x 11.034)? Please use algebraic identities to make the computation simple. -39. What is the value of [226.13² - 98.2² - 127.93²] / [196.4]? -40. What is the square root of [52.3² + 52.1² + 5449.66]? -41. Please find the value of 6001², by using algebraic identities to simplify computation. -42. What is the value of 0.638² + 9.362² + (0.638 x 18.724)? Please use algebraic identities to make the computation simple. -43. What is the value of [203.38² - 103.89² - 99.49²] / [207.78]? -44. What is the square root of [33.2² + 48.9² + 3246.96]? -45. Please find the value of 5003², by using algebraic identities to simplify computation. -46. What is the value of 0.611² + 8.389² + (0.611 x 16.778)? Please use algebraic identities to make the computation simple. -47. What is the value of [178.52² - 84.3² - 94.22²] / [168.6]? -48. What is the square root of [19.1² + 78.6² + 3002.52]? --
49. Please find the value of 3002², by using algebraic identities to simplify computation. -50. What is the value of 0.434² + 8.566² + (0.434 x 17.132)? Please use algebraic identities to make the computation simple. -51. What is the value of [160.69² - 67.54² - 93.15²] / [135.08]? -52. What is the square root of [49.6² + 59.5² + 5902.4]? -53. Please find the value of 801², by using algebraic identities to simplify computation. -54. What is the value of 0.406² + 7.594² + (0.406 x 15.188)? Please use algebraic identities to make the computation simple. -55. What is the value of [198.26² - 82.73² - 115.53²] / [165.46]? -56. What is the square root of [96.6² + 62.9² + 12152.28]? -57. Please find the value of 2001², by using algebraic identities to simplify computation. -58. What is the value of 0.557² + 4.443² + (0.557 x 8.886)? Please use algebraic identities to make the computation simple. -59. What is the value of [163.81² - 84.71² - 79.1²] / [169.42]? -60. What is the square root of [78.3² + 35.5² + 5559.3]? -61. Please find the value of 602², by using algebraic identities to simplify computation.
-62. What is the value of 0.434² + 4.566² + (0.434 x 9.132)? Please use algebraic identities to make the computation simple. -63. What is the value of [195.56² - 125.5² - 70.06²] / [251]? -64. What is the square root of [32.9² + 55.2² + 3632.16]? -65. Please find the value of 301², by using algebraic identities to simplify computation. -66. What is the value of 0.56² + 1.44² + (0.56 x 2.88)? Please use algebraic identities to make the computation simple. -67. What is the value of [194.67² - 88.07² - 106.6²] / [176.14]? -68. What is the square root of [67.4² + 53.5² + 7211.8]? -69. Please find the value of 603², by using algebraic identities to simplify computation. -70. What is the value of 0.671² + 3.329² + (0.671 x 6.658)? Please use algebraic identities to make the computation simple. -71. What is the value of [181.39² - 128.83² - 52.56²] / [257.66]? -72. What is the square root of [73.9² + 32.8² + 4847.84]? -73. Please find the value of 6001², by using algebraic identities to simplify computation. --
74. What is the value of 0.511² + 3.489² + (0.511 x 6.978)? Please use algebraic identities to make the computation simple. -75. What is the value of [202.83² - 126.07² - 76.76²] / [252.14]? -76. What is the square root of [83.8² + 52.3² + 8765.48]? -77. Please find the value of 20001², by using algebraic identities to simplify computation. -78. What is the value of 0.547² + 7.453² + (0.547 x 14.906)? Please use algebraic identities to make the computation simple. -79. What is the value of [125.62² - 50.04² - 75.58²] / [100.08]? -80. What is the square root of [43² + 95.8² + 8238.8]? -81. Please find the value of 403², by using algebraic identities to simplify computation. -82. What is the value of 0.528² + 2.472² + (0.528 x 4.944)? Please use algebraic identities to make the computation simple. -83. What is the value of [151.1² - 89.92² - 61.18²] / [179.84]? -84. What is the square root of [61.4² + 32.6² + 4003.28]? -85. Please find the value of 301², by using algebraic identities to simplify computation. -86. What is the value of 0.41² + 3.59² + (0.41 x 7.18)? Please use algebraic
identities to make the computation simple. -87. What is the value of [229.3² - 125.22² - 104.08²] / [250.44]? -88. What is the square root of [37² + 53.6² + 3966.4]? -89. Please find the value of 6002², by using algebraic identities to simplify computation. -90. What is the value of 0.577² + 3.423² + (0.577 x 6.846)? Please use algebraic identities to make the computation simple. -91. What is the value of [158.93² - 70.02² - 88.91²] / [140.04]? -92. What is the square root of [48² + 38.3² + 3676.8]? -93. Please find the value of 80003², by using algebraic identities to simplify computation. -94. What is the value of 0.583² + 6.417² + (0.583 x 12.834)? Please use algebraic identities to make the computation simple. -95. What is the value of [143.11² - 89.53² - 53.58²] / [179.06]? -96. What is the square root of [92.3² + 46² + 8491.6]? -97. Please find the value of 303², by using algebraic identities to simplify computation. -98. What is the value of 0.403² + 8.597² + (0.403 x 17.194)? Please use algebraic identities to make the computation simple.
-99. What is the value of [218.76² - 101.75² - 117.01²] / [203.5]? -100. What is the square root of [64.6² + 87.1² + 11253.32]? -101. Please find the value of 301², by using algebraic identities to simplify computation. -102. What is the value of 0.49² + 5.51² + (0.49 x 11.02)? Please use algebraic identities to make the computation simple. -103. What is the value of [141.49² - 64.47² - 77.02²] / [128.94]? -104. What is the square root of [28.4² + 33.4² + 1897.12]? -105. Please find the value of 402², by using algebraic identities to simplify computation. -106. What is the value of 0.535² + 1.465² + (0.535 x 2.93)? Please use algebraic identities to make the computation simple. -107. What is the value of [172.12² - 52.02² - 120.1²] / [104.04]? -108. What is the square root of [83.4² + 58.9² + 9824.52]? -109. Please find the value of 60003², by using algebraic identities to simplify computation. -110. What is the value of 0.512² + 4.488² + (0.512 x 8.976)? Please use algebraic identities to make the computation simple. --
111. What is the value of [184.92² - 59.1² - 125.82²] / [118.2]? -112. What is the square root of [64.4² + 67.5² + 8694]? -113. Please find the value of 7001², by using algebraic identities to simplify computation. -114. What is the value of 0.638² + 7.362² + (0.638 x 14.724)? Please use algebraic identities to make the computation simple. -115. What is the value of [185.39² - 91.2² - 94.19²] / [182.4]? -116. What is the square root of [43.7² + 22.2² + 1940.28]? -117. Please find the value of 80002², by using algebraic identities to simplify computation. -118. What is the value of 0.699² + 4.301² + (0.699 x 8.602)? Please use algebraic identities to make the computation simple. -119. What is the value of [180.81² - 70.59² - 110.22²] / [141.18]? -120. What is the square root of [11.1² + 35.3² + 783.66]? -121. Please find the value of 30001², by using algebraic identities to simplify computation. -122. What is the value of 0.517² + 7.483² + (0.517 x 14.966)? Please use algebraic identities to make the computation simple. -123. What is the value of [229.06² - 119.04² - 110.02²] / [238.08]? --
124. What is the square root of [12.6² + 64.3² + 1620.36]? -125. Please find the value of 50003², by using algebraic identities to simplify computation. -126. What is the value of 0.405² + 0.595² + (0.405 x 1.19)? Please use algebraic identities to make the computation simple. -127. What is the value of [195.28² - 110.77² - 84.51²] / [221.54]? -128. What is the square root of [39.1² + 20.9² + 1634.38]? -129. Please find the value of 4002², by using algebraic identities to simplify computation. -130. What is the value of 0.467² + 9.533² + (0.467 x 19.066)? Please use algebraic identities to make the computation simple. -131. What is the value of [175.37² - 84.05² - 91.32²] / [168.1]? -132. What is the square root of [11² + 59² + 1298]? -133. Please find the value of 402², by using algebraic identities to simplify computation. -134. What is the value of 0.687² + 4.313² + (0.687 x 8.626)? Please use algebraic identities to make the computation simple. -135. What is the value of [139.11² - 76.89² - 62.22²] / [153.78]? -136. What is the square root of [88.5² + 23.1² + 4088.7]?
-137. Please find the value of 203², by using algebraic identities to simplify computation. -138. What is the value of 0.589² + 6.411² + (0.589 x 12.822)? Please use algebraic identities to make the computation simple. -139. What is the value of [166.11² - 112.79² - 53.32²] / [225.58]? -140. What is the square root of [32.3² + 13.1² + 846.26]? -141. Please find the value of 2003², by using algebraic identities to simplify computation. -142. What is the value of 0.579² + 2.421² + (0.579 x 4.842)? Please use algebraic identities to make the computation simple. -143. What is the value of [214.46² - 87.8² - 126.66²] / [175.6]? -144. What is the square root of [60.7² + 37.8² + 4588.92]? -145. Please find the value of 60001², by using algebraic identities to simplify computation. -146. What is the value of 0.407² + 2.593² + (0.407 x 5.186)? Please use algebraic identities to make the computation simple. -147. What is the value of [235.1² - 118.4² - 116.7²] / [236.8]? -148. What is the square root of [35.5² + 32² + 2272]? --
149. Please find the value of 4001², by using algebraic identities to simplify computation. -150. What is the value of 0.486² + 4.514² + (0.486 x 9.028)? Please use algebraic identities to make the computation simple. -151. What is the value of [161.19² - 79.1² - 82.09²] / [158.2]? -152. What is the square root of [29.2² + 99.1² + 5787.44]? -153. Please find the value of 2001², by using algebraic identities to simplify computation. -154. What is the value of 0.619² + 9.381² + (0.619 x 18.762)? Please use algebraic identities to make the computation simple. -155. What is the value of [191.23² - 78.89² - 112.34²] / [157.78]? -156. What is the square root of [77.7² + 47.1² + 7319.34]? -157. Please find the value of 6003², by using algebraic identities to simplify computation. -158. What is the value of 0.449² + 5.551² + (0.449 x 11.102)? Please use algebraic identities to make the computation simple. -159. What is the value of [159.44² - 88.36² - 71.08²] / [176.72]? -160. What is the square root of [48.5² + 73.4² + 7119.8]? -161. Please find the value of 703², by using algebraic identities to simplify
computation. -162. What is the value of 0.658² + 8.342² + (0.658 x 16.684)? Please use algebraic identities to make the computation simple. -163. What is the value of [157.97² - 105.61² - 52.36²] / [211.22]? -164. What is the square root of [16.6² + 12.8² + 424.96]? -165. Please find the value of 80003², by using algebraic identities to simplify computation. -166. What is the value of 0.651² + 1.349² + (0.651 x 2.698)? Please use algebraic identities to make the computation simple. -167. What is the value of [175.64² - 92.8² - 82.84²] / [185.6]? -168. What is the square root of [87.9² + 14.2² + 2496.36]? -169. Please find the value of 802², by using algebraic identities to simplify computation. -170. What is the value of 0.548² + 9.452² + (0.548 x 18.904)? Please use algebraic identities to make the computation simple. -171. What is the value of [190.12² - 67.65² - 122.47²] / [135.3]? -172. What is the square root of [14.3² + 42² + 1201.2]? -173. Please find the value of 2002², by using algebraic identities to simplify computation.
-174. What is the value of 0.687² + 4.313² + (0.687 x 8.626)? Please use algebraic identities to make the computation simple. -175. What is the value of [154.66² - 96.22² - 58.44²] / [192.44]? -176. What is the square root of [53.1² + 17.5² + 1858.5]? -177. Please find the value of 4001², by using algebraic identities to simplify computation. -178. What is the value of 0.496² + 2.504² + (0.496 x 5.008)? Please use algebraic identities to make the computation simple. -179. What is the value of [161.85² - 90.79² - 71.06²] / [181.58]? -180. What is the square root of [47.4² + 59.6² + 5650.08]? -181. Please find the value of 703², by using algebraic identities to simplify computation. -182. What is the value of 0.644² + 3.356² + (0.644 x 6.712)? Please use algebraic identities to make the computation simple. -183. What is the value of [138.06² - 60.27² - 77.79²] / [120.54]? -184. What is the square root of [37.1² + 70.4² + 5223.68]? -185. Please find the value of 3002², by using algebraic identities to simplify computation. --
186. What is the value of 0.658² + 9.342² + (0.658 x 18.684)? Please use algebraic identities to make the computation simple. -187. What is the value of [179.61² - 129.07² - 50.54²] / [258.14]? -188. What is the square root of [57.1² + 19.6² + 2238.32]? -189. Please find the value of 703², by using algebraic identities to simplify computation. -190. What is the value of 0.697² + 0.303² + (0.697 x 0.606)? Please use algebraic identities to make the computation simple. -191. What is the value of [153.19² - 99.29² - 53.9²] / [198.58]? -192. What is the square root of [20.3² + 47.6² + 1932.56]? -193. Please find the value of 501², by using algebraic identities to simplify computation. -194. What is the value of 0.562² + 4.438² + (0.562 x 8.876)? Please use algebraic identities to make the computation simple. -195. What is the value of [184.02² - 78.17² - 105.85²] / [156.34]? -196. What is the square root of [35.3² + 41.8² + 2951.08]? -197. Please find the value of 801², by using algebraic identities to simplify computation. -198. What is the value of 0.522² + 4.478² + (0.522 x 8.956)? Please use
algebraic identities to make the computation simple. -199. What is the value of [118.16² - 60.94² - 57.22²] / [121.88]? -200. What is the square root of [26.5² + 20.8² + 1102.4]? -201. Please find the value of 2001², by using algebraic identities to simplify computation. -202. What is the value of 0.417² + 5.583² + (0.417 x 11.166)? Please use algebraic identities to make the computation simple. -203. What is the value of [170.7² - 93.83² - 76.87²] / [187.66]? -204. What is the square root of [63² + 50.2² + 6325.2]? -205. Please find the value of 601², by using algebraic identities to simplify computation. -206. What is the value of 0.533² + 9.467² + (0.533 x 18.934)? Please use algebraic identities to make the computation simple. -207. What is the value of [123.86² - 71.57² - 52.29²] / [143.14]? -208. What is the square root of [72.7² + 80.5² + 11704.7]? -209. Please find the value of 6001², by using algebraic identities to simplify computation. -210. What is the value of 0.414² + 9.586² + (0.414 x 19.172)? Please use algebraic identities to make the computation simple.
-211. What is the value of [166.98² - 73² - 93.98²] / [146]? -212. What is the square root of [89.4² + 69.1² + 12355.08]? -213. Please find the value of 40002², by using algebraic identities to simplify computation. -214. What is the value of 0.672² + 2.328² + (0.672 x 4.656)? Please use algebraic identities to make the computation simple. -215. What is the value of [194.26² - 120.81² - 73.45²] / [241.62]? -216. What is the square root of [33.9² + 10.5² + 711.9]? -217. Please find the value of 60002², by using algebraic identities to simplify computation. -218. What is the value of 0.485² + 7.515² + (0.485 x 15.03)? Please use algebraic identities to make the computation simple. -219. What is the value of [175.6² - 54.55² - 121.05²] / [109.1]? -220. What is the square root of [92.2² + 38.9² + 7173.16]? -221. Please find the value of 70002², by using algebraic identities to simplify computation. -222. What is the value of 0.583² + 5.417² + (0.583 x 10.834)? Please use algebraic identities to make the computation simple. --
223. What is the value of [209.84² - 101.8² - 108.04²] / [203.6]? -224. What is the square root of [38.8² + 70.1² + 5439.76]? -225. Please find the value of 8002², by using algebraic identities to simplify computation. -226. What is the value of 0.547² + 1.453² + (0.547 x 2.906)? Please use algebraic identities to make the computation simple. -227. What is the value of [153.59² - 89.53² - 64.06²] / [179.06]? -228. What is the square root of [17.4² + 49² + 1705.2]? -229. Please find the value of 301², by using algebraic identities to simplify computation. -230. What is the value of 0.576² + 5.424² + (0.576 x 10.848)? Please use algebraic identities to make the computation simple. -231. What is the value of [242.13² - 129.94² - 112.19²] / [259.88]? -232. What is the square root of [48.7² + 43.4² + 4227.16]? -233. Please find the value of 8002², by using algebraic identities to simplify computation. -234. What is the value of 0.625² + 7.375² + (0.625 x 14.75)? Please use algebraic identities to make the computation simple. -235. What is the value of [142.25² - 75.31² - 66.94²] / [150.62]?
-236. What is the square root of [77.2² + 42.6² + 6577.44]? -237. Please find the value of 7002², by using algebraic identities to simplify computation. -238. What is the value of 0.63² + 8.37² + (0.63 x 16.74)? Please use algebraic identities to make the computation simple. -239. What is the value of [191.69² - 92.09² - 99.6²] / [184.18]? -240. What is the square root of [11.8² + 23.1² + 545.16]? -241. Please find the value of 703², by using algebraic identities to simplify computation. -242. What is the value of 0.691² + 4.309² + (0.691 x 8.618)? Please use algebraic identities to make the computation simple. -243. What is the value of [193.79² - 105.23² - 88.56²] / [210.46]? -244. What is the square root of [86.5² + 58.3² + 10085.9]? -245. Please find the value of 6001², by using algebraic identities to simplify computation. -246. What is the value of 0.523² + 3.477² + (0.523 x 6.954)? Please use algebraic identities to make the computation simple. -247. What is the value of [232.1² - 103.92² - 128.18²] / [207.84]? --
248. What is the square root of [93.8² + 40.4² + 7579.04]? -249. Please find the value of 703², by using algebraic identities to simplify computation. -250. What is the value of 0.603² + 6.397² + (0.603 x 12.794)? Please use algebraic identities to make the computation simple. -251. What is the value of [212.68² - 125.12² - 87.56²] / [250.24]? -252. What is the square root of [77.5² + 98.1² + 15205.5]? -253. Please find the value of 20002², by using algebraic identities to simplify computation. -254. What is the value of 0.586² + 5.414² + (0.586 x 10.828)? Please use algebraic identities to make the computation simple. -255. What is the value of [154.39² - 62.54² - 91.85²] / [125.08]? -256. What is the square root of [24.6² + 20² + 984]? -257. Please find the value of 80001², by using algebraic identities to simplify computation. -258. What is the value of 0.582² + 0.418² + (0.582 x 0.836)? Please use algebraic identities to make the computation simple. -259. What is the value of [206.31² - 105.03² - 101.28²] / [210.06]? -260. What is the square root of [41.9² + 88.5² + 7416.3]?
-261. Please find the value of 4001², by using algebraic identities to simplify computation. -262. What is the value of 0.669² + 1.331² + (0.669 x 2.662)? Please use algebraic identities to make the computation simple. -263. What is the value of [172.55² - 63.25² - 109.3²] / [126.5]? -264. What is the square root of [85.8² + 28² + 4804.8]? -265. Please find the value of 2001², by using algebraic identities to simplify computation. -266. What is the value of 0.433² + 8.567² + (0.433 x 17.134)? Please use algebraic identities to make the computation simple. -267. What is the value of [139.14² - 69.88² - 69.26²] / [139.76]? -268. What is the square root of [11.6² + 70.6² + 1637.92]? -269. Please find the value of 7002², by using algebraic identities to simplify computation. -270. What is the value of 0.443² + 3.557² + (0.443 x 7.114)? Please use algebraic identities to make the computation simple. -271. What is the value of [192.48² - 102.79² - 89.69²] / [205.58]? -272. What is the square root of [96.9² + 62.5² + 12112.5]? --
273. Please find the value of 40003², by using algebraic identities to simplify computation. -274. What is the value of 0.656² + 9.344² + (0.656 x 18.688)? Please use algebraic identities to make the computation simple. -275. What is the value of [125.98² - 54.49² - 71.49²] / [108.98]? -276. What is the square root of [51.6² + 61.6² + 6357.12]? -277. Please find the value of 2001², by using algebraic identities to simplify computation. -278. What is the value of 0.549² + 2.451² + (0.549 x 4.902)? Please use algebraic identities to make the computation simple. -279. What is the value of [232.8² - 116.72² - 116.08²] / [233.44]? -280. What is the square root of [43.6² + 99.8² + 8702.56]? -281. Please find the value of 6003², by using algebraic identities to simplify computation. -282. What is the value of 0.466² + 4.534² + (0.466 x 9.068)? Please use algebraic identities to make the computation simple. -283. What is the value of [201.65² - 129.85² - 71.8²] / [259.7]? -284. What is the square root of [24.9² + 40.9² + 2036.82]? -285. Please find the value of 80003², by using algebraic identities to simplify
computation. -286. What is the value of 0.468² + 9.532² + (0.468 x 19.064)? Please use algebraic identities to make the computation simple. -287. What is the value of [235.36² - 128.97² - 106.39²] / [257.94]? -288. What is the square root of [47.7² + 48.5² + 4626.9]? -289. Please find the value of 70003², by using algebraic identities to simplify computation. -290. What is the value of 0.64² + 3.36² + (0.64 x 6.72)? Please use algebraic identities to make the computation simple. -291. What is the value of [186.32² - 69.61² - 116.71²] / [139.22]? -292. What is the square root of [86.7² + 85.3² + 14791.02]? -293. Please find the value of 20003², by using algebraic identities to simplify computation. -294. What is the value of 0.504² + 3.496² + (0.504 x 6.992)? Please use algebraic identities to make the computation simple. -295. What is the value of [230.71² - 118.08² - 112.63²] / [236.16]? -296. What is the square root of [96.8² + 10.8² + 2090.88]? -297. Please find the value of 2003², by using algebraic identities to simplify computation.
-298. What is the value of 0.586² + 9.414² + (0.586 x 18.828)? Please use algebraic identities to make the computation simple. -299. What is the value of [233.44² - 113.45² - 119.99²] / [226.9]? -300. What is the square root of [50.9² + 72.4² + 7370.32]? -301. Please find the value of 60003², by using algebraic identities to simplify computation. -302. What is the value of 0.669² + 1.331² + (0.669 x 2.662)? Please use algebraic identities to make the computation simple. -303. What is the value of [175.56² - 74.88² - 100.68²] / [149.76]? -304. What is the square root of [39.8² + 81.5² + 6487.4]? -305. Please find the value of 702², by using algebraic identities to simplify computation. -306. What is the value of 0.505² + 5.495² + (0.505 x 10.99)? Please use algebraic identities to make the computation simple. -307. What is the value of [179.71² - 51.1² - 128.61²] / [102.2]? -308. What is the square root of [94.4² + 19.6² + 3700.48]? -309. Please find the value of 5002², by using algebraic identities to simplify computation. --
310. What is the value of 0.645² + 1.355² + (0.645 x 2.71)? Please use algebraic identities to make the computation simple. -311. What is the value of [174.53² - 122.09² - 52.44²] / [244.18]? -312. What is the square root of [99.7² + 48.4² + 9650.96]? -313. Please find the value of 7001², by using algebraic identities to simplify computation. -314. What is the value of 0.675² + 0.325² + (0.675 x 0.65)? Please use algebraic identities to make the computation simple. -315. What is the value of [195.56² - 96.01² - 99.55²] / [192.02]? -316. What is the square root of [93.8² + 11² + 2063.6]? -317. Please find the value of 3001², by using algebraic identities to simplify computation. -318. What is the value of 0.661² + 6.339² + (0.661 x 12.678)? Please use algebraic identities to make the computation simple. -319. What is the value of [150.29² - 88.55² - 61.74²] / [177.1]? -320. What is the square root of [88.7² + 95.7² + 16977.18]? -321. Please find the value of 802², by using algebraic identities to simplify computation. -322. What is the value of 0.63² + 1.37² + (0.63 x 2.74)? Please use algebraic
identities to make the computation simple. -323. What is the value of [177.87² - 104.7² - 73.17²] / [209.4]? -324. What is the square root of [18.6² + 65.8² + 2447.76]? -325. Please find the value of 80002², by using algebraic identities to simplify computation. -326. What is the value of 0.651² + 5.349² + (0.651 x 10.698)? Please use algebraic identities to make the computation simple. -327. What is the value of [207.74² - 114.45² - 93.29²] / [228.9]? -328. What is the square root of [61.3² + 25.8² + 3163.08]? -329. Please find the value of 60001², by using algebraic identities to simplify computation. -330. What is the value of 0.542² + 8.458² + (0.542 x 16.916)? Please use algebraic identities to make the computation simple. -331. What is the value of [155.16² - 100.65² - 54.51²] / [201.3]? -332. What is the square root of [49.9² + 75.5² + 7534.9]? -333. Please find the value of 401², by using algebraic identities to simplify computation. -334. What is the value of 0.655² + 1.345² + (0.655 x 2.69)? Please use algebraic identities to make the computation simple.
-335. What is the value of [231.3² - 120.31² - 110.99²] / [240.62]? -336. What is the square root of [81² + 69.3² + 11226.6]? -337. Please find the value of 6002², by using algebraic identities to simplify computation. -338. What is the value of 0.551² + 9.449² + (0.551 x 18.898)? Please use algebraic identities to make the computation simple. -339. What is the value of [164.75² - 94.35² - 70.4²] / [188.7]? -340. What is the square root of [48.2² + 49.6² + 4781.44]? -341. Please find the value of 702², by using algebraic identities to simplify computation. -342. What is the value of 0.693² + 5.307² + (0.693 x 10.614)? Please use algebraic identities to make the computation simple. -343. What is the value of [174.81² - 51.77² - 123.04²] / [103.54]? -344. What is the square root of [10.8² + 83.8² + 1810.08]? -345. Please find the value of 80003², by using algebraic identities to simplify computation. -346. What is the value of 0.623² + 0.377² + (0.623 x 0.754)? Please use algebraic identities to make the computation simple. --
347. What is the value of [202.26² - 81.2² - 121.06²] / [162.4]? -348. What is the square root of [32.2² + 82.1² + 5287.24]? -349. Please find the value of 50001², by using algebraic identities to simplify computation. -350. What is the value of 0.497² + 3.503² + (0.497 x 7.006)? Please use algebraic identities to make the computation simple. -351. What is the value of [161.96² - 96.53² - 65.43²] / [193.06]? -352. What is the square root of [42.9² + 57.4² + 4924.92]? -353. Please find the value of 501², by using algebraic identities to simplify computation. -354. What is the value of 0.583² + 8.417² + (0.583 x 16.834)? Please use algebraic identities to make the computation simple. -355. What is the value of [220.37² - 113.66² - 106.71²] / [227.32]? -356. What is the square root of [47.3² + 69.6² + 6584.16]? -357. Please find the value of 202², by using algebraic identities to simplify computation. -358. What is the value of 0.464² + 7.536² + (0.464 x 15.072)? Please use algebraic identities to make the computation simple. -359. What is the value of [174.22² - 81.38² - 92.84²] / [162.76]?
-360. What is the square root of [71.8² + 78.9² + 11330.04]? -361. Please find the value of 4003², by using algebraic identities to simplify computation. -362. What is the value of 0.445² + 0.555² + (0.445 x 1.11)? Please use algebraic identities to make the computation simple. -363. What is the value of [219.49² - 100.91² - 118.58²] / [201.82]? -364. What is the square root of [63.8² + 62.4² + 7962.24]? -365. Please find the value of 7002², by using algebraic identities to simplify computation. -366. What is the value of 0.541² + 3.459² + (0.541 x 6.918)? Please use algebraic identities to make the computation simple. -367. What is the value of [168.91² - 72.87² - 96.04²] / [145.74]? -368. What is the square root of [45.3² + 29.6² + 2681.76]? -369. Please find the value of 701², by using algebraic identities to simplify computation. -370. What is the value of 0.511² + 3.489² + (0.511 x 6.978)? Please use algebraic identities to make the computation simple. -371. What is the value of [144.77² - 71.39² - 73.38²] / [142.78]? --
372. What is the square root of [51.3² + 51.2² + 5253.12]? -373. Please find the value of 501², by using algebraic identities to simplify computation. -374. What is the value of 0.531² + 7.469² + (0.531 x 14.938)? Please use algebraic identities to make the computation simple. -375. What is the value of [132.92² - 77.16² - 55.76²] / [154.32]? -376. What is the square root of [35.7² + 62.2² + 4441.08]? -377. Please find the value of 3002², by using algebraic identities to simplify computation. -378. What is the value of 0.614² + 4.386² + (0.614 x 8.772)? Please use algebraic identities to make the computation simple. -379. What is the value of [257.17² - 128.07² - 129.1²] / [256.14]? -380. What is the square root of [43.4² + 32² + 2777.6]? -381. Please find the value of 50003², by using algebraic identities to simplify computation. -382. What is the value of 0.565² + 4.435² + (0.565 x 8.87)? Please use algebraic identities to make the computation simple. -383. What is the value of [175.64² - 113.46² - 62.18²] / [226.92]? -384. What is the square root of [60.5² + 70.6² + 8542.6]?
-385. Please find the value of 60003², by using algebraic identities to simplify computation. -386. What is the value of 0.667² + 4.333² + (0.667 x 8.666)? Please use algebraic identities to make the computation simple. -387. What is the value of [151.78² - 50.23² - 101.55²] / [100.46]? -388. What is the square root of [17² + 25.8² + 877.2]? -389. Please find the value of 2002², by using algebraic identities to simplify computation. -390. What is the value of 0.45² + 5.55² + (0.45 x 11.1)? Please use algebraic identities to make the computation simple. -391. What is the value of [220.15² - 126.44² - 93.71²] / [252.88]? -392. What is the square root of [87.1² + 39.3² + 6846.06]? -393. Please find the value of 502², by using algebraic identities to simplify computation. -394. What is the value of 0.41² + 1.59² + (0.41 x 3.18)? Please use algebraic identities to make the computation simple. -395. What is the value of [181.76² - 128.93² - 52.83²] / [257.86]? -396. What is the square root of [35.8² + 52.7² + 3773.32]? --
397. Please find the value of 301², by using algebraic identities to simplify computation. -398. What is the value of 0.486² + 8.514² + (0.486 x 17.028)? Please use algebraic identities to make the computation simple. -399. What is the value of [145.39² - 67.76² - 77.63²] / [135.52]? -400. What is the square root of [27.6² + 15.4² + 850.08]? --
401. Please find the value of 1997², by using algebraic identities to simplify computation. -402. What is the value of 3.251² + 2.051² - (3.251 x 4.102)? Please use algebraic identities to make the computation simple. -403. What is the value of [446.4² + 207.5² - 238.9²] / [892.8]? -404. What is the positive square root of [2638² + 80² - 422080]? -405. Please find the value of 197², by using algebraic identities to simplify computation. -406. What is the value of 2.187² + 1.187² - (2.187 x 2.374)? Please use algebraic identities to make the computation simple. -407. What is the value of [519.1² + 225.9² - 293.2²] / [1038.2]? -408. What is the positive square root of [2961² + 40² - 236880]? --
409. Please find the value of 19999², by using algebraic identities to simplify computation. -410. What is the value of 3.527² + 2.127² - (3.527 x 4.254)? Please use algebraic identities to make the computation simple. -411. What is the value of [420.1² + 295² - 125.1²] / [840.2]? -412. What is the positive square root of [3202² + 70² - 448280]? -413. Please find the value of 69999², by using algebraic identities to simplify computation. -414. What is the value of 3.362² + 2.562² - (3.362 x 5.124)? Please use algebraic identities to make the computation simple. -415. What is the value of [554.4² + 201.6² - 352.8²] / [1108.8]? -416. What is the positive square root of [3023² + 80² - 483680]? -417. Please find the value of 3997², by using algebraic identities to simplify computation. -418. What is the value of 1.462² + 0.462² - (1.462 x 0.924)? Please use algebraic identities to make the computation simple. -419. What is the value of [541.7² + 145.6² - 396.1²] / [1083.4]? -420. What is the positive square root of [2403² + 20² - 96120]? -421. Please find the value of 49997², by using algebraic identities to simplify computation.
-422. What is the value of 1.924² + 0.724² - (1.924 x 1.448)? Please use algebraic identities to make the computation simple. -423. What is the value of [506.3² + 172.3² - 334²] / [1012.6]? -424. What is the positive square root of [3957² + 50² - 395700]? -425. Please find the value of 89999², by using algebraic identities to simplify computation. -426. What is the value of 3.413² + 3.013² - (3.413 x 6.026)? Please use algebraic identities to make the computation simple. -427. What is the value of [531.2² + 179.9² - 351.3²] / [1062.4]? -428. What is the positive square root of [3927² + 30² - 235620]? -429. Please find the value of 199², by using algebraic identities to simplify computation. -430. What is the value of 2.435² + 1.235² - (2.435 x 2.47)? Please use algebraic identities to make the computation simple. -431. What is the value of [400.5² + 142.4² - 258.1²] / [801]? -432. What is the positive square root of [2632² + 40² - 210560]? -433. Please find the value of 497², by using algebraic identities to simplify computation. --
434. What is the value of 2.988² + 2.588² - (2.988 x 5.176)? Please use algebraic identities to make the computation simple. -435. What is the value of [562.5² + 144.8² - 417.7²] / [1125]? -436. What is the positive square root of [3812² + 80² - 609920]? -437. Please find the value of 197², by using algebraic identities to simplify computation. -438. What is the value of 1.596² + 0.396² - (1.596 x 0.792)? Please use algebraic identities to make the computation simple. -439. What is the value of [465.9² + 199.3² - 266.6²] / [931.8]? -440. What is the positive square root of [3203² + 30² - 192180]? -441. Please find the value of 49998², by using algebraic identities to simplify computation. -442. What is the value of 3.61² + 3.01² - (3.61 x 6.02)? Please use algebraic identities to make the computation simple. -443. What is the value of [430² + 106.6² - 323.4²] / [860]? -444. What is the positive square root of [2720² + 50² - 272000]? -445. Please find the value of 299², by using algebraic identities to simplify computation. -446. What is the value of 1.661² + 0.261² - (1.661 x 0.522)? Please use algebraic identities to make the computation simple.
-447. What is the value of [588.9² + 185² - 403.9²] / [1177.8]? -448. What is the positive square root of [2343² + 30² - 140580]? -449. Please find the value of 69997², by using algebraic identities to simplify computation. -450. What is the value of 3.69² + 2.89² - (3.69 x 5.78)? Please use algebraic identities to make the computation simple. -451. What is the value of [585.1² + 139.4² - 445.7²] / [1170.2]? -452. What is the positive square root of [3035² + 50² - 303500]? -453. Please find the value of 198², by using algebraic identities to simplify computation. -454. What is the value of 1.55² + 0.35² - (1.55 x 0.7)? Please use algebraic identities to make the computation simple. -455. What is the value of [516.7² + 250.7² - 266.²] / [1033.4]? -456. What is the positive square root of [3104² + 30² - 186240]? -457. Please find the value of 89997², by using algebraic identities to simplify computation. -458. What is the value of 1.765² + 1.565² - (1.765 x 3.13)? Please use algebraic identities to make the computation simple. --
459. What is the value of [590.1² + 141² - 449.1²] / [1180.2]? -460. What is the positive square root of [2219² + 90² - 399420]? -461. Please find the value of 1997², by using algebraic identities to simplify computation. -462. What is the value of 1.978² + 0.978² - (1.978 x 1.956)? Please use algebraic identities to make the computation simple. -463. What is the value of [525.6² + 102.2² - 423.4²] / [1051.2]? -464. What is the positive square root of [2989² + 90² - 538020]? -465. Please find the value of 8998², by using algebraic identities to simplify computation. -466. What is the value of 3.456² + 2.656² - (3.456 x 5.312)? Please use algebraic identities to make the computation simple. -467. What is the value of [463.9² + 148² - 315.9²] / [927.8]? -468. What is the positive square root of [2121² + 60² - 254520]? -469. Please find the value of 79999², by using algebraic identities to simplify computation. -470. What is the value of 1.826² + 1.626² - (1.826 x 3.252)? Please use algebraic identities to make the computation simple. -471. What is the value of [405.1² + 212.2² - 192.9²] / [810.2]? --
472. What is the positive square root of [3695² + 30² - 221700]? -473. Please find the value of 59997², by using algebraic identities to simplify computation. -474. What is the value of 2.494² + 2.294² - (2.494 x 4.588)? Please use algebraic identities to make the computation simple. -475. What is the value of [450.3² + 142² - 308.3²] / [900.6]? -476. What is the positive square root of [2628² + 70² - 367920]? -477. Please find the value of 49998², by using algebraic identities to simplify computation. -478. What is the value of 1.664² + 1.464² - (1.664 x 2.928)? Please use algebraic identities to make the computation simple. -479. What is the value of [502² + 267.1² - 234.9²] / [1004]? -480. What is the positive square root of [2444² + 70² - 342160]? -481. Please find the value of 49997², by using algebraic identities to simplify computation. -482. What is the value of 3.368² + 2.968² - (3.368 x 5.936)? Please use algebraic identities to make the computation simple. -483. What is the value of [488.7² + 265.9² - 222.8²] / [977.4]? -484. What is the positive square root of [1908² + 20² - 76320]? --
485. Please find the value of 499², by using algebraic identities to simplify computation. -486. What is the value of 2.583² + 1.183² - (2.583 x 2.366)? Please use algebraic identities to make the computation simple. -487. What is the value of [512.1² + 196.3² - 315.8²] / [1024.2]? -488. What is the positive square root of [2779² + 70² - 389060]? -489. Please find the value of 699², by using algebraic identities to simplify computation. -490. What is the value of 3.186² + 2.986² - (3.186 x 5.972)? Please use algebraic identities to make the computation simple. -491. What is the value of [440.1² + 107.5² - 332.6²] / [880.2]? -492. What is the positive square root of [1797² + 50² - 179700]? -493. Please find the value of 699², by using algebraic identities to simplify computation. -494. What is the value of 2.11² + 1.11² - (2.11 x 2.22)? Please use algebraic identities to make the computation simple. -495. What is the value of [573.7² + 211.3² - 362.4²] / [1147.4]? -496. What is the positive square root of [3997² + 40² - 319760]? -497. Please find the value of 897², by using algebraic identities to simplify computation.
-498. What is the value of 1.559² + 0.759² - (1.559 x 1.518)? Please use algebraic identities to make the computation simple. -499. What is the value of [431.6² + 150.9² - 280.7²] / [863.2]? -500. What is the positive square root of [2682² + 40² - 214560]? -501. Please find the value of 19998², by using algebraic identities to simplify computation. -502. What is the value of 3.635² + 3.035² - (3.635 x 6.07)? Please use algebraic identities to make the computation simple. -503. What is the value of [520.1² + 281.2² - 238.9²] / [1040.2]? -504. What is the positive square root of [2915² + 20² - 116600]? -505. Please find the value of 599², by using algebraic identities to simplify computation. -506. What is the value of 2.91² + 1.91² - (2.91 x 3.82)? Please use algebraic identities to make the computation simple. -507. What is the value of [562.6² + 245.1² - 317.5²] / [1125.2]? -508. What is the positive square root of [2443² + 50² - 244300]? -509. Please find the value of 2999², by using algebraic identities to simplify computation. --
510. What is the value of 3.217² + 2.417² - (3.217 x 4.834)? Please use algebraic identities to make the computation simple. -511. What is the value of [488.8² + 119.8² - 369²] / [977.6]? -512. What is the positive square root of [2138² + 90² - 384840]? -513. Please find the value of 299², by using algebraic identities to simplify computation. -514. What is the value of 2.601² + 2.401² - (2.601 x 4.802)? Please use algebraic identities to make the computation simple. -515. What is the value of [563.9² + 191.2² - 372.7²] / [1127.8]? -516. What is the positive square root of [3525² + 40² - 282000]? -517. Please find the value of 1997², by using algebraic identities to simplify computation. -518. What is the value of 3.321² + 2.521² - (3.321 x 5.042)? Please use algebraic identities to make the computation simple. -519. What is the value of [424² + 211.6² - 212.4²] / [848]? -520. What is the positive square root of [3775² + 40² - 302000]? -521. Please find the value of 5999², by using algebraic identities to simplify computation. -522. What is the value of 3.177² + 1.777² - (3.177 x 3.554)? Please use algebraic identities to make the computation simple.
-523. What is the value of [508.1² + 200.3² - 307.8²] / [1016.2]? -524. What is the positive square root of [3708² + 50² - 370800]? -525. Please find the value of 5999², by using algebraic identities to simplify computation. -526. What is the value of 2.127² + 1.727² - (2.127 x 3.454)? Please use algebraic identities to make the computation simple. -527. What is the value of [452.1² + 250² - 202.1²] / [904.2]? -528. What is the positive square root of [3646² + 90² - 656280]? -529. Please find the value of 79997², by using algebraic identities to simplify computation. -530. What is the value of 3.545² + 2.345² - (3.545 x 4.69)? Please use algebraic identities to make the computation simple. -531. What is the value of [439.1² + 219.9² - 219.2²] / [878.2]? -532. What is the positive square root of [3439² + 30² - 206340]? -533. Please find the value of 598², by using algebraic identities to simplify computation. -534. What is the value of 1.832² + 1.632² - (1.832 x 3.264)? Please use algebraic identities to make the computation simple. --
535. What is the value of [475.5² + 215.7² - 259.8²] / [951]? -536. What is the positive square root of [2164² + 60² - 259680]? -537. Please find the value of 897², by using algebraic identities to simplify computation. -538. What is the value of 3.392² + 2.992² - (3.392 x 5.984)? Please use algebraic identities to make the computation simple. -539. What is the value of [582² + 182.4² - 399.6²] / [1164]? -540. What is the positive square root of [2188² + 30² - 131280]? -541. Please find the value of 298², by using algebraic identities to simplify computation. -542. What is the value of 2.263² + 2.063² - (2.263 x 4.126)? Please use algebraic identities to make the computation simple. -543. What is the value of [440.6² + 286.9² - 153.7²] / [881.2]? -544. What is the positive square root of [2028² + 40² - 162240]? -545. Please find the value of 598², by using algebraic identities to simplify computation. -546. What is the value of 3.423² + 2.023² - (3.423 x 4.046)? Please use algebraic identities to make the computation simple. -547. What is the value of [493.4² + 226.2² - 267.2²] / [986.8]? --
548. What is the positive square root of [1932² + 90² - 347760]? -549. Please find the value of 49998², by using algebraic identities to simplify computation. -550. What is the value of 3.287² + 2.487² - (3.287 x 4.974)? Please use algebraic identities to make the computation simple. -551. What is the value of [442.6² + 115.3² - 327.3²] / [885.2]? -552. What is the positive square root of [2864² + 90² - 515520]? -553. Please find the value of 7999², by using algebraic identities to simplify computation. -554. What is the value of 3.567² + 2.967² - (3.567 x 5.934)? Please use algebraic identities to make the computation simple. -555. What is the value of [569.7² + 178.6² - 391.1²] / [1139.4]? -556. What is the positive square root of [1726² + 50² - 172600]? -557. Please find the value of 6999², by using algebraic identities to simplify computation. -558. What is the value of 2.318² + 1.718² - (2.318 x 3.436)? Please use algebraic identities to make the computation simple. -559. What is the value of [489² + 144.2² - 344.8²] / [978]? -560. What is the positive square root of [3716² + 20² - 148640]? --
561. Please find the value of 3999², by using algebraic identities to simplify computation. -562. What is the value of 3.277² + 2.077² - (3.277 x 4.154)? Please use algebraic identities to make the computation simple. -563. What is the value of [595.8² + 222² - 373.8²] / [1191.6]? -564. What is the positive square root of [3895² + 20² - 155800]? -565. Please find the value of 49997², by using algebraic identities to simplify computation. -566. What is the value of 2.71² + 1.31² - (2.71 x 2.62)? Please use algebraic identities to make the computation simple. -567. What is the value of [439.6² + 168.5² - 271.1²] / [879.2]? -568. What is the positive square root of [2802² + 80² - 448320]? -569. Please find the value of 6999², by using algebraic identities to simplify computation. -570. What is the value of 3.049² + 1.649² - (3.049 x 3.298)? Please use algebraic identities to make the computation simple. -571. What is the value of [439.4² + 278.1² - 161.3²] / [878.8]? -572. What is the positive square root of [2817² + 50² - 281700]? -573. Please find the value of 8997², by using algebraic identities to simplify
computation. -574. What is the value of 2.403² + 2.203² - (2.403 x 4.406)? Please use algebraic identities to make the computation simple. -575. What is the value of [454.4² + 199.4² - 255.²] / [908.8]? -576. What is the positive square root of [3201² + 50² - 320100]? -577. Please find the value of 8998², by using algebraic identities to simplify computation. -578. What is the value of 3.381² + 2.581² - (3.381 x 5.162)? Please use algebraic identities to make the computation simple. -579. What is the value of [595² + 100.4² - 494.6²] / [1190]? -580. What is the positive square root of [2446² + 20² - 97840]? -581. Please find the value of 59998², by using algebraic identities to simplify computation. -582. What is the value of 2.871² + 2.471² - (2.871 x 4.942)? Please use algebraic identities to make the computation simple. -583. What is the value of [533.6² + 273.7² - 259.9²] / [1067.2]? -584. What is the positive square root of [3090² + 90² - 556200]? -585. Please find the value of 6998², by using algebraic identities to simplify computation. --
586. What is the value of 2.54² + 2.14² - (2.54 x 4.28)? Please use algebraic identities to make the computation simple. -587. What is the value of [485.1² + 210.6² - 274.5²] / [970.2]? -588. What is the positive square root of [3625² + 90² - 652500]? -589. Please find the value of 698², by using algebraic identities to simplify computation. -590. What is the value of 2.605² + 1.205² - (2.605 x 2.41)? Please use algebraic identities to make the computation simple. -591. What is the value of [436.4² + 258.7² - 177.7²] / [872.8]? -592. What is the positive square root of [3747² + 50² - 374700]? -593. Please find the value of 899², by using algebraic identities to simplify computation. -594. What is the value of 2.308² + 1.708² - (2.308 x 3.416)? Please use algebraic identities to make the computation simple. -595. What is the value of [457.4² + 219.6² - 237.8²] / [914.8]? -596. What is the positive square root of [2937² + 70² - 411180]? -597. Please find the value of 497², by using algebraic identities to simplify computation. -598. What is the value of 1.993² + 1.193² - (1.993 x 2.386)? Please use algebraic identities to make the computation simple.
-599. What is the value of [569.4² + 246.5² - 322.9²] / [1138.8]? -600. What is the positive square root of [3710² + 60² - 445200]? -601. Please find the value of 89997², by using algebraic identities to simplify computation. -602. What is the value of 1.601² + 1.401² - (1.601 x 2.802)? Please use algebraic identities to make the computation simple. -603. What is the value of [549.8² + 156.2² - 393.6²] / [1099.6]? -604. What is the positive square root of [1968² + 50² - 196800]? -605. Please find the value of 7997², by using algebraic identities to simplify computation. -606. What is the value of 3.063² + 1.663² - (3.063 x 3.326)? Please use algebraic identities to make the computation simple. -607. What is the value of [480.2² + 182.8² - 297.4²] / [960.4]? -608. What is the positive square root of [2834² + 80² - 453440]? -609. Please find the value of 59997², by using algebraic identities to simplify computation. -610. What is the value of 2.547² + 2.347² - (2.547 x 4.694)? Please use algebraic identities to make the computation simple. --
611. What is the value of [563.7² + 270.1² - 293.6²] / [1127.4]? -612. What is the positive square root of [2449² + 40² - 195920]? -613. Please find the value of 8997², by using algebraic identities to simplify computation. -614. What is the value of 3.444² + 2.444² - (3.444 x 4.888)? Please use algebraic identities to make the computation simple. -615. What is the value of [571.6² + 224.2² - 347.4²] / [1143.2]? -616. What is the positive square root of [3068² + 80² - 490880]? -617. Please find the value of 49999², by using algebraic identities to simplify computation. -618. What is the value of 1.942² + 0.942² - (1.942 x 1.884)? Please use algebraic identities to make the computation simple. -619. What is the value of [536.7² + 229.3² - 307.4²] / [1073.4]? -620. What is the positive square root of [2287² + 50² - 228700]? -621. Please find the value of 298², by using algebraic identities to simplify computation. -622. What is the value of 1.695² + 1.295² - (1.695 x 2.59)? Please use algebraic identities to make the computation simple. -623. What is the value of [420.1² + 168.4² - 251.7²] / [840.2]? --
624. What is the positive square root of [1845² + 40² - 147600]? -625. Please find the value of 397², by using algebraic identities to simplify computation. -626. What is the value of 1.987² + 1.587² - (1.987 x 3.174)? Please use algebraic identities to make the computation simple. -627. What is the value of [542.1² + 282.3² - 259.8²] / [1084.2]? -628. What is the positive square root of [1749² + 80² - 279840]? -629. Please find the value of 4997², by using algebraic identities to simplify computation. -630. What is the value of 2.174² + 0.974² - (2.174 x 1.948)? Please use algebraic identities to make the computation simple. -631. What is the value of [476.9² + 153.2² - 323.7²] / [953.8]? -632. What is the positive square root of [3840² + 50² - 384000]? -633. Please find the value of 298², by using algebraic identities to simplify computation. -634. What is the value of 2.084² + 1.484² - (2.084 x 2.968)? Please use algebraic identities to make the computation simple. -635. What is the value of [512.8² + 203.9² - 308.9²] / [1025.6]? -636. What is the positive square root of [2359² + 60² - 283080]? --
637. Please find the value of 698², by using algebraic identities to simplify computation. -638. What is the value of 1.861² + 0.861² - (1.861 x 1.722)? Please use algebraic identities to make the computation simple. -639. What is the value of [536.5² + 266.7² - 269.8²] / [1073]? -640. What is the positive square root of [3250² + 70² - 455000]? -641. Please find the value of 798², by using algebraic identities to simplify computation. -642. What is the value of 3.057² + 2.057² - (3.057 x 4.114)? Please use algebraic identities to make the computation simple. -643. What is the value of [456.4² + 294.7² - 161.7²] / [912.8]? -644. What is the positive square root of [2740² + 70² - 383600]? -645. Please find the value of 4998², by using algebraic identities to simplify computation. -646. What is the value of 2.005² + 1.605² - (2.005 x 3.21)? Please use algebraic identities to make the computation simple. -647. What is the value of [431.2² + 165² - 266.2²] / [862.4]? -648. What is the positive square root of [2636² + 20² - 105440]? -649. Please find the value of 2997², by using algebraic identities to simplify
computation. -650. What is the value of 2.758² + 2.158² - (2.758 x 4.316)? Please use algebraic identities to make the computation simple. -651. What is the value of [542.4² + 152.4² - 390²] / [1084.8]? -652. What is the positive square root of [3991² + 50² - 399100]? -653. Please find the value of 397², by using algebraic identities to simplify computation. -654. What is the value of 2.454² + 2.254² - (2.454 x 4.508)? Please use algebraic identities to make the computation simple. -655. What is the value of [489.5² + 241.2² - 248.3²] / [979]? -656. What is the positive square root of [3019² + 50² - 301900]? -657. Please find the value of 7997², by using algebraic identities to simplify computation. -658. What is the value of 1.814² + 1.014² - (1.814 x 2.028)? Please use algebraic identities to make the computation simple. -659. What is the value of [434.2² + 187.1² - 247.1²] / [868.4]? -660. What is the positive square root of [3091² + 60² - 370920]? -661. Please find the value of 297², by using algebraic identities to simplify computation. --
662. What is the value of 1.534² + 1.334² - (1.534 x 2.668)? Please use algebraic identities to make the computation simple. -663. What is the value of [442.9² + 140.1² - 302.8²] / [885.8]? -664. What is the positive square root of [3779² + 90² - 680220]? -665. Please find the value of 4998², by using algebraic identities to simplify computation. -666. What is the value of 3.457² + 3.057² - (3.457 x 6.114)? Please use algebraic identities to make the computation simple. -667. What is the value of [544.8² + 297.1² - 247.7²] / [1089.6]? -668. What is the positive square root of [2373² + 20² - 94920]? -669. Please find the value of 39998², by using algebraic identities to simplify computation. -670. What is the value of 3.413² + 2.613² - (3.413 x 5.226)? Please use algebraic identities to make the computation simple. -671. What is the value of [571.5² + 263.6² - 307.9²] / [1143]? -672. What is the positive square root of [3167² + 60² - 380040]? -673. Please find the value of 49997², by using algebraic identities to simplify computation. -674. What is the value of 3.506² + 3.306² - (3.506 x 6.612)? Please use algebraic identities to make the computation simple.
-675. What is the value of [525.3² + 176.7² - 348.6²] / [1050.6]? -676. What is the positive square root of [1906² + 20² - 76240]? -677. Please find the value of 798², by using algebraic identities to simplify computation. -678. What is the value of 2.612² + 2.012² - (2.612 x 4.024)? Please use algebraic identities to make the computation simple. -679. What is the value of [426.6² + 160.7² - 265.9²] / [853.2]? -680. What is the positive square root of [2986² + 20² - 119440]? -681. Please find the value of 399², by using algebraic identities to simplify computation. -682. What is the value of 1.592² + 1.392² - (1.592 x 2.784)? Please use algebraic identities to make the computation simple. -683. What is the value of [567.8² + 199.4² - 368.4²] / [1135.6]? -684. What is the positive square root of [3168² + 50² - 316800]? -685. Please find the value of 6997², by using algebraic identities to simplify computation. -686. What is the value of 2.219² + 1.419² - (2.219 x 2.838)? Please use algebraic identities to make the computation simple. --
687. What is the value of [473.2² + 292.2² - 181²] / [946.4]? -688. What is the positive square root of [1795² + 30² - 107700]? -689. Please find the value of 6999², by using algebraic identities to simplify computation. -690. What is the value of 2.105² + 1.505² - (2.105 x 3.01)? Please use algebraic identities to make the computation simple. -691. What is the value of [512.7² + 158.3² - 354.4²] / [1025.4]? -692. What is the positive square root of [3399² + 90² - 611820]? -693. Please find the value of 497², by using algebraic identities to simplify computation. -694. What is the value of 2.656² + 1.656² - (2.656 x 3.312)? Please use algebraic identities to make the computation simple. -695. What is the value of [592.8² + 263.6² - 329.2²] / [1185.6]? -696. What is the positive square root of [3142² + 40² - 251360]? -697. Please find the value of 798², by using algebraic identities to simplify computation. -698. What is the value of 3.149² + 2.549² - (3.149 x 5.098)? Please use algebraic identities to make the computation simple. -699. What is the value of [488.4² + 294.5² - 193.9²] / [976.8]? --
700. What is the positive square root of [3394² + 60² - 407280]? -701. Please find the value of 7998², by using algebraic identities to simplify computation. -702. What is the value of 2.059² + 0.859² - (2.059 x 1.718)? Please use algebraic identities to make the computation simple. -703. What is the value of [500.4² + 157.8² - 342.6²] / [1000.8]? -704. What is the positive square root of [2626² + 60² - 315120]? -705. Please find the value of 697², by using algebraic identities to simplify computation. -706. What is the value of 2.625² + 1.825² - (2.625 x 3.65)? Please use algebraic identities to make the computation simple. -707. What is the value of [561.8² + 218² - 343.8²] / [1123.6]? -708. What is the positive square root of [2658² + 60² - 318960]? -709. Please find the value of 49998², by using algebraic identities to simplify computation. -710. What is the value of 3.051² + 2.451² - (3.051 x 4.902)? Please use algebraic identities to make the computation simple. -711. What is the value of [541.5² + 149.7² - 391.8²] / [1083]? -712. What is the positive square root of [2967² + 60² - 356040]? --
713. Please find the value of 6997², by using algebraic identities to simplify computation. -714. What is the value of 1.541² + 0.341² - (1.541 x 0.682)? Please use algebraic identities to make the computation simple. -715. What is the value of [508.4² + 160.2² - 348.2²] / [1016.8]? -716. What is the positive square root of [2921² + 70² - 408940]? -717. Please find the value of 79998², by using algebraic identities to simplify computation. -718. What is the value of 2.496² + 1.896² - (2.496 x 3.792)? Please use algebraic identities to make the computation simple. -719. What is the value of [492.1² + 211.8² - 280.3²] / [984.2]? -720. What is the positive square root of [3079² + 30² - 184740]? -721. Please find the value of 3999², by using algebraic identities to simplify computation. -722. What is the value of 2.653² + 1.653² - (2.653 x 3.306)? Please use algebraic identities to make the computation simple. -723. What is the value of [589.5² + 265.4² - 324.1²] / [1179]? -724. What is the positive square root of [2326² + 30² - 139560]? -725. Please find the value of 29999², by using algebraic identities to simplify computation.
-726. What is the value of 1.48² + 0.28² - (1.48 x 0.56)? Please use algebraic identities to make the computation simple. -727. What is the value of [503.8² + 253.3² - 250.5²] / [1007.6]? -728. What is the positive square root of [2682² + 20² - 107280]? -729. Please find the value of 29998², by using algebraic identities to simplify computation. -730. What is the value of 2.526² + 1.726² - (2.526 x 3.452)? Please use algebraic identities to make the computation simple. -731. What is the value of [499² + 293.5² - 205.5²] / [998]? -732. What is the positive square root of [2310² + 40² - 184800]? -733. Please find the value of 7999², by using algebraic identities to simplify computation. -734. What is the value of 3.159² + 1.959² - (3.159 x 3.918)? Please use algebraic identities to make the computation simple. -735. What is the value of [596.1² + 157.5² - 438.6²] / [1192.2]? -736. What is the positive square root of [3006² + 20² - 120240]? -737. Please find the value of 4998², by using algebraic identities to simplify computation. --
738. What is the value of 1.929² + 1.529² - (1.929 x 3.058)? Please use algebraic identities to make the computation simple. -739. What is the value of [462.6² + 219.9² - 242.7²] / [925.2]? -740. What is the positive square root of [2128² + 30² - 127680]? -741. Please find the value of 7998², by using algebraic identities to simplify computation. -742. What is the value of 3.115² + 1.915² - (3.115 x 3.83)? Please use algebraic identities to make the computation simple. -743. What is the value of [541.3² + 229.8² - 311.5²] / [1082.6]? -744. What is the positive square root of [3039² + 30² - 182340]? -745. Please find the value of 49997², by using algebraic identities to simplify computation. -746. What is the value of 3.188² + 2.188² - (3.188 x 4.376)? Please use algebraic identities to make the computation simple. -747. What is the value of [466² + 272.9² - 193.1²] / [932]? -748. What is the positive square root of [3959² + 40² - 316720]? -749. Please find the value of 8997², by using algebraic identities to simplify computation. -750. What is the value of 1.817² + 0.417² - (1.817 x 0.834)? Please use algebraic identities to make the computation simple.
-751. What is the value of [535.2² + 220.1² - 315.1²] / [1070.4]? -752. What is the positive square root of [3157² + 40² - 252560]? -753. Please find the value of 89997², by using algebraic identities to simplify computation. -754. What is the value of 3.481² + 2.481² - (3.481 x 4.962)? Please use algebraic identities to make the computation simple. -755. What is the value of [591.5² + 208.8² - 382.7²] / [1183]? -756. What is the positive square root of [2037² + 50² - 203700]? -757. Please find the value of 79999², by using algebraic identities to simplify computation. -758. What is the value of 3.561² + 2.761² - (3.561 x 5.522)? Please use algebraic identities to make the computation simple. -759. What is the value of [557.3² + 175.9² - 381.4²] / [1114.6]? -760. What is the positive square root of [2171² + 30² - 130260]? -761. Please find the value of 697², by using algebraic identities to simplify computation. -762. What is the value of 1.985² + 0.785² - (1.985 x 1.57)? Please use algebraic identities to make the computation simple. --
763. What is the value of [554.1² + 175.5² - 378.6²] / [1108.2]? -764. What is the positive square root of [2576² + 20² - 103040]? -765. Please find the value of 1999², by using algebraic identities to simplify computation. -766. What is the value of 3.307² + 2.307² - (3.307 x 4.614)? Please use algebraic identities to make the computation simple. -767. What is the value of [463.3² + 278.6² - 184.7²] / [926.6]? -768. What is the positive square root of [2541² + 90² - 457380]? -769. Please find the value of 39997², by using algebraic identities to simplify computation. -770. What is the value of 2.212² + 1.012² - (2.212 x 2.024)? Please use algebraic identities to make the computation simple. -771. What is the value of [575.1² + 259² - 316.1²] / [1150.2]? -772. What is the positive square root of [3147² + 20² - 125880]? -773. Please find the value of 29997², by using algebraic identities to simplify computation. -774. What is the value of 3.291² + 2.491² - (3.291 x 4.982)? Please use algebraic identities to make the computation simple. -775. What is the value of [507.9² + 162.5² - 345.4²] / [1015.8]? --
776. What is the positive square root of [3621² + 40² - 289680]? -777. Please find the value of 79999², by using algebraic identities to simplify computation. -778. What is the value of 1.756² + 0.756² - (1.756 x 1.512)? Please use algebraic identities to make the computation simple. -779. What is the value of [491.3² + 107.5² - 383.8²] / [982.6]? -780. What is the positive square root of [2575² + 20² - 103000]? -781. Please find the value of 1999², by using algebraic identities to simplify computation. -782. What is the value of 2.759² + 1.959² - (2.759 x 3.918)? Please use algebraic identities to make the computation simple. -783. What is the value of [552.1² + 124.2² - 427.9²] / [1104.2]? -784. What is the positive square root of [2843² + 20² - 113720]? -785. Please find the value of 69997², by using algebraic identities to simplify computation. -786. What is the value of 2.845² + 1.445² - (2.845 x 2.89)? Please use algebraic identities to make the computation simple. -787. What is the value of [473.1² + 238.9² - 234.2²] / [946.2]? -788. What is the positive square root of [3615² + 70² - 506100]? --
789. Please find the value of 899², by using algebraic identities to simplify computation. -790. What is the value of 2.131² + 0.931² - (2.131 x 1.862)? Please use algebraic identities to make the computation simple. -791. What is the value of [403.8² + 194.6² - 209.2²] / [807.6]? -792. What is the positive square root of [1799² + 90² - 323820]? -793. Please find the value of 198², by using algebraic identities to simplify computation. -794. What is the value of 2.802² + 2.202² - (2.802 x 4.404)? Please use algebraic identities to make the computation simple. -795. What is the value of [584² + 222.4² - 361.6²] / [1168]? -796. What is the positive square root of [1906² + 50² - 190600]? -797. Please find the value of 198², by using algebraic identities to simplify computation. -798. What is the value of 2.162² + 1.762² - (2.162 x 3.524)? Please use algebraic identities to make the computation simple. -799. What is the value of [475.4² + 299.7² - 175.7²] / [950.8]? -800. What is the positive square root of [2115² + 60² - 253800]? --
801. Evaluate the expression (8134² - 1866²), by using algebraic identities to simplify computation. -802. Evaluate the expression (90007² - 89993²), by using algebraic identities to simplify computation. -803. Find the value of (50006 x 49994), by using algebraic identities to simplify the calculation. -804. What is the value of the positive square root of [(340 + 213)(340 - 213) + 45369]? -805. Evaluate the expression (9986² - 14²), by using algebraic identities to simplify computation. -806. Evaluate the expression (7008² - 6992²), by using algebraic identities to simplify computation. -807. Find the value of (7006 x 6994), by using algebraic identities to simplify the calculation. -808. What is the value of the positive square root of [(660 + 139)(660 - 139) + 19321]? -809. Evaluate the expression (6459² - 3541²), by using algebraic identities to simplify computation. -810. Evaluate the expression (7004² - 6996²), by using algebraic identities to simplify computation. -811. Find the value of (40009 x 39991), by using algebraic identities to simplify the calculation. --
812. What is the value of the positive square root of [(340 + 255)(340 - 255) + 65025]? -813. Evaluate the expression (7254² - 2746²), by using algebraic identities to simplify computation. -814. Evaluate the expression (30007² - 29993²), by using algebraic identities to simplify computation. -815. Find the value of (70009 x 69991), by using algebraic identities to simplify the calculation. -816. What is the value of the positive square root of [(730 + 222)(730 - 222) + 49284]? -817. Evaluate the expression (6684² - 3316²), by using algebraic identities to simplify computation. -818. Evaluate the expression (8003² - 7997²), by using algebraic identities to simplify computation. -819. Find the value of (2004 x 1996), by using algebraic identities to simplify the calculation. -820. What is the value of the positive square root of [(600 + 175)(600 - 175) + 30625]? -821. Evaluate the expression (9662² - 338²), by using algebraic identities to simplify computation. -822. Evaluate the expression (60004² - 59996²), by using algebraic identities to simplify computation.
-823. Find the value of (40005 x 39995), by using algebraic identities to simplify the calculation. -824. What is the value of the positive square root of [(520 + 247)(520 - 247) + 61009]? -825. Evaluate the expression (9865² - 135²), by using algebraic identities to simplify computation. -826. Evaluate the expression (5008² - 4992²), by using algebraic identities to simplify computation. -827. Find the value of (40008 x 39992), by using algebraic identities to simplify the calculation. -828. What is the value of the positive square root of [(480 + 245)(480 - 245) + 60025]? -829. Evaluate the expression (5597² - 4403²), by using algebraic identities to simplify computation. -830. Evaluate the expression (2002² - 1998²), by using algebraic identities to simplify computation. -831. Find the value of (30009 x 29991), by using algebraic identities to simplify the calculation. -832. What is the value of the positive square root of [(750 + 201)(750 - 201) + 40401]? -833. Evaluate the expression (7906² - 2094²), by using algebraic identities to
simplify computation. -834. Evaluate the expression (5009² - 4991²), by using algebraic identities to simplify computation. -835. Find the value of (90007 x 89993), by using algebraic identities to simplify the calculation. -836. What is the value of the positive square root of [(440 + 293)(440 - 293) + 85849]? -837. Evaluate the expression (7038² - 2962²), by using algebraic identities to simplify computation. -838. Evaluate the expression (8003² - 7997²), by using algebraic identities to simplify computation. -839. Find the value of (2008 x 1992), by using algebraic identities to simplify the calculation. -840. What is the value of the positive square root of [(810 + 281)(810 - 281) + 78961]? -841. Evaluate the expression (9108² - 892²), by using algebraic identities to simplify computation. -842. Evaluate the expression (3009² - 2991²), by using algebraic identities to simplify computation. -843. Find the value of (8009 x 7991), by using algebraic identities to simplify the calculation. --
844. What is the value of the positive square root of [(620 + 130)(620 - 130) + 16900]? -845. Evaluate the expression (6281² - 3719²), by using algebraic identities to simplify computation. -846. Evaluate the expression (40003² - 39997²), by using algebraic identities to simplify computation. -847. Find the value of (9007 x 8993), by using algebraic identities to simplify the calculation. -848. What is the value of the positive square root of [(470 + 165)(470 - 165) + 27225]? -849. Evaluate the expression (9966² - 34²), by using algebraic identities to simplify computation. -850. Evaluate the expression (4007² - 3993²), by using algebraic identities to simplify computation. -851. Find the value of (30005 x 29995), by using algebraic identities to simplify the calculation. -852. What is the value of the positive square root of [(610 + 137)(610 - 137) + 18769]? -853. Evaluate the expression (7615² - 2385²), by using algebraic identities to simplify computation. -854. Evaluate the expression (7008² - 6992²), by using algebraic identities to simplify computation. --
855. Find the value of (80008 x 79992), by using algebraic identities to simplify the calculation. -856. What is the value of the positive square root of [(900 + 275)(900 - 275) + 75625]? -857. Evaluate the expression (5001² - 4999²), by using algebraic identities to simplify computation. -858. Evaluate the expression (40004² - 39996²), by using algebraic identities to simplify computation. -859. Find the value of (40006 x 39994), by using algebraic identities to simplify the calculation. -860. What is the value of the positive square root of [(840 + 109)(840 - 109) + 11881]? -861. Evaluate the expression (7164² - 2836²), by using algebraic identities to simplify computation. -862. Evaluate the expression (20009² - 19991²), by using algebraic identities to simplify computation. -863. Find the value of (80008 x 79992), by using algebraic identities to simplify the calculation. -864. What is the value of the positive square root of [(900 + 179)(900 - 179) + 32041]? -865. Evaluate the expression (6118² - 3882²), by using algebraic identities to simplify computation.
-866. Evaluate the expression (5008² - 4992²), by using algebraic identities to simplify computation. -867. Find the value of (8007 x 7993), by using algebraic identities to simplify the calculation. -868. What is the value of the positive square root of [(870 + 198)(870 - 198) + 39204]? -869. Evaluate the expression (8317² - 1683²), by using algebraic identities to simplify computation. -870. Evaluate the expression (90004² - 89996²), by using algebraic identities to simplify computation. -871. Find the value of (50005 x 49995), by using algebraic identities to simplify the calculation. -872. What is the value of the positive square root of [(820 + 114)(820 - 114) + 12996]? -873. Evaluate the expression (5614² - 4386²), by using algebraic identities to simplify computation. -874. Evaluate the expression (20008² - 19992²), by using algebraic identities to simplify computation. -875. Find the value of (6007 x 5993), by using algebraic identities to simplify the calculation. --
876. What is the value of the positive square root of [(530 + 182)(530 - 182) + 33124]? -877. Evaluate the expression (8554² - 1446²), by using algebraic identities to simplify computation. -878. Evaluate the expression (9003² - 8997²), by using algebraic identities to simplify computation. -879. Find the value of (80005 x 79995), by using algebraic identities to simplify the calculation. -880. What is the value of the positive square root of [(390 + 287)(390 - 287) + 82369]? -881. Evaluate the expression (9323² - 677²), by using algebraic identities to simplify computation. -882. Evaluate the expression (40007² - 39993²), by using algebraic identities to simplify computation. -883. Find the value of (3002 x 2998), by using algebraic identities to simplify the calculation. -884. What is the value of the positive square root of [(810 + 203)(810 - 203) + 41209]? -885. Evaluate the expression (6592² - 3408²), by using algebraic identities to simplify computation. -886. Evaluate the expression (9005² - 8995²), by using algebraic identities to simplify computation.
-887. Find the value of (4009 x 3991), by using algebraic identities to simplify the calculation. -888. What is the value of the positive square root of [(770 + 178)(770 - 178) + 31684]? -889. Evaluate the expression (6145² - 3855²), by using algebraic identities to simplify computation. -890. Evaluate the expression (20007² - 19993²), by using algebraic identities to simplify computation. -891. Find the value of (7003 x 6997), by using algebraic identities to simplify the calculation. -892. What is the value of the positive square root of [(370 + 126)(370 - 126) + 15876]? -893. Evaluate the expression (8969² - 1031²), by using algebraic identities to simplify computation. -894. Evaluate the expression (50005² - 49995²), by using algebraic identities to simplify computation. -895. Find the value of (4008 x 3992), by using algebraic identities to simplify the calculation. -896. What is the value of the positive square root of [(530 + 113)(530 - 113) + 12769]? -897. Evaluate the expression (9735² - 265²), by using algebraic identities to
simplify computation. -898. Evaluate the expression (80008² - 79992²), by using algebraic identities to simplify computation. -899. Find the value of (60007 x 59993), by using algebraic identities to simplify the calculation. -900. What is the value of the positive square root of [(580 + 191)(580 - 191) + 36481]? -901. Evaluate the expression (8146² - 1854²), by using algebraic identities to simplify computation. -902. Evaluate the expression (5004² - 4996²), by using algebraic identities to simplify computation. -903. Find the value of (30008 x 29992), by using algebraic identities to simplify the calculation. -904. What is the value of the positive square root of [(640 + 192)(640 - 192) + 36864]? -905. Evaluate the expression (6690² - 3310²), by using algebraic identities to simplify computation. -906. Evaluate the expression (6007² - 5993²), by using algebraic identities to simplify computation. -907. Find the value of (30004 x 29996), by using algebraic identities to simplify the calculation. --
908. What is the value of the positive square root of [(360 + 226)(360 - 226) + 51076]? -909. Evaluate the expression (9962² - 38²), by using algebraic identities to simplify computation. -910. Evaluate the expression (70004² - 69996²), by using algebraic identities to simplify computation. -911. Find the value of (80003 x 79997), by using algebraic identities to simplify the calculation. -912. What is the value of the positive square root of [(300 + 220)(300 - 220) + 48400]? -913. Evaluate the expression (9365² - 635²), by using algebraic identities to simplify computation. -914. Evaluate the expression (80004² - 79996²), by using algebraic identities to simplify computation. -915. Find the value of (8004 x 7996), by using algebraic identities to simplify the calculation. -916. What is the value of the positive square root of [(690 + 239)(690 - 239) + 57121]? -917. Evaluate the expression (7681² - 2319²), by using algebraic identities to simplify computation. -918. Evaluate the expression (9004² - 8996²), by using algebraic identities to simplify computation. --
919. Find the value of (8003 x 7997), by using algebraic identities to simplify the calculation. -920. What is the value of the positive square root of [(850 + 216)(850 - 216) + 46656]? -921. Evaluate the expression (6130² - 3870²), by using algebraic identities to simplify computation. -922. Evaluate the expression (4007² - 3993²), by using algebraic identities to simplify computation. -923. Find the value of (4003 x 3997), by using algebraic identities to simplify the calculation. -924. What is the value of the positive square root of [(450 + 160)(450 - 160) + 25600]? -925. Evaluate the expression (5592² - 4408²), by using algebraic identities to simplify computation. -926. Evaluate the expression (8002² - 7998²), by using algebraic identities to simplify computation. -927. Find the value of (4004 x 3996), by using algebraic identities to simplify the calculation. -928. What is the value of the positive square root of [(810 + 269)(810 - 269) + 72361]? -929. Evaluate the expression (7672² - 2328²), by using algebraic identities to simplify computation.
-930. Evaluate the expression (20006² - 19994²), by using algebraic identities to simplify computation. -931. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -932. What is the value of the positive square root of [(540 + 150)(540 - 150) + 22500]? -933. Evaluate the expression (8163² - 1837²), by using algebraic identities to simplify computation. -934. Evaluate the expression (7007² - 6993²), by using algebraic identities to simplify computation. -935. Find the value of (6007 x 5993), by using algebraic identities to simplify the calculation. -936. What is the value of the positive square root of [(570 + 179)(570 - 179) + 32041]? -937. Evaluate the expression (6696² - 3304²), by using algebraic identities to simplify computation. -938. Evaluate the expression (90007² - 89993²), by using algebraic identities to simplify computation. -939. Find the value of (20009 x 19991), by using algebraic identities to simplify the calculation. -940. What is the value of the positive square root of [(430 + 200)(430 - 200)
+ 40000]? -941. Evaluate the expression (6457² - 3543²), by using algebraic identities to simplify computation. -942. Evaluate the expression (8008² - 7992²), by using algebraic identities to simplify computation. -943. Find the value of (2002 x 1998), by using algebraic identities to simplify the calculation. -944. What is the value of the positive square root of [(690 + 185)(690 - 185) + 34225]? -945. Evaluate the expression (9461² - 539²), by using algebraic identities to simplify computation. -946. Evaluate the expression (20003² - 19997²), by using algebraic identities to simplify computation. -947. Find the value of (2005 x 1995), by using algebraic identities to simplify the calculation. -948. What is the value of the positive square root of [(510 + 214)(510 - 214) + 45796]? -949. Evaluate the expression (8015² - 1985²), by using algebraic identities to simplify computation. -950. Evaluate the expression (7009² - 6991²), by using algebraic identities to simplify computation. --
951. Find the value of (6008 x 5992), by using algebraic identities to simplify the calculation. -952. What is the value of the positive square root of [(820 + 117)(820 - 117) + 13689]? -953. Evaluate the expression (9062² - 938²), by using algebraic identities to simplify computation. -954. Evaluate the expression (40002² - 39998²), by using algebraic identities to simplify computation. -955. Find the value of (30006 x 29994), by using algebraic identities to simplify the calculation. -956. What is the value of the positive square root of [(510 + 184)(510 - 184) + 33856]? -957. Evaluate the expression (7785² - 2215²), by using algebraic identities to simplify computation. -958. Evaluate the expression (70005² - 69995²), by using algebraic identities to simplify computation. -959. Find the value of (30002 x 29998), by using algebraic identities to simplify the calculation. -960. What is the value of the positive square root of [(690 + 281)(690 - 281) + 78961]? -961. Evaluate the expression (5618² - 4382²), by using algebraic identities to simplify computation. --
962. Evaluate the expression (3009² - 2991²), by using algebraic identities to simplify computation. -963. Find the value of (9009 x 8991), by using algebraic identities to simplify the calculation. -964. What is the value of the positive square root of [(880 + 135)(880 - 135) + 18225]? -965. Evaluate the expression (7067² - 2933²), by using algebraic identities to simplify computation. -966. Evaluate the expression (3008² - 2992²), by using algebraic identities to simplify computation. -967. Find the value of (80009 x 79991), by using algebraic identities to simplify the calculation. -968. What is the value of the positive square root of [(840 + 230)(840 - 230) + 52900]? -969. Evaluate the expression (7001² - 2999²), by using algebraic identities to simplify computation. -970. Evaluate the expression (20002² - 19998²), by using algebraic identities to simplify computation. -971. Find the value of (2007 x 1993), by using algebraic identities to simplify the calculation. -972. What is the value of the positive square root of [(880 + 263)(880 - 263) + 69169]?
-973. Evaluate the expression (5721² - 4279²), by using algebraic identities to simplify computation. -974. Evaluate the expression (7009² - 6991²), by using algebraic identities to simplify computation. -975. Find the value of (60005 x 59995), by using algebraic identities to simplify the calculation. -976. What is the value of the positive square root of [(730 + 154)(730 - 154) + 23716]? -977. Evaluate the expression (7696² - 2304²), by using algebraic identities to simplify computation. -978. Evaluate the expression (5002² - 4998²), by using algebraic identities to simplify computation. -979. Find the value of (7002 x 6998), by using algebraic identities to simplify the calculation. -980. What is the value of the positive square root of [(390 + 278)(390 - 278) + 77284]? -981. Evaluate the expression (6300² - 3700²), by using algebraic identities to simplify computation. -982. Evaluate the expression (90005² - 89995²), by using algebraic identities to simplify computation. -983. Find the value of (3007 x 2993), by using algebraic identities to
simplify the calculation. -984. What is the value of the positive square root of [(690 + 152)(690 - 152) + 23104]? -985. Evaluate the expression (8713² - 1287²), by using algebraic identities to simplify computation. -986. Evaluate the expression (8003² - 7997²), by using algebraic identities to simplify computation. -987. Find the value of (90004 x 89996), by using algebraic identities to simplify the calculation. -988. What is the value of the positive square root of [(310 + 184)(310 - 184) + 33856]? -989. Evaluate the expression (5842² - 4158²), by using algebraic identities to simplify computation. -990. Evaluate the expression (2004² - 1996²), by using algebraic identities to simplify computation. -991. Find the value of (60007 x 59993), by using algebraic identities to simplify the calculation. -992. What is the value of the positive square root of [(540 + 285)(540 - 285) + 81225]? -993. Evaluate the expression (9310² - 690²), by using algebraic identities to simplify computation. --
994. Evaluate the expression (9007² - 8993²), by using algebraic identities to simplify computation. -995. Find the value of (20007 x 19993), by using algebraic identities to simplify the calculation. -996. What is the value of the positive square root of [(580 + 135)(580 - 135) + 18225]? -997. Evaluate the expression (8169² - 1831²), by using algebraic identities to simplify computation. -998. Evaluate the expression (9004² - 8996²), by using algebraic identities to simplify computation. -999. Find the value of (2008 x 1992), by using algebraic identities to simplify the calculation. -1000. What is the value of the positive square root of [(530 + 110)(530 110) + 12100]? -1001. Evaluate the expression (8972² - 1028²), by using algebraic identities to simplify computation. -1002. Evaluate the expression (80008² - 79992²), by using algebraic identities to simplify computation. -1003. Find the value of (9008 x 8992), by using algebraic identities to simplify the calculation. -1004. What is the value of the positive square root of [(770 + 181)(770 181) + 32761]?
-1005. Evaluate the expression (7925² - 2075²), by using algebraic identities to simplify computation. -1006. Evaluate the expression (80002² - 79998²), by using algebraic identities to simplify computation. -1007. Find the value of (2006 x 1994), by using algebraic identities to simplify the calculation. -1008. What is the value of the positive square root of [(670 + 162)(670 162) + 26244]? -1009. Evaluate the expression (5607² - 4393²), by using algebraic identities to simplify computation. -1010. Evaluate the expression (8003² - 7997²), by using algebraic identities to simplify computation. -1011. Find the value of (20007 x 19993), by using algebraic identities to simplify the calculation. -1012. What is the value of the positive square root of [(310 + 229)(310 229) + 52441]? -1013. Evaluate the expression (8793² - 1207²), by using algebraic identities to simplify computation. -1014. Evaluate the expression (7003² - 6997²), by using algebraic identities to simplify computation. -1015. Find the value of (90005 x 89995), by using algebraic identities to
simplify the calculation. -1016. What is the value of the positive square root of [(770 + 211)(770 211) + 44521]? -1017. Evaluate the expression (7078² - 2922²), by using algebraic identities to simplify computation. -1018. Evaluate the expression (40003² - 39997²), by using algebraic identities to simplify computation. -1019. Find the value of (80003 x 79997), by using algebraic identities to simplify the calculation. -1020. What is the value of the positive square root of [(810 + 128)(810 128) + 16384]? -1021. Evaluate the expression (6117² - 3883²), by using algebraic identities to simplify computation. -1022. Evaluate the expression (2006² - 1994²), by using algebraic identities to simplify computation. -1023. Find the value of (6007 x 5993), by using algebraic identities to simplify the calculation. -1024. What is the value of the positive square root of [(420 + 265)(420 265) + 70225]? -1025. Evaluate the expression (6801² - 3199²), by using algebraic identities to simplify computation. --
1026. Evaluate the expression (50002² - 49998²), by using algebraic identities to simplify computation. -1027. Find the value of (5005 x 4995), by using algebraic identities to simplify the calculation. -1028. What is the value of the positive square root of [(470 + 205)(470 205) + 42025]? -1029. Evaluate the expression (5379² - 4621²), by using algebraic identities to simplify computation. -1030. Evaluate the expression (8005² - 7995²), by using algebraic identities to simplify computation. -1031. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -1032. What is the value of the positive square root of [(300 + 103)(300 103) + 10609]? -1033. Evaluate the expression (5490² - 4510²), by using algebraic identities to simplify computation. -1034. Evaluate the expression (9009² - 8991²), by using algebraic identities to simplify computation. -1035. Find the value of (5006 x 4994), by using algebraic identities to simplify the calculation. -1036. What is the value of the positive square root of [(610 + 159)(610 159) + 25281]? --
1037. Evaluate the expression (9042² - 958²), by using algebraic identities to simplify computation. -1038. Evaluate the expression (3006² - 2994²), by using algebraic identities to simplify computation. -1039. Find the value of (4006 x 3994), by using algebraic identities to simplify the calculation. -1040. What is the value of the positive square root of [(670 + 197)(670 197) + 38809]? -1041. Evaluate the expression (7943² - 2057²), by using algebraic identities to simplify computation. -1042. Evaluate the expression (3007² - 2993²), by using algebraic identities to simplify computation. -1043. Find the value of (90003 x 89997), by using algebraic identities to simplify the calculation. -1044. What is the value of the positive square root of [(540 + 188)(540 188) + 35344]? -1045. Evaluate the expression (7030² - 2970²), by using algebraic identities to simplify computation. -1046. Evaluate the expression (3009² - 2991²), by using algebraic identities to simplify computation. -1047. Find the value of (4009 x 3991), by using algebraic identities to simplify the calculation.
-1048. What is the value of the positive square root of [(470 + 160)(470 160) + 25600]? -1049. Evaluate the expression (7643² - 2357²), by using algebraic identities to simplify computation. -1050. Evaluate the expression (20004² - 19996²), by using algebraic identities to simplify computation. -1051. Find the value of (90002 x 89998), by using algebraic identities to simplify the calculation. -1052. What is the value of the positive square root of [(300 + 133)(300 133) + 17689]? -1053. Evaluate the expression (9793² - 207²), by using algebraic identities to simplify computation. -1054. Evaluate the expression (7002² - 6998²), by using algebraic identities to simplify computation. -1055. Find the value of (60009 x 59991), by using algebraic identities to simplify the calculation. -1056. What is the value of the positive square root of [(890 + 277)(890 277) + 76729]? -1057. Evaluate the expression (9578² - 422²), by using algebraic identities to simplify computation. -1058. Evaluate the expression (80005² - 79995²), by using algebraic
identities to simplify computation. -1059. Find the value of (5004 x 4996), by using algebraic identities to simplify the calculation. -1060. What is the value of the positive square root of [(840 + 255)(840 255) + 65025]? -1061. Evaluate the expression (9536² - 464²), by using algebraic identities to simplify computation. -1062. Evaluate the expression (4004² - 3996²), by using algebraic identities to simplify computation. -1063. Find the value of (6005 x 5995), by using algebraic identities to simplify the calculation. -1064. What is the value of the positive square root of [(810 + 197)(810 197) + 38809]? -1065. Evaluate the expression (6169² - 3831²), by using algebraic identities to simplify computation. -1066. Evaluate the expression (70009² - 69991²), by using algebraic identities to simplify computation. -1067. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -1068. What is the value of the positive square root of [(600 + 260)(600 260) + 67600]? --
1069. Evaluate the expression (6410² - 3590²), by using algebraic identities to simplify computation. -1070. Evaluate the expression (90004² - 89996²), by using algebraic identities to simplify computation. -1071. Find the value of (70004 x 69996), by using algebraic identities to simplify the calculation. -1072. What is the value of the positive square root of [(450 + 205)(450 205) + 42025]? -1073. Evaluate the expression (7939² - 2061²), by using algebraic identities to simplify computation. -1074. Evaluate the expression (80007² - 79993²), by using algebraic identities to simplify computation. -1075. Find the value of (90005 x 89995), by using algebraic identities to simplify the calculation. -1076. What is the value of the positive square root of [(680 + 178)(680 178) + 31684]? -1077. Evaluate the expression (9786² - 214²), by using algebraic identities to simplify computation. -1078. Evaluate the expression (30006² - 29994²), by using algebraic identities to simplify computation. -1079. Find the value of (7008 x 6992), by using algebraic identities to simplify the calculation. --
1080. What is the value of the positive square root of [(580 + 198)(580 198) + 39204]? -1081. Evaluate the expression (9171² - 829²), by using algebraic identities to simplify computation. -1082. Evaluate the expression (5006² - 4994²), by using algebraic identities to simplify computation. -1083. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -1084. What is the value of the positive square root of [(580 + 252)(580 252) + 63504]? -1085. Evaluate the expression (9239² - 761²), by using algebraic identities to simplify computation. -1086. Evaluate the expression (3003² - 2997²), by using algebraic identities to simplify computation. -1087. Find the value of (7008 x 6992), by using algebraic identities to simplify the calculation. -1088. What is the value of the positive square root of [(470 + 112)(470 112) + 12544]? -1089. Evaluate the expression (8671² - 1329²), by using algebraic identities to simplify computation. -1090. Evaluate the expression (70008² - 69992²), by using algebraic identities to simplify computation.
-1091. Find the value of (2006 x 1994), by using algebraic identities to simplify the calculation. -1092. What is the value of the positive square root of [(400 + 208)(400 208) + 43264]? -1093. Evaluate the expression (8216² - 1784²), by using algebraic identities to simplify computation. -1094. Evaluate the expression (80007² - 79993²), by using algebraic identities to simplify computation. -1095. Find the value of (50002 x 49998), by using algebraic identities to simplify the calculation. -1096. What is the value of the positive square root of [(430 + 294)(430 294) + 86436]? -1097. Evaluate the expression (7724² - 2276²), by using algebraic identities to simplify computation. -1098. Evaluate the expression (90007² - 89993²), by using algebraic identities to simplify computation. -1099. Find the value of (2005 x 1995), by using algebraic identities to simplify the calculation. -1100. What is the value of the positive square root of [(470 + 257)(470 257) + 66049]? -1101. Evaluate the expression (7505² - 2495²), by using algebraic identities
to simplify computation. -1102. Evaluate the expression (60002² - 59998²), by using algebraic identities to simplify computation. -1103. Find the value of (20004 x 19996), by using algebraic identities to simplify the calculation. -1104. What is the value of the positive square root of [(660 + 146)(660 146) + 21316]? -1105. Evaluate the expression (9831² - 169²), by using algebraic identities to simplify computation. -1106. Evaluate the expression (50009² - 49991²), by using algebraic identities to simplify computation. -1107. Find the value of (9007 x 8993), by using algebraic identities to simplify the calculation. -1108. What is the value of the positive square root of [(900 + 155)(900 155) + 24025]? -1109. Evaluate the expression (8082² - 1918²), by using algebraic identities to simplify computation. -1110. Evaluate the expression (4007² - 3993²), by using algebraic identities to simplify computation. -1111. Find the value of (4006 x 3994), by using algebraic identities to simplify the calculation. --
1112. What is the value of the positive square root of [(620 + 174)(620 174) + 30276]? -1113. Evaluate the expression (7056² - 2944²), by using algebraic identities to simplify computation. -1114. Evaluate the expression (30009² - 29991²), by using algebraic identities to simplify computation. -1115. Find the value of (8004 x 7996), by using algebraic identities to simplify the calculation. -1116. What is the value of the positive square root of [(340 + 113)(340 113) + 12769]? -1117. Evaluate the expression (8508² - 1492²), by using algebraic identities to simplify computation. -1118. Evaluate the expression (6009² - 5991²), by using algebraic identities to simplify computation. -1119. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -1120. What is the value of the positive square root of [(600 + 147)(600 147) + 21609]? -1121. Evaluate the expression (8087² - 1913²), by using algebraic identities to simplify computation. -1122. Evaluate the expression (70003² - 69997²), by using algebraic identities to simplify computation.
-1123. Find the value of (60003 x 59997), by using algebraic identities to simplify the calculation. -1124. What is the value of the positive square root of [(470 + 291)(470 291) + 84681]? -1125. Evaluate the expression (8416² - 1584²), by using algebraic identities to simplify computation. -1126. Evaluate the expression (30003² - 29997²), by using algebraic identities to simplify computation. -1127. Find the value of (8007 x 7993), by using algebraic identities to simplify the calculation. -1128. What is the value of the positive square root of [(750 + 249)(750 249) + 62001]? -1129. Evaluate the expression (7133² - 2867²), by using algebraic identities to simplify computation. -1130. Evaluate the expression (80002² - 79998²), by using algebraic identities to simplify computation. -1131. Find the value of (7002 x 6998), by using algebraic identities to simplify the calculation. -1132. What is the value of the positive square root of [(690 + 191)(690 191) + 36481]? -1133. Evaluate the expression (9147² - 853²), by using algebraic identities to
simplify computation. -1134. Evaluate the expression (3002² - 2998²), by using algebraic identities to simplify computation. -1135. Find the value of (20002 x 19998), by using algebraic identities to simplify the calculation. -1136. What is the value of the positive square root of [(300 + 108)(300 108) + 11664]? -1137. Evaluate the expression (6619² - 3381²), by using algebraic identities to simplify computation. -1138. Evaluate the expression (8008² - 7992²), by using algebraic identities to simplify computation. -1139. Find the value of (2007 x 1993), by using algebraic identities to simplify the calculation. -1140. What is the value of the positive square root of [(410 + 256)(410 256) + 65536]? -1141. Evaluate the expression (8211² - 1789²), by using algebraic identities to simplify computation. -1142. Evaluate the expression (3002² - 2998²), by using algebraic identities to simplify computation. -1143. Find the value of (7005 x 6995), by using algebraic identities to simplify the calculation. --
1144. What is the value of the positive square root of [(510 + 295)(510 295) + 87025]? -1145. Evaluate the expression (5537² - 4463²), by using algebraic identities to simplify computation. -1146. Evaluate the expression (9002² - 8998²), by using algebraic identities to simplify computation. -1147. Find the value of (6004 x 5996), by using algebraic identities to simplify the calculation. -1148. What is the value of the positive square root of [(850 + 126)(850 126) + 15876]? -1149. Evaluate the expression (5849² - 4151²), by using algebraic identities to simplify computation. -1150. Evaluate the expression (50007² - 49993²), by using algebraic identities to simplify computation. -1151. Find the value of (7008 x 6992), by using algebraic identities to simplify the calculation. -1152. What is the value of the positive square root of [(690 + 246)(690 246) + 60516]? -1153. Evaluate the expression (9193² - 807²), by using algebraic identities to simplify computation. -1154. Evaluate the expression (8009² - 7991²), by using algebraic identities to simplify computation. --
1155. Find the value of (7006 x 6994), by using algebraic identities to simplify the calculation. -1156. What is the value of the positive square root of [(320 + 261)(320 261) + 68121]? -1157. Evaluate the expression (7761² - 2239²), by using algebraic identities to simplify computation. -1158. Evaluate the expression (9005² - 8995²), by using algebraic identities to simplify computation. -1159. Find the value of (6005 x 5995), by using algebraic identities to simplify the calculation. -1160. What is the value of the positive square root of [(820 + 163)(820 163) + 26569]? -1161. Evaluate the expression (6544² - 3456²), by using algebraic identities to simplify computation. -1162. Evaluate the expression (70003² - 69997²), by using algebraic identities to simplify computation. -1163. Find the value of (60004 x 59996), by using algebraic identities to simplify the calculation. -1164. What is the value of the positive square root of [(870 + 160)(870 160) + 25600]? -1165. Evaluate the expression (5573² - 4427²), by using algebraic identities to simplify computation.
-1166. Evaluate the expression (50007² - 49993²), by using algebraic identities to simplify computation. -1167. Find the value of (8004 x 7996), by using algebraic identities to simplify the calculation. -1168. What is the value of the positive square root of [(490 + 154)(490 154) + 23716]? -1169. Evaluate the expression (7291² - 2709²), by using algebraic identities to simplify computation. -1170. Evaluate the expression (90002² - 89998²), by using algebraic identities to simplify computation. -1171. Find the value of (20004 x 19996), by using algebraic identities to simplify the calculation. -1172. What is the value of the positive square root of [(370 + 128)(370 128) + 16384]? -1173. Evaluate the expression (9095² - 905²), by using algebraic identities to simplify computation. -1174. Evaluate the expression (6007² - 5993²), by using algebraic identities to simplify computation. -1175. Find the value of (3004 x 2996), by using algebraic identities to simplify the calculation. -1176. What is the value of the positive square root of [(730 + 286)(730 -
286) + 81796]? -1177. Evaluate the expression (5338² - 4662²), by using algebraic identities to simplify computation. -1178. Evaluate the expression (30007² - 29993²), by using algebraic identities to simplify computation. -1179. Find the value of (20003 x 19997), by using algebraic identities to simplify the calculation. -1180. What is the value of the positive square root of [(790 + 194)(790 194) + 37636]? -1181. Evaluate the expression (7274² - 2726²), by using algebraic identities to simplify computation. -1182. Evaluate the expression (70007² - 69993²), by using algebraic identities to simplify computation. -1183. Find the value of (7006 x 6994), by using algebraic identities to simplify the calculation. -1184. What is the value of the positive square root of [(530 + 165)(530 165) + 27225]? -1185. Evaluate the expression (5923² - 4077²), by using algebraic identities to simplify computation. -1186. Evaluate the expression (6004² - 5996²), by using algebraic identities to simplify computation. --
1187. Find the value of (7007 x 6993), by using algebraic identities to simplify the calculation. -1188. What is the value of the positive square root of [(480 + 252)(480 252) + 63504]? -1189. Evaluate the expression (6126² - 3874²), by using algebraic identities to simplify computation. -1190. Evaluate the expression (3009² - 2991²), by using algebraic identities to simplify computation. -1191. Find the value of (30002 x 29998), by using algebraic identities to simplify the calculation. -1192. What is the value of the positive square root of [(350 + 241)(350 241) + 58081]? -1193. Evaluate the expression (5096² - 4904²), by using algebraic identities to simplify computation. -1194. Evaluate the expression (7007² - 6993²), by using algebraic identities to simplify computation. -1195. Find the value of (7008 x 6992), by using algebraic identities to simplify the calculation. -1196. What is the value of the positive square root of [(340 + 110)(340 110) + 12100]? -1197. Evaluate the expression (5092² - 4908²), by using algebraic identities to simplify computation.
-1198. Evaluate the expression (20003² - 19997²), by using algebraic identities to simplify computation. -1199. Find the value of (70008 x 69992), by using algebraic identities to simplify the calculation. -1200. What is the value of the positive square root of [(670 + 182)(670 182) + 33124]? --
Solutions: Detailed Drills
1. Please find the value of 4001², by using algebraic identities to simplify computation. -Solution: Since 4001 is only a little higher than 4000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 4000 and b = 1. After the substitution, we get 4001² = (4000 + 1)² = 4000² + 1² + (2 x 4000 x 1) = 16000000 + 1 + 8000 = 16008001. ---2. What is the value of 0.575² + 0.425² + (0.575 x 0.85)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.85 = 2 x 0.425. Using this fact, we can rewrite the given expression as 0.575² + 0.425² + (2 x 0.575 x 0.425). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.575² + 0.425² + (0.575 x 0.85) = 0.575² + 0.425² + (2 x 0.575 x 0.425) = (0.575 + 0.425)² = 1² = 1. ---3. What is the value of [222.6² - 113.84² - 108.76²] / [227.68]? -Solution: First, we look at the three numbers under the square signs in the numerator: 222.6, 113.84, and 108.76. We note that 222.6 = 113.84 + 108.76. We also note, in the denominator, that 227.68 = 2 x 113.84. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 113.84, and b = 108.76, to get [(113.84 + 108.76)² - 113.84² - 108.76²] / 227.68 = [222.6² - 113.84² 108.76²] / 227.68 = 108.76. The Left Hand Side of this equation is exactly
the expression in our question, so the value of that expression is equal to 108.76. ---4. What is the square root of [42.8² + 38.5² + 3295.6]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 42.8, and b = 38.5, we see that 2ab = 3295.6, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [42.8² + 38.5² + 3295.6] = [42.8 + 38.5]². Therefore, the square root of the expression given is equal to [42.8 + 38.5] = 81.3. ---5. Please find the value of 70002², by using algebraic identities to simplify computation. -Solution: Since 70002 is only a little higher than 70000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 70000 and b = 2. After the substitution, we get 70002² = (70000 + 2)² = 70000² + 2² + (2 x 70000 x 2) = 4900000000 + 4 + 280000 = 4900280004. ---6. What is the value of 0.447² + 7.553² + (0.447 x 15.106)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 15.106 = 2 x 7.553. Using this fact, we can rewrite the given expression as 0.447² + 7.553² + (2 x 0.447 x 7.553). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.447² + 7.553² + (0.447 x 15.106) = 0.447² + 7.553² + (2 x 0.447 x 7.553) = (0.447 + 7.553)² = 8² = 64. ---7. What is the value of [127.08² - 52.72² - 74.36²] / [105.44]? -Solution: First, we look at the three numbers under the square signs in the numerator: 127.08, 52.72, and 74.36. We note that 127.08 = 52.72 + 74.36.
We also note, in the denominator, that 105.44 = 2 x 52.72. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 52.72, and b = 74.36, to get [(52.72 + 74.36)² - 52.72² - 74.36²] / 105.44 = [127.08² - 52.72² - 74.36²] / 105.44 = 74.36. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 74.36. ---8. What is the square root of [55.3² + 79.6² + 8803.76]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 55.3, and b = 79.6, we see that 2ab = 8803.76, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [55.3² + 79.6² + 8803.76] = [55.3 + 79.6]². Therefore, the square root of the expression given is equal to [55.3 + 79.6] = 134.9. ---9. Please find the value of 7001², by using algebraic identities to simplify computation. -Solution: Since 7001 is only a little higher than 7000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 1. After the substitution, we get 7001² = (7000 + 1)² = 7000² + 1² + (2 x 7000 x 1) = 49000000 + 1 + 14000 = 49014001. ---10. What is the value of 0.461² + 8.539² + (0.461 x 17.078)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 17.078 = 2 x 8.539. Using this fact, we can rewrite the given expression as 0.461² + 8.539² + (2 x 0.461 x 8.539). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.461² + 8.539² + (0.461 x 17.078) = 0.461² + 8.539² + (2 x
0.461 x 8.539) = (0.461 + 8.539)² = 9² = 81. ---11. What is the value of [170.52² - 113.14² - 57.38²] / [226.28]? -Solution: First, we look at the three numbers under the square signs in the numerator: 170.52, 113.14, and 57.38. We note that 170.52 = 113.14 + 57.38. We also note, in the denominator, that 226.28 = 2 x 113.14. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 113.14, and b = 57.38, to get [(113.14 + 57.38)² - 113.14² - 57.38²] / 226.28 = [170.52² - 113.14² - 57.38²] / 226.28 = 57.38. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 57.38. ---12. What is the square root of [80.7² + 93² + 15010.2]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 80.7, and b = 93, we see that 2ab = 15010.2, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [80.7² + 93² + 15010.2] = [80.7 + 93]². Therefore, the square root of the expression given is equal to [80.7 + 93] = 173.7. ---13. Please find the value of 203², by using algebraic identities to simplify computation. -Solution: Since 203 is only a little higher than 200, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 200 and b = 3. After the substitution, we get 203² = (200 + 3)² = 200² + 3² + (2 x 200 x 3) = 40000 + 9 + 1200 = 41209. ----
14. What is the value of 0.533² + 2.467² + (0.533 x 4.934)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.934 = 2 x 2.467. Using this fact, we can rewrite the given expression as 0.533² + 2.467² + (2 x 0.533 x 2.467). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.533² + 2.467² + (0.533 x 4.934) = 0.533² + 2.467² + (2 x 0.533 x 2.467) = (0.533 + 2.467)² = 3² = 9. ---15. What is the value of [141.25² - 74.79² - 66.46²] / [149.58]? -Solution: First, we look at the three numbers under the square signs in the numerator: 141.25, 74.79, and 66.46. We note that 141.25 = 74.79 + 66.46. We also note, in the denominator, that 149.58 = 2 x 74.79. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 74.79, and b = 66.46, to get [(74.79 + 66.46)² - 74.79² - 66.46²] / 149.58 = [141.25² - 74.79² - 66.46²] / 149.58 = 66.46. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 66.46. ---16. What is the square root of [75.7² + 65.6² + 9931.84]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 75.7, and b = 65.6, we see that 2ab = 9931.84, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [75.7² + 65.6² + 9931.84] = [75.7 + 65.6]². Therefore, the square root of the expression given is equal to [75.7 + 65.6] = 141.3. ---17. Please find the value of 30001², by using algebraic identities to simplify computation. --
Solution: Since 30001 is only a little higher than 30000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 30000 and b = 1. After the substitution, we get 30001² = (30000 + 1)² = 30000² + 1² + (2 x 30000 x 1) = 900000000 + 1 + 60000 = 900060001. ---18. What is the value of 0.628² + 9.372² + (0.628 x 18.744)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 18.744 = 2 x 9.372. Using this fact, we can rewrite the given expression as 0.628² + 9.372² + (2 x 0.628 x 9.372). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.628² + 9.372² + (0.628 x 18.744) = 0.628² + 9.372² + (2 x 0.628 x 9.372) = (0.628 + 9.372)² = 10² = 100. ---19. What is the value of [250.09² - 129.05² - 121.04²] / [258.1]? -Solution: First, we look at the three numbers under the square signs in the numerator: 250.09, 129.05, and 121.04. We note that 250.09 = 129.05 + 121.04. We also note, in the denominator, that 258.1 = 2 x 129.05. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 129.05, and b = 121.04, to get [(129.05 + 121.04)² - 129.05² - 121.04²] / 258.1 = [250.09² - 129.05² 121.04²] / 258.1 = 121.04. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 121.04. ---20. What is the square root of [88.3² + 69.2² + 12220.72]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 88.3, and b = 69.2, we see that 2ab = 12220.72, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [88.3² + 69.2² + 12220.72] = [88.3 + 69.2]².
Therefore, the square root of the expression given is equal to [88.3 + 69.2] = 157.5. ---21. Please find the value of 602², by using algebraic identities to simplify computation. -Solution: Since 602 is only a little higher than 600, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 600 and b = 2. After the substitution, we get 602² = (600 + 2)² = 600² + 2² + (2 x 600 x 2) = 360000 + 4 + 2400 = 362404. ---22. What is the value of 0.514² + 4.486² + (0.514 x 8.972)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.972 = 2 x 4.486. Using this fact, we can rewrite the given expression as 0.514² + 4.486² + (2 x 0.514 x 4.486). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.514² + 4.486² + (0.514 x 8.972) = 0.514² + 4.486² + (2 x 0.514 x 4.486) = (0.514 + 4.486)² = 5² = 25. ---23. What is the value of [154.15² - 59.18² - 94.97²] / [118.36]? -Solution: First, we look at the three numbers under the square signs in the numerator: 154.15, 59.18, and 94.97. We note that 154.15 = 59.18 + 94.97. We also note, in the denominator, that 118.36 = 2 x 59.18. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 59.18, and b = 94.97, to get [(59.18 + 94.97)² - 59.18² - 94.97²] / 118.36 = [154.15² - 59.18² - 94.97²] / 118.36 = 94.97. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 94.97. ----
24. What is the square root of [82.7² + 13.2² + 2183.28]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 82.7, and b = 13.2, we see that 2ab = 2183.28, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [82.7² + 13.2² + 2183.28] = [82.7 + 13.2]². Therefore, the square root of the expression given is equal to [82.7 + 13.2] = 95.9. ---25. Please find the value of 50001², by using algebraic identities to simplify computation. -Solution: Since 50001 is only a little higher than 50000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 1. After the substitution, we get 50001² = (50000 + 1)² = 50000² + 1² + (2 x 50000 x 1) = 2500000000 + 1 + 100000 = 2500100001. ---26. What is the value of 0.695² + 0.305² + (0.695 x 0.61)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.61 = 2 x 0.305. Using this fact, we can rewrite the given expression as 0.695² + 0.305² + (2 x 0.695 x 0.305). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.695² + 0.305² + (0.695 x 0.61) = 0.695² + 0.305² + (2 x 0.695 x 0.305) = (0.695 + 0.305)² = 1² = 1. ---27. What is the value of [123.06² - 52.33² - 70.73²] / [104.66]? -Solution: First, we look at the three numbers under the square signs in the numerator: 123.06, 52.33, and 70.73. We note that 123.06 = 52.33 + 70.73. We also note, in the denominator, that 104.66 = 2 x 52.33. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression.
After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 52.33, and b = 70.73, to get [(52.33 + 70.73)² - 52.33² - 70.73²] / 104.66 = [123.06² - 52.33² - 70.73²] / 104.66 = 70.73. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 70.73. ---28. What is the square root of [21.8² + 46.3² + 2018.68]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 21.8, and b = 46.3, we see that 2ab = 2018.68, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [21.8² + 46.3² + 2018.68] = [21.8 + 46.3]². Therefore, the square root of the expression given is equal to [21.8 + 46.3] = 68.1. ---29. Please find the value of 602², by using algebraic identities to simplify computation. -Solution: Since 602 is only a little higher than 600, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 600 and b = 2. After the substitution, we get 602² = (600 + 2)² = 600² + 2² + (2 x 600 x 2) = 360000 + 4 + 2400 = 362404. ---30. What is the value of 0.631² + 5.369² + (0.631 x 10.738)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 10.738 = 2 x 5.369. Using this fact, we can rewrite the given expression as 0.631² + 5.369² + (2 x 0.631 x 5.369). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.631² + 5.369² + (0.631 x 10.738) = 0.631² + 5.369² + (2 x 0.631 x 5.369) = (0.631 + 5.369)² = 6² = 36. ---31. What is the value of [185.14² - 96.09² - 89.05²] / [192.18]?
-Solution: First, we look at the three numbers under the square signs in the numerator: 185.14, 96.09, and 89.05. We note that 185.14 = 96.09 + 89.05. We also note, in the denominator, that 192.18 = 2 x 96.09. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 96.09, and b = 89.05, to get [(96.09 + 89.05)² - 96.09² - 89.05²] / 192.18 = [185.14² - 96.09² - 89.05²] / 192.18 = 89.05. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 89.05. ---32. What is the square root of [30.3² + 82.4² + 4993.44]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 30.3, and b = 82.4, we see that 2ab = 4993.44, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [30.3² + 82.4² + 4993.44] = [30.3 + 82.4]². Therefore, the square root of the expression given is equal to [30.3 + 82.4] = 112.7. ---33. Please find the value of 30001², by using algebraic identities to simplify computation. -Solution: Since 30001 is only a little higher than 30000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 30000 and b = 1. After the substitution, we get 30001² = (30000 + 1)² = 30000² + 1² + (2 x 30000 x 1) = 900000000 + 1 + 60000 = 900060001. ---34. What is the value of 0.44² + 8.56² + (0.44 x 17.12)? Please use algebraic identities to make the computation simple. --
Solution: First, we note that 17.12 = 2 x 8.56. Using this fact, we can rewrite the given expression as 0.44² + 8.56² + (2 x 0.44 x 8.56). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.44² + 8.56² + (0.44 x 17.12) = 0.44² + 8.56² + (2 x 0.44 x 8.56) = (0.44 + 8.56)² = 9² = 81. ---35. What is the value of [146.85² - 71.13² - 75.72²] / [142.26]? -Solution: First, we look at the three numbers under the square signs in the numerator: 146.85, 71.13, and 75.72. We note that 146.85 = 71.13 + 75.72. We also note, in the denominator, that 142.26 = 2 x 71.13. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 71.13, and b = 75.72, to get [(71.13 + 75.72)² - 71.13² - 75.72²] / 142.26 = [146.85² - 71.13² - 75.72²] / 142.26 = 75.72. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 75.72. ---36. What is the square root of [35.7² + 85² + 6069.]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 35.7, and b = 85, we see that 2ab = 6069., which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [35.7² + 85² + 6069.] = [35.7 + 85]². Therefore, the square root of the expression given is equal to [35.7 + 85] = 120.7. ---37. Please find the value of 70001², by using algebraic identities to simplify computation. -Solution: Since 70001 is only a little higher than 70000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a =
70000 and b = 1. After the substitution, we get 70001² = (70000 + 1)² = 70000² + 1² + (2 x 70000 x 1) = 4900000000 + 1 + 140000 = 4900140001. ---38. What is the value of 0.483² + 5.517² + (0.483 x 11.034)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 11.034 = 2 x 5.517. Using this fact, we can rewrite the given expression as 0.483² + 5.517² + (2 x 0.483 x 5.517). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.483² + 5.517² + (0.483 x 11.034) = 0.483² + 5.517² + (2 x 0.483 x 5.517) = (0.483 + 5.517)² = 6² = 36. ---39. What is the value of [226.13² - 98.2² - 127.93²] / [196.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 226.13, 98.2, and 127.93. We note that 226.13 = 98.2 + 127.93. We also note, in the denominator, that 196.4 = 2 x 98.2. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 98.2, and b = 127.93, to get [(98.2 + 127.93)² - 98.2² - 127.93²] / 196.4 = [226.13² - 98.2² - 127.93²] / 196.4 = 127.93. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 127.93. ---40. What is the square root of [52.3² + 52.1² + 5449.66]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 52.3, and b = 52.1, we see that 2ab = 5449.66, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [52.3² + 52.1² + 5449.66] = [52.3 + 52.1]². Therefore, the square root of the expression given is equal to [52.3 + 52.1] = 104.4. ----
41. Please find the value of 6001², by using algebraic identities to simplify computation. -Solution: Since 6001 is only a little higher than 6000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 1. After the substitution, we get 6001² = (6000 + 1)² = 6000² + 1² + (2 x 6000 x 1) = 36000000 + 1 + 12000 = 36012001. ---42. What is the value of 0.638² + 9.362² + (0.638 x 18.724)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 18.724 = 2 x 9.362. Using this fact, we can rewrite the given expression as 0.638² + 9.362² + (2 x 0.638 x 9.362). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.638² + 9.362² + (0.638 x 18.724) = 0.638² + 9.362² + (2 x 0.638 x 9.362) = (0.638 + 9.362)² = 10² = 100. ---43. What is the value of [203.38² - 103.89² - 99.49²] / [207.78]? -Solution: First, we look at the three numbers under the square signs in the numerator: 203.38, 103.89, and 99.49. We note that 203.38 = 103.89 + 99.49. We also note, in the denominator, that 207.78 = 2 x 103.89. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 103.89, and b = 99.49, to get [(103.89 + 99.49)² - 103.89² - 99.49²] / 207.78 = [203.38² - 103.89² - 99.49²] / 207.78 = 99.49. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 99.49. ---44. What is the square root of [33.2² + 48.9² + 3246.96]? -Solution: By observation, we can deduce that the given expression looks a
little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 33.2, and b = 48.9, we see that 2ab = 3246.96, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [33.2² + 48.9² + 3246.96] = [33.2 + 48.9]². Therefore, the square root of the expression given is equal to [33.2 + 48.9] = 82.1. ---45. Please find the value of 5003², by using algebraic identities to simplify computation. -Solution: Since 5003 is only a little higher than 5000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 5000 and b = 3. After the substitution, we get 5003² = (5000 + 3)² = 5000² + 3² + (2 x 5000 x 3) = 25000000 + 9 + 30000 = 25030009. ---46. What is the value of 0.611² + 8.389² + (0.611 x 16.778)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 16.778 = 2 x 8.389. Using this fact, we can rewrite the given expression as 0.611² + 8.389² + (2 x 0.611 x 8.389). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.611² + 8.389² + (0.611 x 16.778) = 0.611² + 8.389² + (2 x 0.611 x 8.389) = (0.611 + 8.389)² = 9² = 81. ---47. What is the value of [178.52² - 84.3² - 94.22²] / [168.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 178.52, 84.3, and 94.22. We note that 178.52 = 84.3 + 94.22. We also note, in the denominator, that 168.6 = 2 x 84.3. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 84.3, and b = 94.22, to get [(84.3 + 94.22)² - 84.3² - 94.22²] / 168.6 = [178.52² - 84.3² - 94.22²] / 168.6 = 94.22.
The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 94.22. ---48. What is the square root of [19.1² + 78.6² + 3002.52]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 19.1, and b = 78.6, we see that 2ab = 3002.52, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [19.1² + 78.6² + 3002.52] = [19.1 + 78.6]². Therefore, the square root of the expression given is equal to [19.1 + 78.6] = 97.7. ---49. Please find the value of 3002², by using algebraic identities to simplify computation. -Solution: Since 3002 is only a little higher than 3000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 3000 and b = 2. After the substitution, we get 3002² = (3000 + 2)² = 3000² + 2² + (2 x 3000 x 2) = 9000000 + 4 + 12000 = 9012004. ---50. What is the value of 0.434² + 8.566² + (0.434 x 17.132)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 17.132 = 2 x 8.566. Using this fact, we can rewrite the given expression as 0.434² + 8.566² + (2 x 0.434 x 8.566). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.434² + 8.566² + (0.434 x 17.132) = 0.434² + 8.566² + (2 x 0.434 x 8.566) = (0.434 + 8.566)² = 9² = 81. ---51. What is the value of [160.69² - 67.54² - 93.15²] / [135.08]? -Solution: First, we look at the three numbers under the square signs in the numerator: 160.69, 67.54, and 93.15. We note that 160.69 = 67.54 + 93.15.
We also note, in the denominator, that 135.08 = 2 x 67.54. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 67.54, and b = 93.15, to get [(67.54 + 93.15)² - 67.54² - 93.15²] / 135.08 = [160.69² - 67.54² - 93.15²] / 135.08 = 93.15. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 93.15. ---52. What is the square root of [49.6² + 59.5² + 5902.4]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 49.6, and b = 59.5, we see that 2ab = 5902.4, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [49.6² + 59.5² + 5902.4] = [49.6 + 59.5]². Therefore, the square root of the expression given is equal to [49.6 + 59.5] = 109.1. ---53. Please find the value of 801², by using algebraic identities to simplify computation. -Solution: Since 801 is only a little higher than 800, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 800 and b = 1. After the substitution, we get 801² = (800 + 1)² = 800² + 1² + (2 x 800 x 1) = 640000 + 1 + 1600 = 641601. ---54. What is the value of 0.406² + 7.594² + (0.406 x 15.188)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 15.188 = 2 x 7.594. Using this fact, we can rewrite the given expression as 0.406² + 7.594² + (2 x 0.406 x 7.594). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.406² + 7.594² + (0.406 x 15.188) = 0.406² + 7.594² + (2 x
0.406 x 7.594) = (0.406 + 7.594)² = 8² = 64. ---55. What is the value of [198.26² - 82.73² - 115.53²] / [165.46]? -Solution: First, we look at the three numbers under the square signs in the numerator: 198.26, 82.73, and 115.53. We note that 198.26 = 82.73 + 115.53. We also note, in the denominator, that 165.46 = 2 x 82.73. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 82.73, and b = 115.53, to get [(82.73 + 115.53)² - 82.73² - 115.53²] / 165.46 = [198.26² - 82.73² - 115.53²] / 165.46 = 115.53. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 115.53. ---56. What is the square root of [96.6² + 62.9² + 12152.28]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 96.6, and b = 62.9, we see that 2ab = 12152.28, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [96.6² + 62.9² + 12152.28] = [96.6 + 62.9]². Therefore, the square root of the expression given is equal to [96.6 + 62.9] = 159.5. ---57. Please find the value of 2001², by using algebraic identities to simplify computation. -Solution: Since 2001 is only a little higher than 2000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 1. After the substitution, we get 2001² = (2000 + 1)² = 2000² + 1² + (2 x 2000 x 1) = 4000000 + 1 + 4000 = 4004001. ----
58. What is the value of 0.557² + 4.443² + (0.557 x 8.886)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.886 = 2 x 4.443. Using this fact, we can rewrite the given expression as 0.557² + 4.443² + (2 x 0.557 x 4.443). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.557² + 4.443² + (0.557 x 8.886) = 0.557² + 4.443² + (2 x 0.557 x 4.443) = (0.557 + 4.443)² = 5² = 25. ---59. What is the value of [163.81² - 84.71² - 79.1²] / [169.42]? -Solution: First, we look at the three numbers under the square signs in the numerator: 163.81, 84.71, and 79.1. We note that 163.81 = 84.71 + 79.1. We also note, in the denominator, that 169.42 = 2 x 84.71. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 84.71, and b = 79.1, to get [(84.71 + 79.1)² - 84.71² - 79.1²] / 169.42 = [163.81² - 84.71² - 79.1²] / 169.42 = 79.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 79.1. ---60. What is the square root of [78.3² + 35.5² + 5559.3]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 78.3, and b = 35.5, we see that 2ab = 5559.3, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [78.3² + 35.5² + 5559.3] = [78.3 + 35.5]². Therefore, the square root of the expression given is equal to [78.3 + 35.5] = 113.8. ---61. Please find the value of 602², by using algebraic identities to simplify computation. --
Solution: Since 602 is only a little higher than 600, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 600 and b = 2. After the substitution, we get 602² = (600 + 2)² = 600² + 2² + (2 x 600 x 2) = 360000 + 4 + 2400 = 362404. ---62. What is the value of 0.434² + 4.566² + (0.434 x 9.132)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 9.132 = 2 x 4.566. Using this fact, we can rewrite the given expression as 0.434² + 4.566² + (2 x 0.434 x 4.566). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.434² + 4.566² + (0.434 x 9.132) = 0.434² + 4.566² + (2 x 0.434 x 4.566) = (0.434 + 4.566)² = 5² = 25. ---63. What is the value of [195.56² - 125.5² - 70.06²] / [251]? -Solution: First, we look at the three numbers under the square signs in the numerator: 195.56, 125.5, and 70.06. We note that 195.56 = 125.5 + 70.06. We also note, in the denominator, that 251 = 2 x 125.5. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 125.5, and b = 70.06, to get [(125.5 + 70.06)² - 125.5² - 70.06²] / 251 = [195.56² - 125.5² - 70.06²] / 251 = 70.06. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 70.06. ---64. What is the square root of [32.9² + 55.2² + 3632.16]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 32.9, and b = 55.2, we see that 2ab = 3632.16, which is exactly equal to the absolute value of the third term in the expression.
Hence, we can say that [32.9² + 55.2² + 3632.16] = [32.9 + 55.2]². Therefore, the square root of the expression given is equal to [32.9 + 55.2] = 88.1. ---65. Please find the value of 301², by using algebraic identities to simplify computation. -Solution: Since 301 is only a little higher than 300, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 1. After the substitution, we get 301² = (300 + 1)² = 300² + 1² + (2 x 300 x 1) = 90000 + 1 + 600 = 90601. ---66. What is the value of 0.56² + 1.44² + (0.56 x 2.88)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.88 = 2 x 1.44. Using this fact, we can rewrite the given expression as 0.56² + 1.44² + (2 x 0.56 x 1.44). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.56² + 1.44² + (0.56 x 2.88) = 0.56² + 1.44² + (2 x 0.56 x 1.44) = (0.56 + 1.44)² = 2² = 4. ---67. What is the value of [194.67² - 88.07² - 106.6²] / [176.14]? -Solution: First, we look at the three numbers under the square signs in the numerator: 194.67, 88.07, and 106.6. We note that 194.67 = 88.07 + 106.6. We also note, in the denominator, that 176.14 = 2 x 88.07. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 88.07, and b = 106.6, to get [(88.07 + 106.6)² - 88.07² - 106.6²] / 176.14 = [194.67² - 88.07² - 106.6²] / 176.14 = 106.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 106.6. ----
68. What is the square root of [67.4² + 53.5² + 7211.8]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 67.4, and b = 53.5, we see that 2ab = 7211.8, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [67.4² + 53.5² + 7211.8] = [67.4 + 53.5]². Therefore, the square root of the expression given is equal to [67.4 + 53.5] = 120.9. ---69. Please find the value of 603², by using algebraic identities to simplify computation. -Solution: Since 603 is only a little higher than 600, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 600 and b = 3. After the substitution, we get 603² = (600 + 3)² = 600² + 3² + (2 x 600 x 3) = 360000 + 9 + 3600 = 363609. ---70. What is the value of 0.671² + 3.329² + (0.671 x 6.658)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.658 = 2 x 3.329. Using this fact, we can rewrite the given expression as 0.671² + 3.329² + (2 x 0.671 x 3.329). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.671² + 3.329² + (0.671 x 6.658) = 0.671² + 3.329² + (2 x 0.671 x 3.329) = (0.671 + 3.329)² = 4² = 16. ---71. What is the value of [181.39² - 128.83² - 52.56²] / [257.66]? -Solution: First, we look at the three numbers under the square signs in the numerator: 181.39, 128.83, and 52.56. We note that 181.39 = 128.83 + 52.56. We also note, in the denominator, that 257.66 = 2 x 128.83. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get
something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 128.83, and b = 52.56, to get [(128.83 + 52.56)² - 128.83² - 52.56²] / 257.66 = [181.39² - 128.83² - 52.56²] / 257.66 = 52.56. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 52.56. ---72. What is the square root of [73.9² + 32.8² + 4847.84]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 73.9, and b = 32.8, we see that 2ab = 4847.84, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [73.9² + 32.8² + 4847.84] = [73.9 + 32.8]². Therefore, the square root of the expression given is equal to [73.9 + 32.8] = 106.7. ---73. Please find the value of 6001², by using algebraic identities to simplify computation. -Solution: Since 6001 is only a little higher than 6000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 1. After the substitution, we get 6001² = (6000 + 1)² = 6000² + 1² + (2 x 6000 x 1) = 36000000 + 1 + 12000 = 36012001. ---74. What is the value of 0.511² + 3.489² + (0.511 x 6.978)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.978 = 2 x 3.489. Using this fact, we can rewrite the given expression as 0.511² + 3.489² + (2 x 0.511 x 3.489). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.511² + 3.489² + (0.511 x 6.978) = 0.511² + 3.489² + (2 x 0.511 x 3.489) = (0.511 + 3.489)² = 4² = 16. ----
75. What is the value of [202.83² - 126.07² - 76.76²] / [252.14]? -Solution: First, we look at the three numbers under the square signs in the numerator: 202.83, 126.07, and 76.76. We note that 202.83 = 126.07 + 76.76. We also note, in the denominator, that 252.14 = 2 x 126.07. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 126.07, and b = 76.76, to get [(126.07 + 76.76)² - 126.07² - 76.76²] / 252.14 = [202.83² - 126.07² - 76.76²] / 252.14 = 76.76. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 76.76. ---76. What is the square root of [83.8² + 52.3² + 8765.48]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 83.8, and b = 52.3, we see that 2ab = 8765.48, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [83.8² + 52.3² + 8765.48] = [83.8 + 52.3]². Therefore, the square root of the expression given is equal to [83.8 + 52.3] = 136.1. ---77. Please find the value of 20001², by using algebraic identities to simplify computation. -Solution: Since 20001 is only a little higher than 20000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 20000 and b = 1. After the substitution, we get 20001² = (20000 + 1)² = 20000² + 1² + (2 x 20000 x 1) = 400000000 + 1 + 40000 = 400040001. ---78. What is the value of 0.547² + 7.453² + (0.547 x 14.906)? Please use algebraic identities to make the computation simple. --
Solution: First, we note that 14.906 = 2 x 7.453. Using this fact, we can rewrite the given expression as 0.547² + 7.453² + (2 x 0.547 x 7.453). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.547² + 7.453² + (0.547 x 14.906) = 0.547² + 7.453² + (2 x 0.547 x 7.453) = (0.547 + 7.453)² = 8² = 64. ---79. What is the value of [125.62² - 50.04² - 75.58²] / [100.08]? -Solution: First, we look at the three numbers under the square signs in the numerator: 125.62, 50.04, and 75.58. We note that 125.62 = 50.04 + 75.58. We also note, in the denominator, that 100.08 = 2 x 50.04. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 50.04, and b = 75.58, to get [(50.04 + 75.58)² - 50.04² - 75.58²] / 100.08 = [125.62² - 50.04² - 75.58²] / 100.08 = 75.58. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 75.58. ---80. What is the square root of [43² + 95.8² + 8238.8]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 43, and b = 95.8, we see that 2ab = 8238.8, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [43² + 95.8² + 8238.8] = [43 + 95.8]². Therefore, the square root of the expression given is equal to [43 + 95.8] = 138.8. ---81. Please find the value of 403², by using algebraic identities to simplify computation. -Solution: Since 403 is only a little higher than 400, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 400
and b = 3. After the substitution, we get 403² = (400 + 3)² = 400² + 3² + (2 x 400 x 3) = 160000 + 9 + 2400 = 162409. ---82. What is the value of 0.528² + 2.472² + (0.528 x 4.944)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.944 = 2 x 2.472. Using this fact, we can rewrite the given expression as 0.528² + 2.472² + (2 x 0.528 x 2.472). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.528² + 2.472² + (0.528 x 4.944) = 0.528² + 2.472² + (2 x 0.528 x 2.472) = (0.528 + 2.472)² = 3² = 9. ---83. What is the value of [151.1² - 89.92² - 61.18²] / [179.84]? -Solution: First, we look at the three numbers under the square signs in the numerator: 151.1, 89.92, and 61.18. We note that 151.1 = 89.92 + 61.18. We also note, in the denominator, that 179.84 = 2 x 89.92. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 89.92, and b = 61.18, to get [(89.92 + 61.18)² - 89.92² - 61.18²] / 179.84 = [151.1² - 89.92² - 61.18²] / 179.84 = 61.18. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 61.18. ---84. What is the square root of [61.4² + 32.6² + 4003.28]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 61.4, and b = 32.6, we see that 2ab = 4003.28, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [61.4² + 32.6² + 4003.28] = [61.4 + 32.6]². Therefore, the square root of the expression given is equal to [61.4 + 32.6] = 94. ----
85. Please find the value of 301², by using algebraic identities to simplify computation. -Solution: Since 301 is only a little higher than 300, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 1. After the substitution, we get 301² = (300 + 1)² = 300² + 1² + (2 x 300 x 1) = 90000 + 1 + 600 = 90601. ---86. What is the value of 0.41² + 3.59² + (0.41 x 7.18)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 7.18 = 2 x 3.59. Using this fact, we can rewrite the given expression as 0.41² + 3.59² + (2 x 0.41 x 3.59). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.41² + 3.59² + (0.41 x 7.18) = 0.41² + 3.59² + (2 x 0.41 x 3.59) = (0.41 + 3.59)² = 4² = 16. ---87. What is the value of [229.3² - 125.22² - 104.08²] / [250.44]? -Solution: First, we look at the three numbers under the square signs in the numerator: 229.3, 125.22, and 104.08. We note that 229.3 = 125.22 + 104.08. We also note, in the denominator, that 250.44 = 2 x 125.22. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 125.22, and b = 104.08, to get [(125.22 + 104.08)² - 125.22² - 104.08²] / 250.44 = [229.3² - 125.22² 104.08²] / 250.44 = 104.08. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 104.08. ---88. What is the square root of [37² + 53.6² + 3966.4]? --
Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 37, and b = 53.6, we see that 2ab = 3966.4, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [37² + 53.6² + 3966.4] = [37 + 53.6]². Therefore, the square root of the expression given is equal to [37 + 53.6] = 90.6. ---89. Please find the value of 6002², by using algebraic identities to simplify computation. -Solution: Since 6002 is only a little higher than 6000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 2. After the substitution, we get 6002² = (6000 + 2)² = 6000² + 2² + (2 x 6000 x 2) = 36000000 + 4 + 24000 = 36024004. ---90. What is the value of 0.577² + 3.423² + (0.577 x 6.846)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.846 = 2 x 3.423. Using this fact, we can rewrite the given expression as 0.577² + 3.423² + (2 x 0.577 x 3.423). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.577² + 3.423² + (0.577 x 6.846) = 0.577² + 3.423² + (2 x 0.577 x 3.423) = (0.577 + 3.423)² = 4² = 16. ---91. What is the value of [158.93² - 70.02² - 88.91²] / [140.04]? -Solution: First, we look at the three numbers under the square signs in the numerator: 158.93, 70.02, and 88.91. We note that 158.93 = 70.02 + 88.91. We also note, in the denominator, that 140.04 = 2 x 70.02. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 70.02, and b = 88.91, to get [(70.02 +
88.91)² - 70.02² - 88.91²] / 140.04 = [158.93² - 70.02² - 88.91²] / 140.04 = 88.91. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 88.91. ---92. What is the square root of [48² + 38.3² + 3676.8]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 48, and b = 38.3, we see that 2ab = 3676.8, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [48² + 38.3² + 3676.8] = [48 + 38.3]². Therefore, the square root of the expression given is equal to [48 + 38.3] = 86.3. ---93. Please find the value of 80003², by using algebraic identities to simplify computation. -Solution: Since 80003 is only a little higher than 80000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 3. After the substitution, we get 80003² = (80000 + 3)² = 80000² + 3² + (2 x 80000 x 3) = 6400000000 + 9 + 480000 = 6400480009. ---94. What is the value of 0.583² + 6.417² + (0.583 x 12.834)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 12.834 = 2 x 6.417. Using this fact, we can rewrite the given expression as 0.583² + 6.417² + (2 x 0.583 x 6.417). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.583² + 6.417² + (0.583 x 12.834) = 0.583² + 6.417² + (2 x 0.583 x 6.417) = (0.583 + 6.417)² = 7² = 49. ---95. What is the value of [143.11² - 89.53² - 53.58²] / [179.06]? -Solution: First, we look at the three numbers under the square signs in the
numerator: 143.11, 89.53, and 53.58. We note that 143.11 = 89.53 + 53.58. We also note, in the denominator, that 179.06 = 2 x 89.53. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 89.53, and b = 53.58, to get [(89.53 + 53.58)² - 89.53² - 53.58²] / 179.06 = [143.11² - 89.53² - 53.58²] / 179.06 = 53.58. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 53.58. ---96. What is the square root of [92.3² + 46² + 8491.6]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 92.3, and b = 46, we see that 2ab = 8491.6, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [92.3² + 46² + 8491.6] = [92.3 + 46]². Therefore, the square root of the expression given is equal to [92.3 + 46] = 138.3. ---97. Please find the value of 303², by using algebraic identities to simplify computation. -Solution: Since 303 is only a little higher than 300, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 3. After the substitution, we get 303² = (300 + 3)² = 300² + 3² + (2 x 300 x 3) = 90000 + 9 + 1800 = 91809. ---98. What is the value of 0.403² + 8.597² + (0.403 x 17.194)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 17.194 = 2 x 8.597. Using this fact, we can rewrite the given expression as 0.403² + 8.597² + (2 x 0.403 x 8.597).
Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.403² + 8.597² + (0.403 x 17.194) = 0.403² + 8.597² + (2 x 0.403 x 8.597) = (0.403 + 8.597)² = 9² = 81. ---99. What is the value of [218.76² - 101.75² - 117.01²] / [203.5]? -Solution: First, we look at the three numbers under the square signs in the numerator: 218.76, 101.75, and 117.01. We note that 218.76 = 101.75 + 117.01. We also note, in the denominator, that 203.5 = 2 x 101.75. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 101.75, and b = 117.01, to get [(101.75 + 117.01)² - 101.75² - 117.01²] / 203.5 = [218.76² - 101.75² 117.01²] / 203.5 = 117.01. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 117.01. ---100. What is the square root of [64.6² + 87.1² + 11253.32]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 64.6, and b = 87.1, we see that 2ab = 11253.32, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [64.6² + 87.1² + 11253.32] = [64.6 + 87.1]². Therefore, the square root of the expression given is equal to [64.6 + 87.1] = 151.7. ---101. Please find the value of 301², by using algebraic identities to simplify computation. -Solution: Since 301 is only a little higher than 300, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 1. After the substitution, we get 301² = (300 + 1)² = 300² + 1² + (2 x 300 x 1) = 90000 + 1 + 600 = 90601.
---102. What is the value of 0.49² + 5.51² + (0.49 x 11.02)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 11.02 = 2 x 5.51. Using this fact, we can rewrite the given expression as 0.49² + 5.51² + (2 x 0.49 x 5.51). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.49² + 5.51² + (0.49 x 11.02) = 0.49² + 5.51² + (2 x 0.49 x 5.51) = (0.49 + 5.51)² = 6² = 36. ---103. What is the value of [141.49² - 64.47² - 77.02²] / [128.94]? -Solution: First, we look at the three numbers under the square signs in the numerator: 141.49, 64.47, and 77.02. We note that 141.49 = 64.47 + 77.02. We also note, in the denominator, that 128.94 = 2 x 64.47. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 64.47, and b = 77.02, to get [(64.47 + 77.02)² - 64.47² - 77.02²] / 128.94 = [141.49² - 64.47² - 77.02²] / 128.94 = 77.02. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 77.02. ---104. What is the square root of [28.4² + 33.4² + 1897.12]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 28.4, and b = 33.4, we see that 2ab = 1897.12, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [28.4² + 33.4² + 1897.12] = [28.4 + 33.4]². Therefore, the square root of the expression given is equal to [28.4 + 33.4] = 61.8. ---105. Please find the value of 402², by using algebraic identities to simplify
computation. -Solution: Since 402 is only a little higher than 400, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 400 and b = 2. After the substitution, we get 402² = (400 + 2)² = 400² + 2² + (2 x 400 x 2) = 160000 + 4 + 1600 = 161604. ---106. What is the value of 0.535² + 1.465² + (0.535 x 2.93)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.93 = 2 x 1.465. Using this fact, we can rewrite the given expression as 0.535² + 1.465² + (2 x 0.535 x 1.465). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.535² + 1.465² + (0.535 x 2.93) = 0.535² + 1.465² + (2 x 0.535 x 1.465) = (0.535 + 1.465)² = 2² = 4. ---107. What is the value of [172.12² - 52.02² - 120.1²] / [104.04]? -Solution: First, we look at the three numbers under the square signs in the numerator: 172.12, 52.02, and 120.1. We note that 172.12 = 52.02 + 120.1. We also note, in the denominator, that 104.04 = 2 x 52.02. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 52.02, and b = 120.1, to get [(52.02 + 120.1)² - 52.02² - 120.1²] / 104.04 = [172.12² - 52.02² - 120.1²] / 104.04 = 120.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 120.1. ---108. What is the square root of [83.4² + 58.9² + 9824.52]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set
a = 83.4, and b = 58.9, we see that 2ab = 9824.52, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [83.4² + 58.9² + 9824.52] = [83.4 + 58.9]². Therefore, the square root of the expression given is equal to [83.4 + 58.9] = 142.3. ---109. Please find the value of 60003², by using algebraic identities to simplify computation. -Solution: Since 60003 is only a little higher than 60000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 60000 and b = 3. After the substitution, we get 60003² = (60000 + 3)² = 60000² + 3² + (2 x 60000 x 3) = 3600000000 + 9 + 360000 = 3600360009. ---110. What is the value of 0.512² + 4.488² + (0.512 x 8.976)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.976 = 2 x 4.488. Using this fact, we can rewrite the given expression as 0.512² + 4.488² + (2 x 0.512 x 4.488). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.512² + 4.488² + (0.512 x 8.976) = 0.512² + 4.488² + (2 x 0.512 x 4.488) = (0.512 + 4.488)² = 5² = 25. ---111. What is the value of [184.92² - 59.1² - 125.82²] / [118.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 184.92, 59.1, and 125.82. We note that 184.92 = 59.1 + 125.82. We also note, in the denominator, that 118.2 = 2 x 59.1. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 59.1, and b = 125.82, to get [(59.1 + 125.82)² - 59.1² - 125.82²] / 118.2 = [184.92² - 59.1² - 125.82²] / 118.2 = 125.82. The Left Hand Side of this equation is exactly the expression in our
question, so the value of that expression is equal to 125.82. ---112. What is the square root of [64.4² + 67.5² + 8694]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 64.4, and b = 67.5, we see that 2ab = 8694, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [64.4² + 67.5² + 8694] = [64.4 + 67.5]². Therefore, the square root of the expression given is equal to [64.4 + 67.5] = 131.9. ---113. Please find the value of 7001², by using algebraic identities to simplify computation. -Solution: Since 7001 is only a little higher than 7000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 1. After the substitution, we get 7001² = (7000 + 1)² = 7000² + 1² + (2 x 7000 x 1) = 49000000 + 1 + 14000 = 49014001. ---114. What is the value of 0.638² + 7.362² + (0.638 x 14.724)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 14.724 = 2 x 7.362. Using this fact, we can rewrite the given expression as 0.638² + 7.362² + (2 x 0.638 x 7.362). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.638² + 7.362² + (0.638 x 14.724) = 0.638² + 7.362² + (2 x 0.638 x 7.362) = (0.638 + 7.362)² = 8² = 64. ---115. What is the value of [185.39² - 91.2² - 94.19²] / [182.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 185.39, 91.2, and 94.19. We note that 185.39 = 91.2 + 94.19. We also note, in the denominator, that 182.4 = 2 x 91.2.
Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 91.2, and b = 94.19, to get [(91.2 + 94.19)² - 91.2² - 94.19²] / 182.4 = [185.39² - 91.2² - 94.19²] / 182.4 = 94.19. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 94.19. ---116. What is the square root of [43.7² + 22.2² + 1940.28]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 43.7, and b = 22.2, we see that 2ab = 1940.28, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [43.7² + 22.2² + 1940.28] = [43.7 + 22.2]². Therefore, the square root of the expression given is equal to [43.7 + 22.2] = 65.9. ---117. Please find the value of 80002², by using algebraic identities to simplify computation. -Solution: Since 80002 is only a little higher than 80000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 2. After the substitution, we get 80002² = (80000 + 2)² = 80000² + 2² + (2 x 80000 x 2) = 6400000000 + 4 + 320000 = 6400320004. ---118. What is the value of 0.699² + 4.301² + (0.699 x 8.602)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.602 = 2 x 4.301. Using this fact, we can rewrite the given expression as 0.699² + 4.301² + (2 x 0.699 x 4.301). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.699² + 4.301² + (0.699 x 8.602) = 0.699² + 4.301² + (2 x 0.699 x 4.301) = (0.699 + 4.301)² = 5² = 25.
---119. What is the value of [180.81² - 70.59² - 110.22²] / [141.18]? -Solution: First, we look at the three numbers under the square signs in the numerator: 180.81, 70.59, and 110.22. We note that 180.81 = 70.59 + 110.22. We also note, in the denominator, that 141.18 = 2 x 70.59. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 70.59, and b = 110.22, to get [(70.59 + 110.22)² - 70.59² - 110.22²] / 141.18 = [180.81² - 70.59² - 110.22²] / 141.18 = 110.22. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 110.22. ---120. What is the square root of [11.1² + 35.3² + 783.66]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 11.1, and b = 35.3, we see that 2ab = 783.66, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [11.1² + 35.3² + 783.66] = [11.1 + 35.3]². Therefore, the square root of the expression given is equal to [11.1 + 35.3] = 46.4. ---121. Please find the value of 30001², by using algebraic identities to simplify computation. -Solution: Since 30001 is only a little higher than 30000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 30000 and b = 1. After the substitution, we get 30001² = (30000 + 1)² = 30000² + 1² + (2 x 30000 x 1) = 900000000 + 1 + 60000 = 900060001. ---122. What is the value of 0.517² + 7.483² + (0.517 x 14.966)? Please use
algebraic identities to make the computation simple. -Solution: First, we note that 14.966 = 2 x 7.483. Using this fact, we can rewrite the given expression as 0.517² + 7.483² + (2 x 0.517 x 7.483). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.517² + 7.483² + (0.517 x 14.966) = 0.517² + 7.483² + (2 x 0.517 x 7.483) = (0.517 + 7.483)² = 8² = 64. ---123. What is the value of [229.06² - 119.04² - 110.02²] / [238.08]? -Solution: First, we look at the three numbers under the square signs in the numerator: 229.06, 119.04, and 110.02. We note that 229.06 = 119.04 + 110.02. We also note, in the denominator, that 238.08 = 2 x 119.04. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 119.04, and b = 110.02, to get [(119.04 + 110.02)² - 119.04² - 110.02²] / 238.08 = [229.06² - 119.04² 110.02²] / 238.08 = 110.02. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 110.02. ---124. What is the square root of [12.6² + 64.3² + 1620.36]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 12.6, and b = 64.3, we see that 2ab = 1620.36, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [12.6² + 64.3² + 1620.36] = [12.6 + 64.3]². Therefore, the square root of the expression given is equal to [12.6 + 64.3] = 76.9. ---125. Please find the value of 50003², by using algebraic identities to simplify computation. --
Solution: Since 50003 is only a little higher than 50000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 3. After the substitution, we get 50003² = (50000 + 3)² = 50000² + 3² + (2 x 50000 x 3) = 2500000000 + 9 + 300000 = 2500300009. ---126. What is the value of 0.405² + 0.595² + (0.405 x 1.19)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.19 = 2 x 0.595. Using this fact, we can rewrite the given expression as 0.405² + 0.595² + (2 x 0.405 x 0.595). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.405² + 0.595² + (0.405 x 1.19) = 0.405² + 0.595² + (2 x 0.405 x 0.595) = (0.405 + 0.595)² = 1² = 1. ---127. What is the value of [195.28² - 110.77² - 84.51²] / [221.54]? -Solution: First, we look at the three numbers under the square signs in the numerator: 195.28, 110.77, and 84.51. We note that 195.28 = 110.77 + 84.51. We also note, in the denominator, that 221.54 = 2 x 110.77. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 110.77, and b = 84.51, to get [(110.77 + 84.51)² - 110.77² - 84.51²] / 221.54 = [195.28² - 110.77² - 84.51²] / 221.54 = 84.51. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 84.51. ---128. What is the square root of [39.1² + 20.9² + 1634.38]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 39.1, and b = 20.9, we see that 2ab = 1634.38, which is exactly equal to the absolute value of the third term in the expression.
Hence, we can say that [39.1² + 20.9² + 1634.38] = [39.1 + 20.9]². Therefore, the square root of the expression given is equal to [39.1 + 20.9] = 60. ---129. Please find the value of 4002², by using algebraic identities to simplify computation. -Solution: Since 4002 is only a little higher than 4000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 4000 and b = 2. After the substitution, we get 4002² = (4000 + 2)² = 4000² + 2² + (2 x 4000 x 2) = 16000000 + 4 + 16000 = 16016004. ---130. What is the value of 0.467² + 9.533² + (0.467 x 19.066)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 19.066 = 2 x 9.533. Using this fact, we can rewrite the given expression as 0.467² + 9.533² + (2 x 0.467 x 9.533). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.467² + 9.533² + (0.467 x 19.066) = 0.467² + 9.533² + (2 x 0.467 x 9.533) = (0.467 + 9.533)² = 10² = 100. ---131. What is the value of [175.37² - 84.05² - 91.32²] / [168.1]? -Solution: First, we look at the three numbers under the square signs in the numerator: 175.37, 84.05, and 91.32. We note that 175.37 = 84.05 + 91.32. We also note, in the denominator, that 168.1 = 2 x 84.05. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 84.05, and b = 91.32, to get [(84.05 + 91.32)² - 84.05² - 91.32²] / 168.1 = [175.37² - 84.05² - 91.32²] / 168.1 = 91.32. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 91.32. ----
132. What is the square root of [11² + 59² + 1298]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 11, and b = 59, we see that 2ab = 1298, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [11² + 59² + 1298] = [11 + 59]². Therefore, the square root of the expression given is equal to [11 + 59] = 70. ---133. Please find the value of 402², by using algebraic identities to simplify computation. -Solution: Since 402 is only a little higher than 400, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 400 and b = 2. After the substitution, we get 402² = (400 + 2)² = 400² + 2² + (2 x 400 x 2) = 160000 + 4 + 1600 = 161604. ---134. What is the value of 0.687² + 4.313² + (0.687 x 8.626)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.626 = 2 x 4.313. Using this fact, we can rewrite the given expression as 0.687² + 4.313² + (2 x 0.687 x 4.313). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.687² + 4.313² + (0.687 x 8.626) = 0.687² + 4.313² + (2 x 0.687 x 4.313) = (0.687 + 4.313)² = 5² = 25. ---135. What is the value of [139.11² - 76.89² - 62.22²] / [153.78]? -Solution: First, we look at the three numbers under the square signs in the numerator: 139.11, 76.89, and 62.22. We note that 139.11 = 76.89 + 62.22. We also note, in the denominator, that 153.78 = 2 x 76.89. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression.
After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 76.89, and b = 62.22, to get [(76.89 + 62.22)² - 76.89² - 62.22²] / 153.78 = [139.11² - 76.89² - 62.22²] / 153.78 = 62.22. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 62.22. ---136. What is the square root of [88.5² + 23.1² + 4088.7]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 88.5, and b = 23.1, we see that 2ab = 4088.7, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [88.5² + 23.1² + 4088.7] = [88.5 + 23.1]². Therefore, the square root of the expression given is equal to [88.5 + 23.1] = 111.6. ---137. Please find the value of 203², by using algebraic identities to simplify computation. -Solution: Since 203 is only a little higher than 200, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 200 and b = 3. After the substitution, we get 203² = (200 + 3)² = 200² + 3² + (2 x 200 x 3) = 40000 + 9 + 1200 = 41209. ---138. What is the value of 0.589² + 6.411² + (0.589 x 12.822)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 12.822 = 2 x 6.411. Using this fact, we can rewrite the given expression as 0.589² + 6.411² + (2 x 0.589 x 6.411). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.589² + 6.411² + (0.589 x 12.822) = 0.589² + 6.411² + (2 x 0.589 x 6.411) = (0.589 + 6.411)² = 7² = 49. ---139. What is the value of [166.11² - 112.79² - 53.32²] / [225.58]?
-Solution: First, we look at the three numbers under the square signs in the numerator: 166.11, 112.79, and 53.32. We note that 166.11 = 112.79 + 53.32. We also note, in the denominator, that 225.58 = 2 x 112.79. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 112.79, and b = 53.32, to get [(112.79 + 53.32)² - 112.79² - 53.32²] / 225.58 = [166.11² - 112.79² - 53.32²] / 225.58 = 53.32. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 53.32. ---140. What is the square root of [32.3² + 13.1² + 846.26]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 32.3, and b = 13.1, we see that 2ab = 846.26, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [32.3² + 13.1² + 846.26] = [32.3 + 13.1]². Therefore, the square root of the expression given is equal to [32.3 + 13.1] = 45.4. ---141. Please find the value of 2003², by using algebraic identities to simplify computation. -Solution: Since 2003 is only a little higher than 2000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 3. After the substitution, we get 2003² = (2000 + 3)² = 2000² + 3² + (2 x 2000 x 3) = 4000000 + 9 + 12000 = 4012009. ---142. What is the value of 0.579² + 2.421² + (0.579 x 4.842)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.842 = 2 x 2.421. Using this fact, we can
rewrite the given expression as 0.579² + 2.421² + (2 x 0.579 x 2.421). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.579² + 2.421² + (0.579 x 4.842) = 0.579² + 2.421² + (2 x 0.579 x 2.421) = (0.579 + 2.421)² = 3² = 9. ---143. What is the value of [214.46² - 87.8² - 126.66²] / [175.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 214.46, 87.8, and 126.66. We note that 214.46 = 87.8 + 126.66. We also note, in the denominator, that 175.6 = 2 x 87.8. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 87.8, and b = 126.66, to get [(87.8 + 126.66)² - 87.8² - 126.66²] / 175.6 = [214.46² - 87.8² - 126.66²] / 175.6 = 126.66. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 126.66. ---144. What is the square root of [60.7² + 37.8² + 4588.92]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 60.7, and b = 37.8, we see that 2ab = 4588.92, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [60.7² + 37.8² + 4588.92] = [60.7 + 37.8]². Therefore, the square root of the expression given is equal to [60.7 + 37.8] = 98.5. ---145. Please find the value of 60001², by using algebraic identities to simplify computation. -Solution: Since 60001 is only a little higher than 60000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 60000 and b = 1. After the substitution, we get 60001² = (60000 + 1)² =
60000² + 1² + (2 x 60000 x 1) = 3600000000 + 1 + 120000 = 3600120001. ---146. What is the value of 0.407² + 2.593² + (0.407 x 5.186)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.186 = 2 x 2.593. Using this fact, we can rewrite the given expression as 0.407² + 2.593² + (2 x 0.407 x 2.593). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.407² + 2.593² + (0.407 x 5.186) = 0.407² + 2.593² + (2 x 0.407 x 2.593) = (0.407 + 2.593)² = 3² = 9. ---147. What is the value of [235.1² - 118.4² - 116.7²] / [236.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 235.1, 118.4, and 116.7. We note that 235.1 = 118.4 + 116.7. We also note, in the denominator, that 236.8 = 2 x 118.4. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 118.4, and b = 116.7, to get [(118.4 + 116.7)² - 118.4² - 116.7²] / 236.8 = [235.1² - 118.4² - 116.7²] / 236.8 = 116.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 116.7. ---148. What is the square root of [35.5² + 32² + 2272]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 35.5, and b = 32, we see that 2ab = 2272, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [35.5² + 32² + 2272] = [35.5 + 32]². Therefore, the square root of the expression given is equal to [35.5 + 32] = 67.5. ----
149. Please find the value of 4001², by using algebraic identities to simplify computation. -Solution: Since 4001 is only a little higher than 4000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 4000 and b = 1. After the substitution, we get 4001² = (4000 + 1)² = 4000² + 1² + (2 x 4000 x 1) = 16000000 + 1 + 8000 = 16008001. ---150. What is the value of 0.486² + 4.514² + (0.486 x 9.028)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 9.028 = 2 x 4.514. Using this fact, we can rewrite the given expression as 0.486² + 4.514² + (2 x 0.486 x 4.514). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.486² + 4.514² + (0.486 x 9.028) = 0.486² + 4.514² + (2 x 0.486 x 4.514) = (0.486 + 4.514)² = 5² = 25. ---151. What is the value of [161.19² - 79.1² - 82.09²] / [158.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 161.19, 79.1, and 82.09. We note that 161.19 = 79.1 + 82.09. We also note, in the denominator, that 158.2 = 2 x 79.1. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 79.1, and b = 82.09, to get [(79.1 + 82.09)² - 79.1² - 82.09²] / 158.2 = [161.19² - 79.1² - 82.09²] / 158.2 = 82.09. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 82.09. ---152. What is the square root of [29.2² + 99.1² + 5787.44]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set
a = 29.2, and b = 99.1, we see that 2ab = 5787.44, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [29.2² + 99.1² + 5787.44] = [29.2 + 99.1]². Therefore, the square root of the expression given is equal to [29.2 + 99.1] = 128.3. ---153. Please find the value of 2001², by using algebraic identities to simplify computation. -Solution: Since 2001 is only a little higher than 2000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 1. After the substitution, we get 2001² = (2000 + 1)² = 2000² + 1² + (2 x 2000 x 1) = 4000000 + 1 + 4000 = 4004001. ---154. What is the value of 0.619² + 9.381² + (0.619 x 18.762)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 18.762 = 2 x 9.381. Using this fact, we can rewrite the given expression as 0.619² + 9.381² + (2 x 0.619 x 9.381). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.619² + 9.381² + (0.619 x 18.762) = 0.619² + 9.381² + (2 x 0.619 x 9.381) = (0.619 + 9.381)² = 10² = 100. ---155. What is the value of [191.23² - 78.89² - 112.34²] / [157.78]? -Solution: First, we look at the three numbers under the square signs in the numerator: 191.23, 78.89, and 112.34. We note that 191.23 = 78.89 + 112.34. We also note, in the denominator, that 157.78 = 2 x 78.89. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 78.89, and b = 112.34, to get [(78.89 + 112.34)² - 78.89² - 112.34²] / 157.78 = [191.23² - 78.89² - 112.34²] / 157.78 = 112.34. The Left Hand Side of this equation is exactly the expression in
our question, so the value of that expression is equal to 112.34. ---156. What is the square root of [77.7² + 47.1² + 7319.34]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 77.7, and b = 47.1, we see that 2ab = 7319.34, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [77.7² + 47.1² + 7319.34] = [77.7 + 47.1]². Therefore, the square root of the expression given is equal to [77.7 + 47.1] = 124.8. ---157. Please find the value of 6003², by using algebraic identities to simplify computation. -Solution: Since 6003 is only a little higher than 6000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 3. After the substitution, we get 6003² = (6000 + 3)² = 6000² + 3² + (2 x 6000 x 3) = 36000000 + 9 + 36000 = 36036009. ---158. What is the value of 0.449² + 5.551² + (0.449 x 11.102)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 11.102 = 2 x 5.551. Using this fact, we can rewrite the given expression as 0.449² + 5.551² + (2 x 0.449 x 5.551). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.449² + 5.551² + (0.449 x 11.102) = 0.449² + 5.551² + (2 x 0.449 x 5.551) = (0.449 + 5.551)² = 6² = 36. ---159. What is the value of [159.44² - 88.36² - 71.08²] / [176.72]? -Solution: First, we look at the three numbers under the square signs in the numerator: 159.44, 88.36, and 71.08. We note that 159.44 = 88.36 + 71.08. We also note, in the denominator, that 176.72 = 2 x 88.36.
Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 88.36, and b = 71.08, to get [(88.36 + 71.08)² - 88.36² - 71.08²] / 176.72 = [159.44² - 88.36² - 71.08²] / 176.72 = 71.08. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 71.08. ---160. What is the square root of [48.5² + 73.4² + 7119.8]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 48.5, and b = 73.4, we see that 2ab = 7119.8, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [48.5² + 73.4² + 7119.8] = [48.5 + 73.4]². Therefore, the square root of the expression given is equal to [48.5 + 73.4] = 121.9. ---161. Please find the value of 703², by using algebraic identities to simplify computation. -Solution: Since 703 is only a little higher than 700, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 3. After the substitution, we get 703² = (700 + 3)² = 700² + 3² + (2 x 700 x 3) = 490000 + 9 + 4200 = 494209. ---162. What is the value of 0.658² + 8.342² + (0.658 x 16.684)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 16.684 = 2 x 8.342. Using this fact, we can rewrite the given expression as 0.658² + 8.342² + (2 x 0.658 x 8.342). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.658² + 8.342² + (0.658 x 16.684) = 0.658² + 8.342² + (2 x 0.658 x 8.342) = (0.658 + 8.342)² = 9² = 81.
---163. What is the value of [157.97² - 105.61² - 52.36²] / [211.22]? -Solution: First, we look at the three numbers under the square signs in the numerator: 157.97, 105.61, and 52.36. We note that 157.97 = 105.61 + 52.36. We also note, in the denominator, that 211.22 = 2 x 105.61. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 105.61, and b = 52.36, to get [(105.61 + 52.36)² - 105.61² - 52.36²] / 211.22 = [157.97² - 105.61² - 52.36²] / 211.22 = 52.36. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 52.36. ---164. What is the square root of [16.6² + 12.8² + 424.96]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 16.6, and b = 12.8, we see that 2ab = 424.96, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [16.6² + 12.8² + 424.96] = [16.6 + 12.8]². Therefore, the square root of the expression given is equal to [16.6 + 12.8] = 29.4. ---165. Please find the value of 80003², by using algebraic identities to simplify computation. -Solution: Since 80003 is only a little higher than 80000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 3. After the substitution, we get 80003² = (80000 + 3)² = 80000² + 3² + (2 x 80000 x 3) = 6400000000 + 9 + 480000 = 6400480009. ---166. What is the value of 0.651² + 1.349² + (0.651 x 2.698)? Please use
algebraic identities to make the computation simple. -Solution: First, we note that 2.698 = 2 x 1.349. Using this fact, we can rewrite the given expression as 0.651² + 1.349² + (2 x 0.651 x 1.349). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.651² + 1.349² + (0.651 x 2.698) = 0.651² + 1.349² + (2 x 0.651 x 1.349) = (0.651 + 1.349)² = 2² = 4. ---167. What is the value of [175.64² - 92.8² - 82.84²] / [185.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 175.64, 92.8, and 82.84. We note that 175.64 = 92.8 + 82.84. We also note, in the denominator, that 185.6 = 2 x 92.8. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 92.8, and b = 82.84, to get [(92.8 + 82.84)² - 92.8² - 82.84²] / 185.6 = [175.64² - 92.8² - 82.84²] / 185.6 = 82.84. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 82.84. ---168. What is the square root of [87.9² + 14.2² + 2496.36]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 87.9, and b = 14.2, we see that 2ab = 2496.36, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [87.9² + 14.2² + 2496.36] = [87.9 + 14.2]². Therefore, the square root of the expression given is equal to [87.9 + 14.2] = 102.1. ---169. Please find the value of 802², by using algebraic identities to simplify computation. -Solution: Since 802 is only a little higher than 800, we can deduce that the
identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 800 and b = 2. After the substitution, we get 802² = (800 + 2)² = 800² + 2² + (2 x 800 x 2) = 640000 + 4 + 3200 = 643204. ---170. What is the value of 0.548² + 9.452² + (0.548 x 18.904)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 18.904 = 2 x 9.452. Using this fact, we can rewrite the given expression as 0.548² + 9.452² + (2 x 0.548 x 9.452). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.548² + 9.452² + (0.548 x 18.904) = 0.548² + 9.452² + (2 x 0.548 x 9.452) = (0.548 + 9.452)² = 10² = 100. ---171. What is the value of [190.12² - 67.65² - 122.47²] / [135.3]? -Solution: First, we look at the three numbers under the square signs in the numerator: 190.12, 67.65, and 122.47. We note that 190.12 = 67.65 + 122.47. We also note, in the denominator, that 135.3 = 2 x 67.65. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 67.65, and b = 122.47, to get [(67.65 + 122.47)² - 67.65² - 122.47²] / 135.3 = [190.12² - 67.65² - 122.47²] / 135.3 = 122.47. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 122.47. ---172. What is the square root of [14.3² + 42² + 1201.2]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 14.3, and b = 42, we see that 2ab = 1201.2, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [14.3² + 42² + 1201.2] = [14.3 + 42]².
Therefore, the square root of the expression given is equal to [14.3 + 42] = 56.3. ---173. Please find the value of 2002², by using algebraic identities to simplify computation. -Solution: Since 2002 is only a little higher than 2000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 2. After the substitution, we get 2002² = (2000 + 2)² = 2000² + 2² + (2 x 2000 x 2) = 4000000 + 4 + 8000 = 4008004. ---174. What is the value of 0.687² + 4.313² + (0.687 x 8.626)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.626 = 2 x 4.313. Using this fact, we can rewrite the given expression as 0.687² + 4.313² + (2 x 0.687 x 4.313). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.687² + 4.313² + (0.687 x 8.626) = 0.687² + 4.313² + (2 x 0.687 x 4.313) = (0.687 + 4.313)² = 5² = 25. ---175. What is the value of [154.66² - 96.22² - 58.44²] / [192.44]? -Solution: First, we look at the three numbers under the square signs in the numerator: 154.66, 96.22, and 58.44. We note that 154.66 = 96.22 + 58.44. We also note, in the denominator, that 192.44 = 2 x 96.22. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 96.22, and b = 58.44, to get [(96.22 + 58.44)² - 96.22² - 58.44²] / 192.44 = [154.66² - 96.22² - 58.44²] / 192.44 = 58.44. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 58.44. ----
176. What is the square root of [53.1² + 17.5² + 1858.5]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 53.1, and b = 17.5, we see that 2ab = 1858.5, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [53.1² + 17.5² + 1858.5] = [53.1 + 17.5]². Therefore, the square root of the expression given is equal to [53.1 + 17.5] = 70.6. ---177. Please find the value of 4001², by using algebraic identities to simplify computation. -Solution: Since 4001 is only a little higher than 4000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 4000 and b = 1. After the substitution, we get 4001² = (4000 + 1)² = 4000² + 1² + (2 x 4000 x 1) = 16000000 + 1 + 8000 = 16008001. ---178. What is the value of 0.496² + 2.504² + (0.496 x 5.008)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.008 = 2 x 2.504. Using this fact, we can rewrite the given expression as 0.496² + 2.504² + (2 x 0.496 x 2.504). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.496² + 2.504² + (0.496 x 5.008) = 0.496² + 2.504² + (2 x 0.496 x 2.504) = (0.496 + 2.504)² = 3² = 9. ---179. What is the value of [161.85² - 90.79² - 71.06²] / [181.58]? -Solution: First, we look at the three numbers under the square signs in the numerator: 161.85, 90.79, and 71.06. We note that 161.85 = 90.79 + 71.06. We also note, in the denominator, that 181.58 = 2 x 90.79. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression.
After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 90.79, and b = 71.06, to get [(90.79 + 71.06)² - 90.79² - 71.06²] / 181.58 = [161.85² - 90.79² - 71.06²] / 181.58 = 71.06. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 71.06. ---180. What is the square root of [47.4² + 59.6² + 5650.08]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 47.4, and b = 59.6, we see that 2ab = 5650.08, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [47.4² + 59.6² + 5650.08] = [47.4 + 59.6]². Therefore, the square root of the expression given is equal to [47.4 + 59.6] = 107. ---181. Please find the value of 703², by using algebraic identities to simplify computation. -Solution: Since 703 is only a little higher than 700, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 3. After the substitution, we get 703² = (700 + 3)² = 700² + 3² + (2 x 700 x 3) = 490000 + 9 + 4200 = 494209. ---182. What is the value of 0.644² + 3.356² + (0.644 x 6.712)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.712 = 2 x 3.356. Using this fact, we can rewrite the given expression as 0.644² + 3.356² + (2 x 0.644 x 3.356). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.644² + 3.356² + (0.644 x 6.712) = 0.644² + 3.356² + (2 x 0.644 x 3.356) = (0.644 + 3.356)² = 4² = 16. ---183. What is the value of [138.06² - 60.27² - 77.79²] / [120.54]?
-Solution: First, we look at the three numbers under the square signs in the numerator: 138.06, 60.27, and 77.79. We note that 138.06 = 60.27 + 77.79. We also note, in the denominator, that 120.54 = 2 x 60.27. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 60.27, and b = 77.79, to get [(60.27 + 77.79)² - 60.27² - 77.79²] / 120.54 = [138.06² - 60.27² - 77.79²] / 120.54 = 77.79. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 77.79. ---184. What is the square root of [37.1² + 70.4² + 5223.68]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 37.1, and b = 70.4, we see that 2ab = 5223.68, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [37.1² + 70.4² + 5223.68] = [37.1 + 70.4]². Therefore, the square root of the expression given is equal to [37.1 + 70.4] = 107.5. ---185. Please find the value of 3002², by using algebraic identities to simplify computation. -Solution: Since 3002 is only a little higher than 3000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 3000 and b = 2. After the substitution, we get 3002² = (3000 + 2)² = 3000² + 2² + (2 x 3000 x 2) = 9000000 + 4 + 12000 = 9012004. ---186. What is the value of 0.658² + 9.342² + (0.658 x 18.684)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 18.684 = 2 x 9.342. Using this fact, we can
rewrite the given expression as 0.658² + 9.342² + (2 x 0.658 x 9.342). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.658² + 9.342² + (0.658 x 18.684) = 0.658² + 9.342² + (2 x 0.658 x 9.342) = (0.658 + 9.342)² = 10² = 100. ---187. What is the value of [179.61² - 129.07² - 50.54²] / [258.14]? -Solution: First, we look at the three numbers under the square signs in the numerator: 179.61, 129.07, and 50.54. We note that 179.61 = 129.07 + 50.54. We also note, in the denominator, that 258.14 = 2 x 129.07. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 129.07, and b = 50.54, to get [(129.07 + 50.54)² - 129.07² - 50.54²] / 258.14 = [179.61² - 129.07² - 50.54²] / 258.14 = 50.54. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 50.54. ---188. What is the square root of [57.1² + 19.6² + 2238.32]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 57.1, and b = 19.6, we see that 2ab = 2238.32, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [57.1² + 19.6² + 2238.32] = [57.1 + 19.6]². Therefore, the square root of the expression given is equal to [57.1 + 19.6] = 76.7. ---189. Please find the value of 703², by using algebraic identities to simplify computation. -Solution: Since 703 is only a little higher than 700, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 3. After the substitution, we get 703² = (700 + 3)² = 700² + 3² + (2 x
700 x 3) = 490000 + 9 + 4200 = 494209. ---190. What is the value of 0.697² + 0.303² + (0.697 x 0.606)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.606 = 2 x 0.303. Using this fact, we can rewrite the given expression as 0.697² + 0.303² + (2 x 0.697 x 0.303). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.697² + 0.303² + (0.697 x 0.606) = 0.697² + 0.303² + (2 x 0.697 x 0.303) = (0.697 + 0.303)² = 1² = 1. ---191. What is the value of [153.19² - 99.29² - 53.9²] / [198.58]? -Solution: First, we look at the three numbers under the square signs in the numerator: 153.19, 99.29, and 53.9. We note that 153.19 = 99.29 + 53.9. We also note, in the denominator, that 198.58 = 2 x 99.29. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 99.29, and b = 53.9, to get [(99.29 + 53.9)² - 99.29² - 53.9²] / 198.58 = [153.19² - 99.29² - 53.9²] / 198.58 = 53.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 53.9. ---192. What is the square root of [20.3² + 47.6² + 1932.56]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 20.3, and b = 47.6, we see that 2ab = 1932.56, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [20.3² + 47.6² + 1932.56] = [20.3 + 47.6]². Therefore, the square root of the expression given is equal to [20.3 + 47.6] = 67.9. ----
193. Please find the value of 501², by using algebraic identities to simplify computation. -Solution: Since 501 is only a little higher than 500, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 500 and b = 1. After the substitution, we get 501² = (500 + 1)² = 500² + 1² + (2 x 500 x 1) = 250000 + 1 + 1000 = 251001. ---194. What is the value of 0.562² + 4.438² + (0.562 x 8.876)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.876 = 2 x 4.438. Using this fact, we can rewrite the given expression as 0.562² + 4.438² + (2 x 0.562 x 4.438). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.562² + 4.438² + (0.562 x 8.876) = 0.562² + 4.438² + (2 x 0.562 x 4.438) = (0.562 + 4.438)² = 5² = 25. ---195. What is the value of [184.02² - 78.17² - 105.85²] / [156.34]? -Solution: First, we look at the three numbers under the square signs in the numerator: 184.02, 78.17, and 105.85. We note that 184.02 = 78.17 + 105.85. We also note, in the denominator, that 156.34 = 2 x 78.17. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 78.17, and b = 105.85, to get [(78.17 + 105.85)² - 78.17² - 105.85²] / 156.34 = [184.02² - 78.17² - 105.85²] / 156.34 = 105.85. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 105.85. ---196. What is the square root of [35.3² + 41.8² + 2951.08]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set
a = 35.3, and b = 41.8, we see that 2ab = 2951.08, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [35.3² + 41.8² + 2951.08] = [35.3 + 41.8]². Therefore, the square root of the expression given is equal to [35.3 + 41.8] = 77.1. ---197. Please find the value of 801², by using algebraic identities to simplify computation. -Solution: Since 801 is only a little higher than 800, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 800 and b = 1. After the substitution, we get 801² = (800 + 1)² = 800² + 1² + (2 x 800 x 1) = 640000 + 1 + 1600 = 641601. ---198. What is the value of 0.522² + 4.478² + (0.522 x 8.956)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.956 = 2 x 4.478. Using this fact, we can rewrite the given expression as 0.522² + 4.478² + (2 x 0.522 x 4.478). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.522² + 4.478² + (0.522 x 8.956) = 0.522² + 4.478² + (2 x 0.522 x 4.478) = (0.522 + 4.478)² = 5² = 25. ---199. What is the value of [118.16² - 60.94² - 57.22²] / [121.88]? -Solution: First, we look at the three numbers under the square signs in the numerator: 118.16, 60.94, and 57.22. We note that 118.16 = 60.94 + 57.22. We also note, in the denominator, that 121.88 = 2 x 60.94. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 60.94, and b = 57.22, to get [(60.94 + 57.22)² - 60.94² - 57.22²] / 121.88 = [118.16² - 60.94² - 57.22²] / 121.88 =
57.22. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 57.22. ---200. What is the square root of [26.5² + 20.8² + 1102.4]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 26.5, and b = 20.8, we see that 2ab = 1102.4, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [26.5² + 20.8² + 1102.4] = [26.5 + 20.8]². Therefore, the square root of the expression given is equal to [26.5 + 20.8] = 47.3. ---201. Please find the value of 2001², by using algebraic identities to simplify computation. -Solution: Since 2001 is only a little higher than 2000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 1. After the substitution, we get 2001² = (2000 + 1)² = 2000² + 1² + (2 x 2000 x 1) = 4000000 + 1 + 4000 = 4004001. ---202. What is the value of 0.417² + 5.583² + (0.417 x 11.166)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 11.166 = 2 x 5.583. Using this fact, we can rewrite the given expression as 0.417² + 5.583² + (2 x 0.417 x 5.583). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.417² + 5.583² + (0.417 x 11.166) = 0.417² + 5.583² + (2 x 0.417 x 5.583) = (0.417 + 5.583)² = 6² = 36. ---203. What is the value of [170.7² - 93.83² - 76.87²] / [187.66]? -Solution: First, we look at the three numbers under the square signs in the numerator: 170.7, 93.83, and 76.87. We note that 170.7 = 93.83 + 76.87. We
also note, in the denominator, that 187.66 = 2 x 93.83. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 93.83, and b = 76.87, to get [(93.83 + 76.87)² - 93.83² - 76.87²] / 187.66 = [170.7² - 93.83² - 76.87²] / 187.66 = 76.87. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 76.87. ---204. What is the square root of [63² + 50.2² + 6325.2]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 63, and b = 50.2, we see that 2ab = 6325.2, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [63² + 50.2² + 6325.2] = [63 + 50.2]². Therefore, the square root of the expression given is equal to [63 + 50.2] = 113.2. ---205. Please find the value of 601², by using algebraic identities to simplify computation. -Solution: Since 601 is only a little higher than 600, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 600 and b = 1. After the substitution, we get 601² = (600 + 1)² = 600² + 1² + (2 x 600 x 1) = 360000 + 1 + 1200 = 361201. ---206. What is the value of 0.533² + 9.467² + (0.533 x 18.934)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 18.934 = 2 x 9.467. Using this fact, we can rewrite the given expression as 0.533² + 9.467² + (2 x 0.533 x 9.467). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² +
2ab), we get 0.533² + 9.467² + (0.533 x 18.934) = 0.533² + 9.467² + (2 x 0.533 x 9.467) = (0.533 + 9.467)² = 10² = 100. ---207. What is the value of [123.86² - 71.57² - 52.29²] / [143.14]? -Solution: First, we look at the three numbers under the square signs in the numerator: 123.86, 71.57, and 52.29. We note that 123.86 = 71.57 + 52.29. We also note, in the denominator, that 143.14 = 2 x 71.57. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 71.57, and b = 52.29, to get [(71.57 + 52.29)² - 71.57² - 52.29²] / 143.14 = [123.86² - 71.57² - 52.29²] / 143.14 = 52.29. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 52.29. ---208. What is the square root of [72.7² + 80.5² + 11704.7]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 72.7, and b = 80.5, we see that 2ab = 11704.7, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [72.7² + 80.5² + 11704.7] = [72.7 + 80.5]². Therefore, the square root of the expression given is equal to [72.7 + 80.5] = 153.2. ---209. Please find the value of 6001², by using algebraic identities to simplify computation. -Solution: Since 6001 is only a little higher than 6000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 1. After the substitution, we get 6001² = (6000 + 1)² = 6000² + 1² + (2 x 6000 x 1) = 36000000 + 1 + 12000 = 36012001.
---210. What is the value of 0.414² + 9.586² + (0.414 x 19.172)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 19.172 = 2 x 9.586. Using this fact, we can rewrite the given expression as 0.414² + 9.586² + (2 x 0.414 x 9.586). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.414² + 9.586² + (0.414 x 19.172) = 0.414² + 9.586² + (2 x 0.414 x 9.586) = (0.414 + 9.586)² = 10² = 100. ---211. What is the value of [166.98² - 73² - 93.98²] / [146]? -Solution: First, we look at the three numbers under the square signs in the numerator: 166.98, 73, and 93.98. We note that 166.98 = 73 + 93.98. We also note, in the denominator, that 146 = 2 x 73. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 73, and b = 93.98, to get [(73 + 93.98)² - 73² - 93.98²] / 146 = [166.98² - 73² - 93.98²] / 146 = 93.98. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 93.98. ---212. What is the square root of [89.4² + 69.1² + 12355.08]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 89.4, and b = 69.1, we see that 2ab = 12355.08, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [89.4² + 69.1² + 12355.08] = [89.4 + 69.1]². Therefore, the square root of the expression given is equal to [89.4 + 69.1] = 158.5. ---213. Please find the value of 40002², by using algebraic identities to simplify
computation. -Solution: Since 40002 is only a little higher than 40000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 40000 and b = 2. After the substitution, we get 40002² = (40000 + 2)² = 40000² + 2² + (2 x 40000 x 2) = 1600000000 + 4 + 160000 = 1600160004. ---214. What is the value of 0.672² + 2.328² + (0.672 x 4.656)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.656 = 2 x 2.328. Using this fact, we can rewrite the given expression as 0.672² + 2.328² + (2 x 0.672 x 2.328). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.672² + 2.328² + (0.672 x 4.656) = 0.672² + 2.328² + (2 x 0.672 x 2.328) = (0.672 + 2.328)² = 3² = 9. ---215. What is the value of [194.26² - 120.81² - 73.45²] / [241.62]? -Solution: First, we look at the three numbers under the square signs in the numerator: 194.26, 120.81, and 73.45. We note that 194.26 = 120.81 + 73.45. We also note, in the denominator, that 241.62 = 2 x 120.81. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 120.81, and b = 73.45, to get [(120.81 + 73.45)² - 120.81² - 73.45²] / 241.62 = [194.26² - 120.81² - 73.45²] / 241.62 = 73.45. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 73.45. ---216. What is the square root of [33.9² + 10.5² + 711.9]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set
a = 33.9, and b = 10.5, we see that 2ab = 711.9, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [33.9² + 10.5² + 711.9] = [33.9 + 10.5]². Therefore, the square root of the expression given is equal to [33.9 + 10.5] = 44.4. ---217. Please find the value of 60002², by using algebraic identities to simplify computation. -Solution: Since 60002 is only a little higher than 60000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 60000 and b = 2. After the substitution, we get 60002² = (60000 + 2)² = 60000² + 2² + (2 x 60000 x 2) = 3600000000 + 4 + 240000 = 3600240004. ---218. What is the value of 0.485² + 7.515² + (0.485 x 15.03)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 15.03 = 2 x 7.515. Using this fact, we can rewrite the given expression as 0.485² + 7.515² + (2 x 0.485 x 7.515). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.485² + 7.515² + (0.485 x 15.03) = 0.485² + 7.515² + (2 x 0.485 x 7.515) = (0.485 + 7.515)² = 8² = 64. ---219. What is the value of [175.6² - 54.55² - 121.05²] / [109.1]? -Solution: First, we look at the three numbers under the square signs in the numerator: 175.6, 54.55, and 121.05. We note that 175.6 = 54.55 + 121.05. We also note, in the denominator, that 109.1 = 2 x 54.55. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 54.55, and b = 121.05, to get [(54.55 + 121.05)² - 54.55² - 121.05²] / 109.1 = [175.6² - 54.55² - 121.05²] / 109.1 =
121.05. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 121.05. ---220. What is the square root of [92.2² + 38.9² + 7173.16]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 92.2, and b = 38.9, we see that 2ab = 7173.16, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [92.2² + 38.9² + 7173.16] = [92.2 + 38.9]². Therefore, the square root of the expression given is equal to [92.2 + 38.9] = 131.1. ---221. Please find the value of 70002², by using algebraic identities to simplify computation. -Solution: Since 70002 is only a little higher than 70000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 70000 and b = 2. After the substitution, we get 70002² = (70000 + 2)² = 70000² + 2² + (2 x 70000 x 2) = 4900000000 + 4 + 280000 = 4900280004. ---222. What is the value of 0.583² + 5.417² + (0.583 x 10.834)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 10.834 = 2 x 5.417. Using this fact, we can rewrite the given expression as 0.583² + 5.417² + (2 x 0.583 x 5.417). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.583² + 5.417² + (0.583 x 10.834) = 0.583² + 5.417² + (2 x 0.583 x 5.417) = (0.583 + 5.417)² = 6² = 36. ---223. What is the value of [209.84² - 101.8² - 108.04²] / [203.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 209.84, 101.8, and 108.04. We note that 209.84 = 101.8 +
108.04. We also note, in the denominator, that 203.6 = 2 x 101.8. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 101.8, and b = 108.04, to get [(101.8 + 108.04)² - 101.8² - 108.04²] / 203.6 = [209.84² - 101.8² - 108.04²] / 203.6 = 108.04. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 108.04. ---224. What is the square root of [38.8² + 70.1² + 5439.76]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 38.8, and b = 70.1, we see that 2ab = 5439.76, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [38.8² + 70.1² + 5439.76] = [38.8 + 70.1]². Therefore, the square root of the expression given is equal to [38.8 + 70.1] = 108.9. ---225. Please find the value of 8002², by using algebraic identities to simplify computation. -Solution: Since 8002 is only a little higher than 8000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 8000 and b = 2. After the substitution, we get 8002² = (8000 + 2)² = 8000² + 2² + (2 x 8000 x 2) = 64000000 + 4 + 32000 = 64032004. ---226. What is the value of 0.547² + 1.453² + (0.547 x 2.906)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.906 = 2 x 1.453. Using this fact, we can rewrite the given expression as 0.547² + 1.453² + (2 x 0.547 x 1.453). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.547² + 1.453² + (0.547 x 2.906) = 0.547² + 1.453² + (2 x
0.547 x 1.453) = (0.547 + 1.453)² = 2² = 4. ---227. What is the value of [153.59² - 89.53² - 64.06²] / [179.06]? -Solution: First, we look at the three numbers under the square signs in the numerator: 153.59, 89.53, and 64.06. We note that 153.59 = 89.53 + 64.06. We also note, in the denominator, that 179.06 = 2 x 89.53. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 89.53, and b = 64.06, to get [(89.53 + 64.06)² - 89.53² - 64.06²] / 179.06 = [153.59² - 89.53² - 64.06²] / 179.06 = 64.06. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 64.06. ---228. What is the square root of [17.4² + 49² + 1705.2]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 17.4, and b = 49, we see that 2ab = 1705.2, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [17.4² + 49² + 1705.2] = [17.4 + 49]². Therefore, the square root of the expression given is equal to [17.4 + 49] = 66.4. ---229. Please find the value of 301², by using algebraic identities to simplify computation. -Solution: Since 301 is only a little higher than 300, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 1. After the substitution, we get 301² = (300 + 1)² = 300² + 1² + (2 x 300 x 1) = 90000 + 1 + 600 = 90601. ----
230. What is the value of 0.576² + 5.424² + (0.576 x 10.848)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 10.848 = 2 x 5.424. Using this fact, we can rewrite the given expression as 0.576² + 5.424² + (2 x 0.576 x 5.424). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.576² + 5.424² + (0.576 x 10.848) = 0.576² + 5.424² + (2 x 0.576 x 5.424) = (0.576 + 5.424)² = 6² = 36. ---231. What is the value of [242.13² - 129.94² - 112.19²] / [259.88]? -Solution: First, we look at the three numbers under the square signs in the numerator: 242.13, 129.94, and 112.19. We note that 242.13 = 129.94 + 112.19. We also note, in the denominator, that 259.88 = 2 x 129.94. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 129.94, and b = 112.19, to get [(129.94 + 112.19)² - 129.94² - 112.19²] / 259.88 = [242.13² - 129.94² 112.19²] / 259.88 = 112.19. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 112.19. ---232. What is the square root of [48.7² + 43.4² + 4227.16]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 48.7, and b = 43.4, we see that 2ab = 4227.16, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [48.7² + 43.4² + 4227.16] = [48.7 + 43.4]². Therefore, the square root of the expression given is equal to [48.7 + 43.4] = 92.1. ---233. Please find the value of 8002², by using algebraic identities to simplify computation.
-Solution: Since 8002 is only a little higher than 8000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 8000 and b = 2. After the substitution, we get 8002² = (8000 + 2)² = 8000² + 2² + (2 x 8000 x 2) = 64000000 + 4 + 32000 = 64032004. ---234. What is the value of 0.625² + 7.375² + (0.625 x 14.75)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 14.75 = 2 x 7.375. Using this fact, we can rewrite the given expression as 0.625² + 7.375² + (2 x 0.625 x 7.375). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.625² + 7.375² + (0.625 x 14.75) = 0.625² + 7.375² + (2 x 0.625 x 7.375) = (0.625 + 7.375)² = 8² = 64. ---235. What is the value of [142.25² - 75.31² - 66.94²] / [150.62]? -Solution: First, we look at the three numbers under the square signs in the numerator: 142.25, 75.31, and 66.94. We note that 142.25 = 75.31 + 66.94. We also note, in the denominator, that 150.62 = 2 x 75.31. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 75.31, and b = 66.94, to get [(75.31 + 66.94)² - 75.31² - 66.94²] / 150.62 = [142.25² - 75.31² - 66.94²] / 150.62 = 66.94. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 66.94. ---236. What is the square root of [77.2² + 42.6² + 6577.44]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 77.2, and b = 42.6, we see that 2ab = 6577.44, which is exactly equal to
the absolute value of the third term in the expression. Hence, we can say that [77.2² + 42.6² + 6577.44] = [77.2 + 42.6]². Therefore, the square root of the expression given is equal to [77.2 + 42.6] = 119.8. ---237. Please find the value of 7002², by using algebraic identities to simplify computation. -Solution: Since 7002 is only a little higher than 7000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 2. After the substitution, we get 7002² = (7000 + 2)² = 7000² + 2² + (2 x 7000 x 2) = 49000000 + 4 + 28000 = 49028004. ---238. What is the value of 0.63² + 8.37² + (0.63 x 16.74)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 16.74 = 2 x 8.37. Using this fact, we can rewrite the given expression as 0.63² + 8.37² + (2 x 0.63 x 8.37). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.63² + 8.37² + (0.63 x 16.74) = 0.63² + 8.37² + (2 x 0.63 x 8.37) = (0.63 + 8.37)² = 9² = 81. ---239. What is the value of [191.69² - 92.09² - 99.6²] / [184.18]? -Solution: First, we look at the three numbers under the square signs in the numerator: 191.69, 92.09, and 99.6. We note that 191.69 = 92.09 + 99.6. We also note, in the denominator, that 184.18 = 2 x 92.09. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 92.09, and b = 99.6, to get [(92.09 + 99.6)² - 92.09² - 99.6²] / 184.18 = [191.69² - 92.09² - 99.6²] / 184.18 = 99.6. The Left Hand Side of this equation is exactly the expression in our
question, so the value of that expression is equal to 99.6. ---240. What is the square root of [11.8² + 23.1² + 545.16]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 11.8, and b = 23.1, we see that 2ab = 545.16, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [11.8² + 23.1² + 545.16] = [11.8 + 23.1]². Therefore, the square root of the expression given is equal to [11.8 + 23.1] = 34.9. ---241. Please find the value of 703², by using algebraic identities to simplify computation. -Solution: Since 703 is only a little higher than 700, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 3. After the substitution, we get 703² = (700 + 3)² = 700² + 3² + (2 x 700 x 3) = 490000 + 9 + 4200 = 494209. ---242. What is the value of 0.691² + 4.309² + (0.691 x 8.618)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.618 = 2 x 4.309. Using this fact, we can rewrite the given expression as 0.691² + 4.309² + (2 x 0.691 x 4.309). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.691² + 4.309² + (0.691 x 8.618) = 0.691² + 4.309² + (2 x 0.691 x 4.309) = (0.691 + 4.309)² = 5² = 25. ---243. What is the value of [193.79² - 105.23² - 88.56²] / [210.46]? -Solution: First, we look at the three numbers under the square signs in the numerator: 193.79, 105.23, and 88.56. We note that 193.79 = 105.23 +
88.56. We also note, in the denominator, that 210.46 = 2 x 105.23. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 105.23, and b = 88.56, to get [(105.23 + 88.56)² - 105.23² - 88.56²] / 210.46 = [193.79² - 105.23² - 88.56²] / 210.46 = 88.56. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 88.56. ---244. What is the square root of [86.5² + 58.3² + 10085.9]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 86.5, and b = 58.3, we see that 2ab = 10085.9, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [86.5² + 58.3² + 10085.9] = [86.5 + 58.3]². Therefore, the square root of the expression given is equal to [86.5 + 58.3] = 144.8. ---245. Please find the value of 6001², by using algebraic identities to simplify computation. -Solution: Since 6001 is only a little higher than 6000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 1. After the substitution, we get 6001² = (6000 + 1)² = 6000² + 1² + (2 x 6000 x 1) = 36000000 + 1 + 12000 = 36012001. ---246. What is the value of 0.523² + 3.477² + (0.523 x 6.954)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.954 = 2 x 3.477. Using this fact, we can rewrite the given expression as 0.523² + 3.477² + (2 x 0.523 x 3.477). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² +
2ab), we get 0.523² + 3.477² + (0.523 x 6.954) = 0.523² + 3.477² + (2 x 0.523 x 3.477) = (0.523 + 3.477)² = 4² = 16. ---247. What is the value of [232.1² - 103.92² - 128.18²] / [207.84]? -Solution: First, we look at the three numbers under the square signs in the numerator: 232.1, 103.92, and 128.18. We note that 232.1 = 103.92 + 128.18. We also note, in the denominator, that 207.84 = 2 x 103.92. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 103.92, and b = 128.18, to get [(103.92 + 128.18)² - 103.92² - 128.18²] / 207.84 = [232.1² - 103.92² 128.18²] / 207.84 = 128.18. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 128.18. ---248. What is the square root of [93.8² + 40.4² + 7579.04]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 93.8, and b = 40.4, we see that 2ab = 7579.04, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [93.8² + 40.4² + 7579.04] = [93.8 + 40.4]². Therefore, the square root of the expression given is equal to [93.8 + 40.4] = 134.2. ---249. Please find the value of 703², by using algebraic identities to simplify computation. -Solution: Since 703 is only a little higher than 700, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 3. After the substitution, we get 703² = (700 + 3)² = 700² + 3² + (2 x
700 x 3) = 490000 + 9 + 4200 = 494209. ---250. What is the value of 0.603² + 6.397² + (0.603 x 12.794)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 12.794 = 2 x 6.397. Using this fact, we can rewrite the given expression as 0.603² + 6.397² + (2 x 0.603 x 6.397). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.603² + 6.397² + (0.603 x 12.794) = 0.603² + 6.397² + (2 x 0.603 x 6.397) = (0.603 + 6.397)² = 7² = 49. ---251. What is the value of [212.68² - 125.12² - 87.56²] / [250.24]? -Solution: First, we look at the three numbers under the square signs in the numerator: 212.68, 125.12, and 87.56. We note that 212.68 = 125.12 + 87.56. We also note, in the denominator, that 250.24 = 2 x 125.12. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 125.12, and b = 87.56, to get [(125.12 + 87.56)² - 125.12² - 87.56²] / 250.24 = [212.68² - 125.12² - 87.56²] / 250.24 = 87.56. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 87.56. ---252. What is the square root of [77.5² + 98.1² + 15205.5]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 77.5, and b = 98.1, we see that 2ab = 15205.5, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [77.5² + 98.1² + 15205.5] = [77.5 + 98.1]². Therefore, the square root of the expression given is equal to [77.5 + 98.1] = 175.6. ----
253. Please find the value of 20002², by using algebraic identities to simplify computation. -Solution: Since 20002 is only a little higher than 20000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 20000 and b = 2. After the substitution, we get 20002² = (20000 + 2)² = 20000² + 2² + (2 x 20000 x 2) = 400000000 + 4 + 80000 = 400080004. ---254. What is the value of 0.586² + 5.414² + (0.586 x 10.828)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 10.828 = 2 x 5.414. Using this fact, we can rewrite the given expression as 0.586² + 5.414² + (2 x 0.586 x 5.414). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.586² + 5.414² + (0.586 x 10.828) = 0.586² + 5.414² + (2 x 0.586 x 5.414) = (0.586 + 5.414)² = 6² = 36. ---255. What is the value of [154.39² - 62.54² - 91.85²] / [125.08]? -Solution: First, we look at the three numbers under the square signs in the numerator: 154.39, 62.54, and 91.85. We note that 154.39 = 62.54 + 91.85. We also note, in the denominator, that 125.08 = 2 x 62.54. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 62.54, and b = 91.85, to get [(62.54 + 91.85)² - 62.54² - 91.85²] / 125.08 = [154.39² - 62.54² - 91.85²] / 125.08 = 91.85. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 91.85. ---256. What is the square root of [24.6² + 20² + 984]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set
a = 24.6, and b = 20, we see that 2ab = 984, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [24.6² + 20² + 984] = [24.6 + 20]². Therefore, the square root of the expression given is equal to [24.6 + 20] = 44.6. ---257. Please find the value of 80001², by using algebraic identities to simplify computation. -Solution: Since 80001 is only a little higher than 80000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 1. After the substitution, we get 80001² = (80000 + 1)² = 80000² + 1² + (2 x 80000 x 1) = 6400000000 + 1 + 160000 = 6400160001. ---258. What is the value of 0.582² + 0.418² + (0.582 x 0.836)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.836 = 2 x 0.418. Using this fact, we can rewrite the given expression as 0.582² + 0.418² + (2 x 0.582 x 0.418). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.582² + 0.418² + (0.582 x 0.836) = 0.582² + 0.418² + (2 x 0.582 x 0.418) = (0.582 + 0.418)² = 1² = 1. ---259. What is the value of [206.31² - 105.03² - 101.28²] / [210.06]? -Solution: First, we look at the three numbers under the square signs in the numerator: 206.31, 105.03, and 101.28. We note that 206.31 = 105.03 + 101.28. We also note, in the denominator, that 210.06 = 2 x 105.03. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 105.03, and b = 101.28, to get [(105.03 + 101.28)² - 105.03² - 101.28²] / 210.06 = [206.31² - 105.03² -
101.28²] / 210.06 = 101.28. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 101.28. ---260. What is the square root of [41.9² + 88.5² + 7416.3]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 41.9, and b = 88.5, we see that 2ab = 7416.3, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [41.9² + 88.5² + 7416.3] = [41.9 + 88.5]². Therefore, the square root of the expression given is equal to [41.9 + 88.5] = 130.4. ---261. Please find the value of 4001², by using algebraic identities to simplify computation. -Solution: Since 4001 is only a little higher than 4000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 4000 and b = 1. After the substitution, we get 4001² = (4000 + 1)² = 4000² + 1² + (2 x 4000 x 1) = 16000000 + 1 + 8000 = 16008001. ---262. What is the value of 0.669² + 1.331² + (0.669 x 2.662)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.662 = 2 x 1.331. Using this fact, we can rewrite the given expression as 0.669² + 1.331² + (2 x 0.669 x 1.331). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.669² + 1.331² + (0.669 x 2.662) = 0.669² + 1.331² + (2 x 0.669 x 1.331) = (0.669 + 1.331)² = 2² = 4. ---263. What is the value of [172.55² - 63.25² - 109.3²] / [126.5]? -Solution: First, we look at the three numbers under the square signs in the
numerator: 172.55, 63.25, and 109.3. We note that 172.55 = 63.25 + 109.3. We also note, in the denominator, that 126.5 = 2 x 63.25. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 63.25, and b = 109.3, to get [(63.25 + 109.3)² - 63.25² - 109.3²] / 126.5 = [172.55² - 63.25² - 109.3²] / 126.5 = 109.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 109.3. ---264. What is the square root of [85.8² + 28² + 4804.8]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 85.8, and b = 28, we see that 2ab = 4804.8, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [85.8² + 28² + 4804.8] = [85.8 + 28]². Therefore, the square root of the expression given is equal to [85.8 + 28] = 113.8. ---265. Please find the value of 2001², by using algebraic identities to simplify computation. -Solution: Since 2001 is only a little higher than 2000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 1. After the substitution, we get 2001² = (2000 + 1)² = 2000² + 1² + (2 x 2000 x 1) = 4000000 + 1 + 4000 = 4004001. ---266. What is the value of 0.433² + 8.567² + (0.433 x 17.134)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 17.134 = 2 x 8.567. Using this fact, we can rewrite the given expression as 0.433² + 8.567² + (2 x 0.433 x 8.567).
Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.433² + 8.567² + (0.433 x 17.134) = 0.433² + 8.567² + (2 x 0.433 x 8.567) = (0.433 + 8.567)² = 9² = 81. ---267. What is the value of [139.14² - 69.88² - 69.26²] / [139.76]? -Solution: First, we look at the three numbers under the square signs in the numerator: 139.14, 69.88, and 69.26. We note that 139.14 = 69.88 + 69.26. We also note, in the denominator, that 139.76 = 2 x 69.88. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 69.88, and b = 69.26, to get [(69.88 + 69.26)² - 69.88² - 69.26²] / 139.76 = [139.14² - 69.88² - 69.26²] / 139.76 = 69.26. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 69.26. ---268. What is the square root of [11.6² + 70.6² + 1637.92]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 11.6, and b = 70.6, we see that 2ab = 1637.92, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [11.6² + 70.6² + 1637.92] = [11.6 + 70.6]². Therefore, the square root of the expression given is equal to [11.6 + 70.6] = 82.2. ---269. Please find the value of 7002², by using algebraic identities to simplify computation. -Solution: Since 7002 is only a little higher than 7000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 2. After the substitution, we get 7002² = (7000 + 2)² = 7000² + 2² +
(2 x 7000 x 2) = 49000000 + 4 + 28000 = 49028004. ---270. What is the value of 0.443² + 3.557² + (0.443 x 7.114)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 7.114 = 2 x 3.557. Using this fact, we can rewrite the given expression as 0.443² + 3.557² + (2 x 0.443 x 3.557). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.443² + 3.557² + (0.443 x 7.114) = 0.443² + 3.557² + (2 x 0.443 x 3.557) = (0.443 + 3.557)² = 4² = 16. ---271. What is the value of [192.48² - 102.79² - 89.69²] / [205.58]? -Solution: First, we look at the three numbers under the square signs in the numerator: 192.48, 102.79, and 89.69. We note that 192.48 = 102.79 + 89.69. We also note, in the denominator, that 205.58 = 2 x 102.79. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 102.79, and b = 89.69, to get [(102.79 + 89.69)² - 102.79² - 89.69²] / 205.58 = [192.48² - 102.79² - 89.69²] / 205.58 = 89.69. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 89.69. ---272. What is the square root of [96.9² + 62.5² + 12112.5]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 96.9, and b = 62.5, we see that 2ab = 12112.5, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [96.9² + 62.5² + 12112.5] = [96.9 + 62.5]². Therefore, the square root of the expression given is equal to [96.9 + 62.5] = 159.4. ----
273. Please find the value of 40003², by using algebraic identities to simplify computation. -Solution: Since 40003 is only a little higher than 40000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 40000 and b = 3. After the substitution, we get 40003² = (40000 + 3)² = 40000² + 3² + (2 x 40000 x 3) = 1600000000 + 9 + 240000 = 1600240009. ---274. What is the value of 0.656² + 9.344² + (0.656 x 18.688)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 18.688 = 2 x 9.344. Using this fact, we can rewrite the given expression as 0.656² + 9.344² + (2 x 0.656 x 9.344). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.656² + 9.344² + (0.656 x 18.688) = 0.656² + 9.344² + (2 x 0.656 x 9.344) = (0.656 + 9.344)² = 10² = 100. ---275. What is the value of [125.98² - 54.49² - 71.49²] / [108.98]? -Solution: First, we look at the three numbers under the square signs in the numerator: 125.98, 54.49, and 71.49. We note that 125.98 = 54.49 + 71.49. We also note, in the denominator, that 108.98 = 2 x 54.49. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 54.49, and b = 71.49, to get [(54.49 + 71.49)² - 54.49² - 71.49²] / 108.98 = [125.98² - 54.49² - 71.49²] / 108.98 = 71.49. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 71.49. ---276. What is the square root of [51.6² + 61.6² + 6357.12]? -Solution: By observation, we can deduce that the given expression looks a
little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 51.6, and b = 61.6, we see that 2ab = 6357.12, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [51.6² + 61.6² + 6357.12] = [51.6 + 61.6]². Therefore, the square root of the expression given is equal to [51.6 + 61.6] = 113.2. ---277. Please find the value of 2001², by using algebraic identities to simplify computation. -Solution: Since 2001 is only a little higher than 2000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 1. After the substitution, we get 2001² = (2000 + 1)² = 2000² + 1² + (2 x 2000 x 1) = 4000000 + 1 + 4000 = 4004001. ---278. What is the value of 0.549² + 2.451² + (0.549 x 4.902)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.902 = 2 x 2.451. Using this fact, we can rewrite the given expression as 0.549² + 2.451² + (2 x 0.549 x 2.451). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.549² + 2.451² + (0.549 x 4.902) = 0.549² + 2.451² + (2 x 0.549 x 2.451) = (0.549 + 2.451)² = 3² = 9. ---279. What is the value of [232.8² - 116.72² - 116.08²] / [233.44]? -Solution: First, we look at the three numbers under the square signs in the numerator: 232.8, 116.72, and 116.08. We note that 232.8 = 116.72 + 116.08. We also note, in the denominator, that 233.44 = 2 x 116.72. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 116.72, and b = 116.08, to get
[(116.72 + 116.08)² - 116.72² - 116.08²] / 233.44 = [232.8² - 116.72² 116.08²] / 233.44 = 116.08. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 116.08. ---280. What is the square root of [43.6² + 99.8² + 8702.56]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 43.6, and b = 99.8, we see that 2ab = 8702.56, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [43.6² + 99.8² + 8702.56] = [43.6 + 99.8]². Therefore, the square root of the expression given is equal to [43.6 + 99.8] = 143.4. ---281. Please find the value of 6003², by using algebraic identities to simplify computation. -Solution: Since 6003 is only a little higher than 6000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 3. After the substitution, we get 6003² = (6000 + 3)² = 6000² + 3² + (2 x 6000 x 3) = 36000000 + 9 + 36000 = 36036009. ---282. What is the value of 0.466² + 4.534² + (0.466 x 9.068)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 9.068 = 2 x 4.534. Using this fact, we can rewrite the given expression as 0.466² + 4.534² + (2 x 0.466 x 4.534). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.466² + 4.534² + (0.466 x 9.068) = 0.466² + 4.534² + (2 x 0.466 x 4.534) = (0.466 + 4.534)² = 5² = 25. ---283. What is the value of [201.65² - 129.85² - 71.8²] / [259.7]? --
Solution: First, we look at the three numbers under the square signs in the numerator: 201.65, 129.85, and 71.8. We note that 201.65 = 129.85 + 71.8. We also note, in the denominator, that 259.7 = 2 x 129.85. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 129.85, and b = 71.8, to get [(129.85 + 71.8)² - 129.85² - 71.8²] / 259.7 = [201.65² - 129.85² - 71.8²] / 259.7 = 71.8. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 71.8. ---284. What is the square root of [24.9² + 40.9² + 2036.82]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 24.9, and b = 40.9, we see that 2ab = 2036.82, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [24.9² + 40.9² + 2036.82] = [24.9 + 40.9]². Therefore, the square root of the expression given is equal to [24.9 + 40.9] = 65.8. ---285. Please find the value of 80003², by using algebraic identities to simplify computation. -Solution: Since 80003 is only a little higher than 80000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 3. After the substitution, we get 80003² = (80000 + 3)² = 80000² + 3² + (2 x 80000 x 3) = 6400000000 + 9 + 480000 = 6400480009. ---286. What is the value of 0.468² + 9.532² + (0.468 x 19.064)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 19.064 = 2 x 9.532. Using this fact, we can rewrite the given expression as 0.468² + 9.532² + (2 x 0.468 x 9.532).
Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.468² + 9.532² + (0.468 x 19.064) = 0.468² + 9.532² + (2 x 0.468 x 9.532) = (0.468 + 9.532)² = 10² = 100. ---287. What is the value of [235.36² - 128.97² - 106.39²] / [257.94]? -Solution: First, we look at the three numbers under the square signs in the numerator: 235.36, 128.97, and 106.39. We note that 235.36 = 128.97 + 106.39. We also note, in the denominator, that 257.94 = 2 x 128.97. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 128.97, and b = 106.39, to get [(128.97 + 106.39)² - 128.97² - 106.39²] / 257.94 = [235.36² - 128.97² 106.39²] / 257.94 = 106.39. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 106.39. ---288. What is the square root of [47.7² + 48.5² + 4626.9]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 47.7, and b = 48.5, we see that 2ab = 4626.9, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [47.7² + 48.5² + 4626.9] = [47.7 + 48.5]². Therefore, the square root of the expression given is equal to [47.7 + 48.5] = 96.2. ---289. Please find the value of 70003², by using algebraic identities to simplify computation. -Solution: Since 70003 is only a little higher than 70000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a =
70000 and b = 3. After the substitution, we get 70003² = (70000 + 3)² = 70000² + 3² + (2 x 70000 x 3) = 4900000000 + 9 + 420000 = 4900420009. ---290. What is the value of 0.64² + 3.36² + (0.64 x 6.72)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.72 = 2 x 3.36. Using this fact, we can rewrite the given expression as 0.64² + 3.36² + (2 x 0.64 x 3.36). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.64² + 3.36² + (0.64 x 6.72) = 0.64² + 3.36² + (2 x 0.64 x 3.36) = (0.64 + 3.36)² = 4² = 16. ---291. What is the value of [186.32² - 69.61² - 116.71²] / [139.22]? -Solution: First, we look at the three numbers under the square signs in the numerator: 186.32, 69.61, and 116.71. We note that 186.32 = 69.61 + 116.71. We also note, in the denominator, that 139.22 = 2 x 69.61. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 69.61, and b = 116.71, to get [(69.61 + 116.71)² - 69.61² - 116.71²] / 139.22 = [186.32² - 69.61² - 116.71²] / 139.22 = 116.71. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 116.71. ---292. What is the square root of [86.7² + 85.3² + 14791.02]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 86.7, and b = 85.3, we see that 2ab = 14791.02, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [86.7² + 85.3² + 14791.02] = [86.7 + 85.3]². Therefore, the square root of the expression given is equal to [86.7 + 85.3] = 172. ----
293. Please find the value of 20003², by using algebraic identities to simplify computation. -Solution: Since 20003 is only a little higher than 20000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 20000 and b = 3. After the substitution, we get 20003² = (20000 + 3)² = 20000² + 3² + (2 x 20000 x 3) = 400000000 + 9 + 120000 = 400120009. ---294. What is the value of 0.504² + 3.496² + (0.504 x 6.992)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.992 = 2 x 3.496. Using this fact, we can rewrite the given expression as 0.504² + 3.496² + (2 x 0.504 x 3.496). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.504² + 3.496² + (0.504 x 6.992) = 0.504² + 3.496² + (2 x 0.504 x 3.496) = (0.504 + 3.496)² = 4² = 16. ---295. What is the value of [230.71² - 118.08² - 112.63²] / [236.16]? -Solution: First, we look at the three numbers under the square signs in the numerator: 230.71, 118.08, and 112.63. We note that 230.71 = 118.08 + 112.63. We also note, in the denominator, that 236.16 = 2 x 118.08. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 118.08, and b = 112.63, to get [(118.08 + 112.63)² - 118.08² - 112.63²] / 236.16 = [230.71² - 118.08² 112.63²] / 236.16 = 112.63. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 112.63. ---296. What is the square root of [96.8² + 10.8² + 2090.88]? --
Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 96.8, and b = 10.8, we see that 2ab = 2090.88, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [96.8² + 10.8² + 2090.88] = [96.8 + 10.8]². Therefore, the square root of the expression given is equal to [96.8 + 10.8] = 107.6. ---297. Please find the value of 2003², by using algebraic identities to simplify computation. -Solution: Since 2003 is only a little higher than 2000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 3. After the substitution, we get 2003² = (2000 + 3)² = 2000² + 3² + (2 x 2000 x 3) = 4000000 + 9 + 12000 = 4012009. ---298. What is the value of 0.586² + 9.414² + (0.586 x 18.828)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 18.828 = 2 x 9.414. Using this fact, we can rewrite the given expression as 0.586² + 9.414² + (2 x 0.586 x 9.414). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.586² + 9.414² + (0.586 x 18.828) = 0.586² + 9.414² + (2 x 0.586 x 9.414) = (0.586 + 9.414)² = 10² = 100. ---299. What is the value of [233.44² - 113.45² - 119.99²] / [226.9]? -Solution: First, we look at the three numbers under the square signs in the numerator: 233.44, 113.45, and 119.99. We note that 233.44 = 113.45 + 119.99. We also note, in the denominator, that 226.9 = 2 x 113.45. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 113.45, and b = 119.99, to get
[(113.45 + 119.99)² - 113.45² - 119.99²] / 226.9 = [233.44² - 113.45² 119.99²] / 226.9 = 119.99. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 119.99. ---300. What is the square root of [50.9² + 72.4² + 7370.32]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 50.9, and b = 72.4, we see that 2ab = 7370.32, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [50.9² + 72.4² + 7370.32] = [50.9 + 72.4]². Therefore, the square root of the expression given is equal to [50.9 + 72.4] = 123.3. ---301. Please find the value of 60003², by using algebraic identities to simplify computation. -Solution: Since 60003 is only a little higher than 60000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 60000 and b = 3. After the substitution, we get 60003² = (60000 + 3)² = 60000² + 3² + (2 x 60000 x 3) = 3600000000 + 9 + 360000 = 3600360009. ---302. What is the value of 0.669² + 1.331² + (0.669 x 2.662)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.662 = 2 x 1.331. Using this fact, we can rewrite the given expression as 0.669² + 1.331² + (2 x 0.669 x 1.331). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.669² + 1.331² + (0.669 x 2.662) = 0.669² + 1.331² + (2 x 0.669 x 1.331) = (0.669 + 1.331)² = 2² = 4. ---303. What is the value of [175.56² - 74.88² - 100.68²] / [149.76]? -Solution: First, we look at the three numbers under the square signs in the
numerator: 175.56, 74.88, and 100.68. We note that 175.56 = 74.88 + 100.68. We also note, in the denominator, that 149.76 = 2 x 74.88. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 74.88, and b = 100.68, to get [(74.88 + 100.68)² - 74.88² - 100.68²] / 149.76 = [175.56² - 74.88² - 100.68²] / 149.76 = 100.68. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 100.68. ---304. What is the square root of [39.8² + 81.5² + 6487.4]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 39.8, and b = 81.5, we see that 2ab = 6487.4, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [39.8² + 81.5² + 6487.4] = [39.8 + 81.5]². Therefore, the square root of the expression given is equal to [39.8 + 81.5] = 121.3. ---305. Please find the value of 702², by using algebraic identities to simplify computation. -Solution: Since 702 is only a little higher than 700, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 2. After the substitution, we get 702² = (700 + 2)² = 700² + 2² + (2 x 700 x 2) = 490000 + 4 + 2800 = 492804. ---306. What is the value of 0.505² + 5.495² + (0.505 x 10.99)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 10.99 = 2 x 5.495. Using this fact, we can rewrite the given expression as 0.505² + 5.495² + (2 x 0.505 x 5.495).
Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.505² + 5.495² + (0.505 x 10.99) = 0.505² + 5.495² + (2 x 0.505 x 5.495) = (0.505 + 5.495)² = 6² = 36. ---307. What is the value of [179.71² - 51.1² - 128.61²] / [102.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 179.71, 51.1, and 128.61. We note that 179.71 = 51.1 + 128.61. We also note, in the denominator, that 102.2 = 2 x 51.1. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 51.1, and b = 128.61, to get [(51.1 + 128.61)² - 51.1² - 128.61²] / 102.2 = [179.71² - 51.1² - 128.61²] / 102.2 = 128.61. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 128.61. ---308. What is the square root of [94.4² + 19.6² + 3700.48]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 94.4, and b = 19.6, we see that 2ab = 3700.48, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [94.4² + 19.6² + 3700.48] = [94.4 + 19.6]². Therefore, the square root of the expression given is equal to [94.4 + 19.6] = 114. ---309. Please find the value of 5002², by using algebraic identities to simplify computation. -Solution: Since 5002 is only a little higher than 5000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 5000 and b = 2. After the substitution, we get 5002² = (5000 + 2)² = 5000² + 2² + (2 x 5000 x 2) = 25000000 + 4 + 20000 = 25020004.
---310. What is the value of 0.645² + 1.355² + (0.645 x 2.71)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.71 = 2 x 1.355. Using this fact, we can rewrite the given expression as 0.645² + 1.355² + (2 x 0.645 x 1.355). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.645² + 1.355² + (0.645 x 2.71) = 0.645² + 1.355² + (2 x 0.645 x 1.355) = (0.645 + 1.355)² = 2² = 4. ---311. What is the value of [174.53² - 122.09² - 52.44²] / [244.18]? -Solution: First, we look at the three numbers under the square signs in the numerator: 174.53, 122.09, and 52.44. We note that 174.53 = 122.09 + 52.44. We also note, in the denominator, that 244.18 = 2 x 122.09. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 122.09, and b = 52.44, to get [(122.09 + 52.44)² - 122.09² - 52.44²] / 244.18 = [174.53² - 122.09² - 52.44²] / 244.18 = 52.44. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 52.44. ---312. What is the square root of [99.7² + 48.4² + 9650.96]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 99.7, and b = 48.4, we see that 2ab = 9650.96, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [99.7² + 48.4² + 9650.96] = [99.7 + 48.4]². Therefore, the square root of the expression given is equal to [99.7 + 48.4] = 148.1. ----
313. Please find the value of 7001², by using algebraic identities to simplify computation. -Solution: Since 7001 is only a little higher than 7000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 1. After the substitution, we get 7001² = (7000 + 1)² = 7000² + 1² + (2 x 7000 x 1) = 49000000 + 1 + 14000 = 49014001. ---314. What is the value of 0.675² + 0.325² + (0.675 x 0.65)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.65 = 2 x 0.325. Using this fact, we can rewrite the given expression as 0.675² + 0.325² + (2 x 0.675 x 0.325). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.675² + 0.325² + (0.675 x 0.65) = 0.675² + 0.325² + (2 x 0.675 x 0.325) = (0.675 + 0.325)² = 1² = 1. ---315. What is the value of [195.56² - 96.01² - 99.55²] / [192.02]? -Solution: First, we look at the three numbers under the square signs in the numerator: 195.56, 96.01, and 99.55. We note that 195.56 = 96.01 + 99.55. We also note, in the denominator, that 192.02 = 2 x 96.01. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 96.01, and b = 99.55, to get [(96.01 + 99.55)² - 96.01² - 99.55²] / 192.02 = [195.56² - 96.01² - 99.55²] / 192.02 = 99.55. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 99.55. ---316. What is the square root of [93.8² + 11² + 2063.6]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set
a = 93.8, and b = 11, we see that 2ab = 2063.6, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [93.8² + 11² + 2063.6] = [93.8 + 11]². Therefore, the square root of the expression given is equal to [93.8 + 11] = 104.8. ---317. Please find the value of 3001², by using algebraic identities to simplify computation. -Solution: Since 3001 is only a little higher than 3000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 3000 and b = 1. After the substitution, we get 3001² = (3000 + 1)² = 3000² + 1² + (2 x 3000 x 1) = 9000000 + 1 + 6000 = 9006001. ---318. What is the value of 0.661² + 6.339² + (0.661 x 12.678)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 12.678 = 2 x 6.339. Using this fact, we can rewrite the given expression as 0.661² + 6.339² + (2 x 0.661 x 6.339). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.661² + 6.339² + (0.661 x 12.678) = 0.661² + 6.339² + (2 x 0.661 x 6.339) = (0.661 + 6.339)² = 7² = 49. ---319. What is the value of [150.29² - 88.55² - 61.74²] / [177.1]? -Solution: First, we look at the three numbers under the square signs in the numerator: 150.29, 88.55, and 61.74. We note that 150.29 = 88.55 + 61.74. We also note, in the denominator, that 177.1 = 2 x 88.55. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 88.55, and b = 61.74, to get [(88.55 + 61.74)² - 88.55² - 61.74²] / 177.1 = [150.29² - 88.55² - 61.74²] / 177.1 =
61.74. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 61.74. ---320. What is the square root of [88.7² + 95.7² + 16977.18]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 88.7, and b = 95.7, we see that 2ab = 16977.18, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [88.7² + 95.7² + 16977.18] = [88.7 + 95.7]². Therefore, the square root of the expression given is equal to [88.7 + 95.7] = 184.4. ---321. Please find the value of 802², by using algebraic identities to simplify computation. -Solution: Since 802 is only a little higher than 800, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 800 and b = 2. After the substitution, we get 802² = (800 + 2)² = 800² + 2² + (2 x 800 x 2) = 640000 + 4 + 3200 = 643204. ---322. What is the value of 0.63² + 1.37² + (0.63 x 2.74)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.74 = 2 x 1.37. Using this fact, we can rewrite the given expression as 0.63² + 1.37² + (2 x 0.63 x 1.37). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.63² + 1.37² + (0.63 x 2.74) = 0.63² + 1.37² + (2 x 0.63 x 1.37) = (0.63 + 1.37)² = 2² = 4. ---323. What is the value of [177.87² - 104.7² - 73.17²] / [209.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 177.87, 104.7, and 73.17. We note that 177.87 = 104.7 + 73.17.
We also note, in the denominator, that 209.4 = 2 x 104.7. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 104.7, and b = 73.17, to get [(104.7 + 73.17)² - 104.7² - 73.17²] / 209.4 = [177.87² - 104.7² - 73.17²] / 209.4 = 73.17. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 73.17. ---324. What is the square root of [18.6² + 65.8² + 2447.76]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 18.6, and b = 65.8, we see that 2ab = 2447.76, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [18.6² + 65.8² + 2447.76] = [18.6 + 65.8]². Therefore, the square root of the expression given is equal to [18.6 + 65.8] = 84.4. ---325. Please find the value of 80002², by using algebraic identities to simplify computation. -Solution: Since 80002 is only a little higher than 80000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 2. After the substitution, we get 80002² = (80000 + 2)² = 80000² + 2² + (2 x 80000 x 2) = 6400000000 + 4 + 320000 = 6400320004. ---326. What is the value of 0.651² + 5.349² + (0.651 x 10.698)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 10.698 = 2 x 5.349. Using this fact, we can rewrite the given expression as 0.651² + 5.349² + (2 x 0.651 x 5.349). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.651² + 5.349² + (0.651 x 10.698) = 0.651² + 5.349² + (2 x
0.651 x 5.349) = (0.651 + 5.349)² = 6² = 36. ---327. What is the value of [207.74² - 114.45² - 93.29²] / [228.9]? -Solution: First, we look at the three numbers under the square signs in the numerator: 207.74, 114.45, and 93.29. We note that 207.74 = 114.45 + 93.29. We also note, in the denominator, that 228.9 = 2 x 114.45. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 114.45, and b = 93.29, to get [(114.45 + 93.29)² - 114.45² - 93.29²] / 228.9 = [207.74² - 114.45² - 93.29²] / 228.9 = 93.29. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 93.29. ---328. What is the square root of [61.3² + 25.8² + 3163.08]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 61.3, and b = 25.8, we see that 2ab = 3163.08, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [61.3² + 25.8² + 3163.08] = [61.3 + 25.8]². Therefore, the square root of the expression given is equal to [61.3 + 25.8] = 87.1. ---329. Please find the value of 60001², by using algebraic identities to simplify computation. -Solution: Since 60001 is only a little higher than 60000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 60000 and b = 1. After the substitution, we get 60001² = (60000 + 1)² = 60000² + 1² + (2 x 60000 x 1) = 3600000000 + 1 + 120000 = 3600120001. ----
330. What is the value of 0.542² + 8.458² + (0.542 x 16.916)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 16.916 = 2 x 8.458. Using this fact, we can rewrite the given expression as 0.542² + 8.458² + (2 x 0.542 x 8.458). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.542² + 8.458² + (0.542 x 16.916) = 0.542² + 8.458² + (2 x 0.542 x 8.458) = (0.542 + 8.458)² = 9² = 81. ---331. What is the value of [155.16² - 100.65² - 54.51²] / [201.3]? -Solution: First, we look at the three numbers under the square signs in the numerator: 155.16, 100.65, and 54.51. We note that 155.16 = 100.65 + 54.51. We also note, in the denominator, that 201.3 = 2 x 100.65. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 100.65, and b = 54.51, to get [(100.65 + 54.51)² - 100.65² - 54.51²] / 201.3 = [155.16² - 100.65² - 54.51²] / 201.3 = 54.51. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 54.51. ---332. What is the square root of [49.9² + 75.5² + 7534.9]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 49.9, and b = 75.5, we see that 2ab = 7534.9, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [49.9² + 75.5² + 7534.9] = [49.9 + 75.5]². Therefore, the square root of the expression given is equal to [49.9 + 75.5] = 125.4. ---333. Please find the value of 401², by using algebraic identities to simplify computation.
-Solution: Since 401 is only a little higher than 400, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 400 and b = 1. After the substitution, we get 401² = (400 + 1)² = 400² + 1² + (2 x 400 x 1) = 160000 + 1 + 800 = 160801. ---334. What is the value of 0.655² + 1.345² + (0.655 x 2.69)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.69 = 2 x 1.345. Using this fact, we can rewrite the given expression as 0.655² + 1.345² + (2 x 0.655 x 1.345). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.655² + 1.345² + (0.655 x 2.69) = 0.655² + 1.345² + (2 x 0.655 x 1.345) = (0.655 + 1.345)² = 2² = 4. ---335. What is the value of [231.3² - 120.31² - 110.99²] / [240.62]? -Solution: First, we look at the three numbers under the square signs in the numerator: 231.3, 120.31, and 110.99. We note that 231.3 = 120.31 + 110.99. We also note, in the denominator, that 240.62 = 2 x 120.31. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 120.31, and b = 110.99, to get [(120.31 + 110.99)² - 120.31² - 110.99²] / 240.62 = [231.3² - 120.31² 110.99²] / 240.62 = 110.99. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 110.99. ---336. What is the square root of [81² + 69.3² + 11226.6]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set
a = 81, and b = 69.3, we see that 2ab = 11226.6, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [81² + 69.3² + 11226.6] = [81 + 69.3]². Therefore, the square root of the expression given is equal to [81 + 69.3] = 150.3. ---337. Please find the value of 6002², by using algebraic identities to simplify computation. -Solution: Since 6002 is only a little higher than 6000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 2. After the substitution, we get 6002² = (6000 + 2)² = 6000² + 2² + (2 x 6000 x 2) = 36000000 + 4 + 24000 = 36024004. ---338. What is the value of 0.551² + 9.449² + (0.551 x 18.898)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 18.898 = 2 x 9.449. Using this fact, we can rewrite the given expression as 0.551² + 9.449² + (2 x 0.551 x 9.449). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.551² + 9.449² + (0.551 x 18.898) = 0.551² + 9.449² + (2 x 0.551 x 9.449) = (0.551 + 9.449)² = 10² = 100. ---339. What is the value of [164.75² - 94.35² - 70.4²] / [188.7]? -Solution: First, we look at the three numbers under the square signs in the numerator: 164.75, 94.35, and 70.4. We note that 164.75 = 94.35 + 70.4. We also note, in the denominator, that 188.7 = 2 x 94.35. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 94.35, and b = 70.4, to get [(94.35 + 70.4)² - 94.35² - 70.4²] / 188.7 = [164.75² - 94.35² - 70.4²] / 188.7 = 70.4.
The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 70.4. ---340. What is the square root of [48.2² + 49.6² + 4781.44]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 48.2, and b = 49.6, we see that 2ab = 4781.44, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [48.2² + 49.6² + 4781.44] = [48.2 + 49.6]². Therefore, the square root of the expression given is equal to [48.2 + 49.6] = 97.8. ---341. Please find the value of 702², by using algebraic identities to simplify computation. -Solution: Since 702 is only a little higher than 700, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 2. After the substitution, we get 702² = (700 + 2)² = 700² + 2² + (2 x 700 x 2) = 490000 + 4 + 2800 = 492804. ---342. What is the value of 0.693² + 5.307² + (0.693 x 10.614)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 10.614 = 2 x 5.307. Using this fact, we can rewrite the given expression as 0.693² + 5.307² + (2 x 0.693 x 5.307). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.693² + 5.307² + (0.693 x 10.614) = 0.693² + 5.307² + (2 x 0.693 x 5.307) = (0.693 + 5.307)² = 6² = 36. ---343. What is the value of [174.81² - 51.77² - 123.04²] / [103.54]? -Solution: First, we look at the three numbers under the square signs in the
numerator: 174.81, 51.77, and 123.04. We note that 174.81 = 51.77 + 123.04. We also note, in the denominator, that 103.54 = 2 x 51.77. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 51.77, and b = 123.04, to get [(51.77 + 123.04)² - 51.77² - 123.04²] / 103.54 = [174.81² - 51.77² - 123.04²] / 103.54 = 123.04. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 123.04. ---344. What is the square root of [10.8² + 83.8² + 1810.08]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 10.8, and b = 83.8, we see that 2ab = 1810.08, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [10.8² + 83.8² + 1810.08] = [10.8 + 83.8]². Therefore, the square root of the expression given is equal to [10.8 + 83.8] = 94.6. ---345. Please find the value of 80003², by using algebraic identities to simplify computation. -Solution: Since 80003 is only a little higher than 80000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 3. After the substitution, we get 80003² = (80000 + 3)² = 80000² + 3² + (2 x 80000 x 3) = 6400000000 + 9 + 480000 = 6400480009. ---346. What is the value of 0.623² + 0.377² + (0.623 x 0.754)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.754 = 2 x 0.377. Using this fact, we can rewrite the given expression as 0.623² + 0.377² + (2 x 0.623 x 0.377).
Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.623² + 0.377² + (0.623 x 0.754) = 0.623² + 0.377² + (2 x 0.623 x 0.377) = (0.623 + 0.377)² = 1² = 1. ---347. What is the value of [202.26² - 81.2² - 121.06²] / [162.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 202.26, 81.2, and 121.06. We note that 202.26 = 81.2 + 121.06. We also note, in the denominator, that 162.4 = 2 x 81.2. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 81.2, and b = 121.06, to get [(81.2 + 121.06)² - 81.2² - 121.06²] / 162.4 = [202.26² - 81.2² - 121.06²] / 162.4 = 121.06. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 121.06. ---348. What is the square root of [32.2² + 82.1² + 5287.24]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 32.2, and b = 82.1, we see that 2ab = 5287.24, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [32.2² + 82.1² + 5287.24] = [32.2 + 82.1]². Therefore, the square root of the expression given is equal to [32.2 + 82.1] = 114.3. ---349. Please find the value of 50001², by using algebraic identities to simplify computation. -Solution: Since 50001 is only a little higher than 50000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 1. After the substitution, we get 50001² = (50000 + 1)² = 50000² + 1² + (2 x 50000 x 1) = 2500000000 + 1 + 100000 = 2500100001.
---350. What is the value of 0.497² + 3.503² + (0.497 x 7.006)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 7.006 = 2 x 3.503. Using this fact, we can rewrite the given expression as 0.497² + 3.503² + (2 x 0.497 x 3.503). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.497² + 3.503² + (0.497 x 7.006) = 0.497² + 3.503² + (2 x 0.497 x 3.503) = (0.497 + 3.503)² = 4² = 16. ---351. What is the value of [161.96² - 96.53² - 65.43²] / [193.06]? -Solution: First, we look at the three numbers under the square signs in the numerator: 161.96, 96.53, and 65.43. We note that 161.96 = 96.53 + 65.43. We also note, in the denominator, that 193.06 = 2 x 96.53. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 96.53, and b = 65.43, to get [(96.53 + 65.43)² - 96.53² - 65.43²] / 193.06 = [161.96² - 96.53² - 65.43²] / 193.06 = 65.43. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 65.43. ---352. What is the square root of [42.9² + 57.4² + 4924.92]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 42.9, and b = 57.4, we see that 2ab = 4924.92, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [42.9² + 57.4² + 4924.92] = [42.9 + 57.4]². Therefore, the square root of the expression given is equal to [42.9 + 57.4] = 100.3. ---353. Please find the value of 501², by using algebraic identities to simplify
computation. -Solution: Since 501 is only a little higher than 500, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 500 and b = 1. After the substitution, we get 501² = (500 + 1)² = 500² + 1² + (2 x 500 x 1) = 250000 + 1 + 1000 = 251001. ---354. What is the value of 0.583² + 8.417² + (0.583 x 16.834)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 16.834 = 2 x 8.417. Using this fact, we can rewrite the given expression as 0.583² + 8.417² + (2 x 0.583 x 8.417). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.583² + 8.417² + (0.583 x 16.834) = 0.583² + 8.417² + (2 x 0.583 x 8.417) = (0.583 + 8.417)² = 9² = 81. ---355. What is the value of [220.37² - 113.66² - 106.71²] / [227.32]? -Solution: First, we look at the three numbers under the square signs in the numerator: 220.37, 113.66, and 106.71. We note that 220.37 = 113.66 + 106.71. We also note, in the denominator, that 227.32 = 2 x 113.66. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 113.66, and b = 106.71, to get [(113.66 + 106.71)² - 113.66² - 106.71²] / 227.32 = [220.37² - 113.66² 106.71²] / 227.32 = 106.71. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 106.71. ---356. What is the square root of [47.3² + 69.6² + 6584.16]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set
a = 47.3, and b = 69.6, we see that 2ab = 6584.16, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [47.3² + 69.6² + 6584.16] = [47.3 + 69.6]². Therefore, the square root of the expression given is equal to [47.3 + 69.6] = 116.9. ---357. Please find the value of 202², by using algebraic identities to simplify computation. -Solution: Since 202 is only a little higher than 200, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 200 and b = 2. After the substitution, we get 202² = (200 + 2)² = 200² + 2² + (2 x 200 x 2) = 40000 + 4 + 800 = 40804. ---358. What is the value of 0.464² + 7.536² + (0.464 x 15.072)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 15.072 = 2 x 7.536. Using this fact, we can rewrite the given expression as 0.464² + 7.536² + (2 x 0.464 x 7.536). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.464² + 7.536² + (0.464 x 15.072) = 0.464² + 7.536² + (2 x 0.464 x 7.536) = (0.464 + 7.536)² = 8² = 64. ---359. What is the value of [174.22² - 81.38² - 92.84²] / [162.76]? -Solution: First, we look at the three numbers under the square signs in the numerator: 174.22, 81.38, and 92.84. We note that 174.22 = 81.38 + 92.84. We also note, in the denominator, that 162.76 = 2 x 81.38. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 81.38, and b = 92.84, to get [(81.38 + 92.84)² - 81.38² - 92.84²] / 162.76 = [174.22² - 81.38² - 92.84²] / 162.76 =
92.84. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 92.84. ---360. What is the square root of [71.8² + 78.9² + 11330.04]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 71.8, and b = 78.9, we see that 2ab = 11330.04, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [71.8² + 78.9² + 11330.04] = [71.8 + 78.9]². Therefore, the square root of the expression given is equal to [71.8 + 78.9] = 150.7. ---361. Please find the value of 4003², by using algebraic identities to simplify computation. -Solution: Since 4003 is only a little higher than 4000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 4000 and b = 3. After the substitution, we get 4003² = (4000 + 3)² = 4000² + 3² + (2 x 4000 x 3) = 16000000 + 9 + 24000 = 16024009. ---362. What is the value of 0.445² + 0.555² + (0.445 x 1.11)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.11 = 2 x 0.555. Using this fact, we can rewrite the given expression as 0.445² + 0.555² + (2 x 0.445 x 0.555). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.445² + 0.555² + (0.445 x 1.11) = 0.445² + 0.555² + (2 x 0.445 x 0.555) = (0.445 + 0.555)² = 1² = 1. ---363. What is the value of [219.49² - 100.91² - 118.58²] / [201.82]? -Solution: First, we look at the three numbers under the square signs in the numerator: 219.49, 100.91, and 118.58. We note that 219.49 = 100.91 +
118.58. We also note, in the denominator, that 201.82 = 2 x 100.91. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 100.91, and b = 118.58, to get [(100.91 + 118.58)² - 100.91² - 118.58²] / 201.82 = [219.49² - 100.91² 118.58²] / 201.82 = 118.58. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 118.58. ---364. What is the square root of [63.8² + 62.4² + 7962.24]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 63.8, and b = 62.4, we see that 2ab = 7962.24, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [63.8² + 62.4² + 7962.24] = [63.8 + 62.4]². Therefore, the square root of the expression given is equal to [63.8 + 62.4] = 126.2. ---365. Please find the value of 7002², by using algebraic identities to simplify computation. -Solution: Since 7002 is only a little higher than 7000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 2. After the substitution, we get 7002² = (7000 + 2)² = 7000² + 2² + (2 x 7000 x 2) = 49000000 + 4 + 28000 = 49028004. ---366. What is the value of 0.541² + 3.459² + (0.541 x 6.918)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.918 = 2 x 3.459. Using this fact, we can rewrite the given expression as 0.541² + 3.459² + (2 x 0.541 x 3.459).
Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.541² + 3.459² + (0.541 x 6.918) = 0.541² + 3.459² + (2 x 0.541 x 3.459) = (0.541 + 3.459)² = 4² = 16. ---367. What is the value of [168.91² - 72.87² - 96.04²] / [145.74]? -Solution: First, we look at the three numbers under the square signs in the numerator: 168.91, 72.87, and 96.04. We note that 168.91 = 72.87 + 96.04. We also note, in the denominator, that 145.74 = 2 x 72.87. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 72.87, and b = 96.04, to get [(72.87 + 96.04)² - 72.87² - 96.04²] / 145.74 = [168.91² - 72.87² - 96.04²] / 145.74 = 96.04. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 96.04. ---368. What is the square root of [45.3² + 29.6² + 2681.76]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 45.3, and b = 29.6, we see that 2ab = 2681.76, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [45.3² + 29.6² + 2681.76] = [45.3 + 29.6]². Therefore, the square root of the expression given is equal to [45.3 + 29.6] = 74.9. ---369. Please find the value of 701², by using algebraic identities to simplify computation. -Solution: Since 701 is only a little higher than 700, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 1. After the substitution, we get 701² = (700 + 1)² = 700² + 1² + (2 x
700 x 1) = 490000 + 1 + 1400 = 491401. ---370. What is the value of 0.511² + 3.489² + (0.511 x 6.978)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.978 = 2 x 3.489. Using this fact, we can rewrite the given expression as 0.511² + 3.489² + (2 x 0.511 x 3.489). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.511² + 3.489² + (0.511 x 6.978) = 0.511² + 3.489² + (2 x 0.511 x 3.489) = (0.511 + 3.489)² = 4² = 16. ---371. What is the value of [144.77² - 71.39² - 73.38²] / [142.78]? -Solution: First, we look at the three numbers under the square signs in the numerator: 144.77, 71.39, and 73.38. We note that 144.77 = 71.39 + 73.38. We also note, in the denominator, that 142.78 = 2 x 71.39. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 71.39, and b = 73.38, to get [(71.39 + 73.38)² - 71.39² - 73.38²] / 142.78 = [144.77² - 71.39² - 73.38²] / 142.78 = 73.38. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 73.38. ---372. What is the square root of [51.3² + 51.2² + 5253.12]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 51.3, and b = 51.2, we see that 2ab = 5253.12, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [51.3² + 51.2² + 5253.12] = [51.3 + 51.2]². Therefore, the square root of the expression given is equal to [51.3 + 51.2] = 102.5. ----
373. Please find the value of 501², by using algebraic identities to simplify computation. -Solution: Since 501 is only a little higher than 500, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 500 and b = 1. After the substitution, we get 501² = (500 + 1)² = 500² + 1² + (2 x 500 x 1) = 250000 + 1 + 1000 = 251001. ---374. What is the value of 0.531² + 7.469² + (0.531 x 14.938)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 14.938 = 2 x 7.469. Using this fact, we can rewrite the given expression as 0.531² + 7.469² + (2 x 0.531 x 7.469). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.531² + 7.469² + (0.531 x 14.938) = 0.531² + 7.469² + (2 x 0.531 x 7.469) = (0.531 + 7.469)² = 8² = 64. ---375. What is the value of [132.92² - 77.16² - 55.76²] / [154.32]? -Solution: First, we look at the three numbers under the square signs in the numerator: 132.92, 77.16, and 55.76. We note that 132.92 = 77.16 + 55.76. We also note, in the denominator, that 154.32 = 2 x 77.16. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 77.16, and b = 55.76, to get [(77.16 + 55.76)² - 77.16² - 55.76²] / 154.32 = [132.92² - 77.16² - 55.76²] / 154.32 = 55.76. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 55.76. ---376. What is the square root of [35.7² + 62.2² + 4441.08]? -Solution: By observation, we can deduce that the given expression looks a
little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 35.7, and b = 62.2, we see that 2ab = 4441.08, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [35.7² + 62.2² + 4441.08] = [35.7 + 62.2]². Therefore, the square root of the expression given is equal to [35.7 + 62.2] = 97.9. ---377. Please find the value of 3002², by using algebraic identities to simplify computation. -Solution: Since 3002 is only a little higher than 3000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 3000 and b = 2. After the substitution, we get 3002² = (3000 + 2)² = 3000² + 2² + (2 x 3000 x 2) = 9000000 + 4 + 12000 = 9012004. ---378. What is the value of 0.614² + 4.386² + (0.614 x 8.772)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.772 = 2 x 4.386. Using this fact, we can rewrite the given expression as 0.614² + 4.386² + (2 x 0.614 x 4.386). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.614² + 4.386² + (0.614 x 8.772) = 0.614² + 4.386² + (2 x 0.614 x 4.386) = (0.614 + 4.386)² = 5² = 25. ---379. What is the value of [257.17² - 128.07² - 129.1²] / [256.14]? -Solution: First, we look at the three numbers under the square signs in the numerator: 257.17, 128.07, and 129.1. We note that 257.17 = 128.07 + 129.1. We also note, in the denominator, that 256.14 = 2 x 128.07. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 128.07, and b = 129.1, to get [(128.07 + 129.1)² - 128.07² - 129.1²] / 256.14 = [257.17² - 128.07² - 129.1²] / 256.14
= 129.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 129.1. ---380. What is the square root of [43.4² + 32² + 2777.6]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 43.4, and b = 32, we see that 2ab = 2777.6, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [43.4² + 32² + 2777.6] = [43.4 + 32]². Therefore, the square root of the expression given is equal to [43.4 + 32] = 75.4. ---381. Please find the value of 50003², by using algebraic identities to simplify computation. -Solution: Since 50003 is only a little higher than 50000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 3. After the substitution, we get 50003² = (50000 + 3)² = 50000² + 3² + (2 x 50000 x 3) = 2500000000 + 9 + 300000 = 2500300009. ---382. What is the value of 0.565² + 4.435² + (0.565 x 8.87)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.87 = 2 x 4.435. Using this fact, we can rewrite the given expression as 0.565² + 4.435² + (2 x 0.565 x 4.435). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.565² + 4.435² + (0.565 x 8.87) = 0.565² + 4.435² + (2 x 0.565 x 4.435) = (0.565 + 4.435)² = 5² = 25. ---383. What is the value of [175.64² - 113.46² - 62.18²] / [226.92]? -Solution: First, we look at the three numbers under the square signs in the
numerator: 175.64, 113.46, and 62.18. We note that 175.64 = 113.46 + 62.18. We also note, in the denominator, that 226.92 = 2 x 113.46. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 113.46, and b = 62.18, to get [(113.46 + 62.18)² - 113.46² - 62.18²] / 226.92 = [175.64² - 113.46² - 62.18²] / 226.92 = 62.18. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 62.18. ---384. What is the square root of [60.5² + 70.6² + 8542.6]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 60.5, and b = 70.6, we see that 2ab = 8542.6, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [60.5² + 70.6² + 8542.6] = [60.5 + 70.6]². Therefore, the square root of the expression given is equal to [60.5 + 70.6] = 131.1. ---385. Please find the value of 60003², by using algebraic identities to simplify computation. -Solution: Since 60003 is only a little higher than 60000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 60000 and b = 3. After the substitution, we get 60003² = (60000 + 3)² = 60000² + 3² + (2 x 60000 x 3) = 3600000000 + 9 + 360000 = 3600360009. ---386. What is the value of 0.667² + 4.333² + (0.667 x 8.666)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 8.666 = 2 x 4.333. Using this fact, we can rewrite the given expression as 0.667² + 4.333² + (2 x 0.667 x 4.333). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² +
2ab), we get 0.667² + 4.333² + (0.667 x 8.666) = 0.667² + 4.333² + (2 x 0.667 x 4.333) = (0.667 + 4.333)² = 5² = 25. ---387. What is the value of [151.78² - 50.23² - 101.55²] / [100.46]? -Solution: First, we look at the three numbers under the square signs in the numerator: 151.78, 50.23, and 101.55. We note that 151.78 = 50.23 + 101.55. We also note, in the denominator, that 100.46 = 2 x 50.23. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 50.23, and b = 101.55, to get [(50.23 + 101.55)² - 50.23² - 101.55²] / 100.46 = [151.78² - 50.23² - 101.55²] / 100.46 = 101.55. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 101.55. ---388. What is the square root of [17² + 25.8² + 877.2]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 17, and b = 25.8, we see that 2ab = 877.2, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [17² + 25.8² + 877.2] = [17 + 25.8]². Therefore, the square root of the expression given is equal to [17 + 25.8] = 42.8. ---389. Please find the value of 2002², by using algebraic identities to simplify computation. -Solution: Since 2002 is only a little higher than 2000, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 2. After the substitution, we get 2002² = (2000 + 2)² = 2000² + 2² + (2 x 2000 x 2) = 4000000 + 4 + 8000 = 4008004.
---390. What is the value of 0.45² + 5.55² + (0.45 x 11.1)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 11.1 = 2 x 5.55. Using this fact, we can rewrite the given expression as 0.45² + 5.55² + (2 x 0.45 x 5.55). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.45² + 5.55² + (0.45 x 11.1) = 0.45² + 5.55² + (2 x 0.45 x 5.55) = (0.45 + 5.55)² = 6² = 36. ---391. What is the value of [220.15² - 126.44² - 93.71²] / [252.88]? -Solution: First, we look at the three numbers under the square signs in the numerator: 220.15, 126.44, and 93.71. We note that 220.15 = 126.44 + 93.71. We also note, in the denominator, that 252.88 = 2 x 126.44. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 126.44, and b = 93.71, to get [(126.44 + 93.71)² - 126.44² - 93.71²] / 252.88 = [220.15² - 126.44² - 93.71²] / 252.88 = 93.71. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 93.71. ---392. What is the square root of [87.1² + 39.3² + 6846.06]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 87.1, and b = 39.3, we see that 2ab = 6846.06, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [87.1² + 39.3² + 6846.06] = [87.1 + 39.3]². Therefore, the square root of the expression given is equal to [87.1 + 39.3] = 126.4. ---393. Please find the value of 502², by using algebraic identities to simplify
computation. -Solution: Since 502 is only a little higher than 500, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 500 and b = 2. After the substitution, we get 502² = (500 + 2)² = 500² + 2² + (2 x 500 x 2) = 250000 + 4 + 2000 = 252004. ---394. What is the value of 0.41² + 1.59² + (0.41 x 3.18)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.18 = 2 x 1.59. Using this fact, we can rewrite the given expression as 0.41² + 1.59² + (2 x 0.41 x 1.59). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.41² + 1.59² + (0.41 x 3.18) = 0.41² + 1.59² + (2 x 0.41 x 1.59) = (0.41 + 1.59)² = 2² = 4. ---395. What is the value of [181.76² - 128.93² - 52.83²] / [257.86]? -Solution: First, we look at the three numbers under the square signs in the numerator: 181.76, 128.93, and 52.83. We note that 181.76 = 128.93 + 52.83. We also note, in the denominator, that 257.86 = 2 x 128.93. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 128.93, and b = 52.83, to get [(128.93 + 52.83)² - 128.93² - 52.83²] / 257.86 = [181.76² - 128.93² - 52.83²] / 257.86 = 52.83. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 52.83. ---396. What is the square root of [35.8² + 52.7² + 3773.32]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set
a = 35.8, and b = 52.7, we see that 2ab = 3773.32, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [35.8² + 52.7² + 3773.32] = [35.8 + 52.7]². Therefore, the square root of the expression given is equal to [35.8 + 52.7] = 88.5. ---397. Please find the value of 301², by using algebraic identities to simplify computation. -Solution: Since 301 is only a little higher than 300, we can deduce that the identity (a + b)² = (a² + b² + 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 1. After the substitution, we get 301² = (300 + 1)² = 300² + 1² + (2 x 300 x 1) = 90000 + 1 + 600 = 90601. ---398. What is the value of 0.486² + 8.514² + (0.486 x 17.028)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 17.028 = 2 x 8.514. Using this fact, we can rewrite the given expression as 0.486² + 8.514² + (2 x 0.486 x 8.514). Comparing this with the Right Hand Side of the identity (a + b)² = (a² + b² + 2ab), we get 0.486² + 8.514² + (0.486 x 17.028) = 0.486² + 8.514² + (2 x 0.486 x 8.514) = (0.486 + 8.514)² = 9² = 81. ---399. What is the value of [145.39² - 67.76² - 77.63²] / [135.52]? -Solution: First, we look at the three numbers under the square signs in the numerator: 145.39, 67.76, and 77.63. We note that 145.39 = 67.76 + 77.63. We also note, in the denominator, that 135.52 = 2 x 67.76. Hence, we can guess that the algebraic identity (a + b)² = (a² + b² + 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [(a + b)² - a² - b²] / 2a = b. In this rearranged equation, we put a = 67.76, and b = 77.63, to get [(67.76 + 77.63)² - 67.76² - 77.63²] / 135.52 = [145.39² - 67.76² - 77.63²] / 135.52 = 77.63. The Left Hand Side of this equation is exactly the expression in our
question, so the value of that expression is equal to 77.63. ---400. What is the square root of [27.6² + 15.4² + 850.08]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a + b)² = (a² + b² + 2ab). Indeed, if we set a = 27.6, and b = 15.4, we see that 2ab = 850.08, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [27.6² + 15.4² + 850.08] = [27.6 + 15.4]². Therefore, the square root of the expression given is equal to [27.6 + 15.4] = 43. ----
401. Please find the value of 1997², by using algebraic identities to simplify computation. -Solution: Since 1997 is only a little lower than 2000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 3. After the substitution, we get 1997² = (2000 - 3)² = 2000² + 3² - (2 x 2000 x 3) = 4000000 + 9 - 12000 = 3988009. ---402. What is the value of 3.251² + 2.051² - (3.251 x 4.102)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.102 = 2 x 2.051. Using this fact, we can rewrite the given expression as 3.251² + 2.051² - (2 x 3.251 x 2.051). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.251² + 2.051² - (3.251 x 4.102) = 3.251² + 2.051² - (2 x 3.251 x 2.051) = (3.251 - 2.051)² = 1.2² = 1.44. ---403. What is the value of [446.4² + 207.5² - 238.9²] / [892.8]? -Solution: First, we look at the three numbers under the square signs in the
numerator: 446.4, 207.5, and 238.9. We note that 238.9 = 446.4 - 207.5. We also note, in the denominator, that 892.8 = 2 x 446.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 446.4, and b = 207.5, to get [(446.4² + 207.5² - (446.4 - 207.5)²] / 892.8 = [446.4² + 207.5² - 238.9²] / 892.8 = 207.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 207.5. ---404. What is the positive square root of [2638² + 80² - 422080]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2638, and b = 80, we see that 2ab = 422080, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2638² + 80² - 422080] = [2638 - 80]². Therefore, the positive square root of the expression given is equal to [2638 80] = 2558. ---405. Please find the value of 197², by using algebraic identities to simplify computation. -Solution: Since 197 is only a little lower than 200, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 200 and b = 3. After the substitution, we get 197² = (200 - 3)² = 200² + 3² - (2 x 200 x 3) = 40000 + 9 - 1200 = 38809. ---406. What is the value of 2.187² + 1.187² - (2.187 x 2.374)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.374 = 2 x 1.187. Using this fact, we can rewrite the given expression as 2.187² + 1.187² - (2 x 2.187 x 1.187). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² -
2ab), we get 2.187² + 1.187² - (2.187 x 2.374) = 2.187² + 1.187² - (2 x 2.187 x 1.187) = (2.187 - 1.187)² = 1² = 1. ---407. What is the value of [519.1² + 225.9² - 293.2²] / [1038.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 519.1, 225.9, and 293.2. We note that 293.2 = 519.1 - 225.9. We also note, in the denominator, that 1038.2 = 2 x 519.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 519.1, and b = 225.9, to get [(519.1² + 225.9² - (519.1 - 225.9)²] / 1038.2 = [519.1² + 225.9² - 293.2²] / 1038.2 = 225.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 225.9. ---408. What is the positive square root of [2961² + 40² - 236880]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2961, and b = 40, we see that 2ab = 236880, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2961² + 40² - 236880] = [2961 - 40]². Therefore, the positive square root of the expression given is equal to [2961 40] = 2921. ---409. Please find the value of 19999², by using algebraic identities to simplify computation. -Solution: Since 19999 is only a little lower than 20000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 20000 and b = 1. After the substitution, we get 19999² = (20000 - 1)² = 20000² + 1² - (2 x 20000 x 1) = 400000000 + 1 - 40000 = 399960001. ----
410. What is the value of 3.527² + 2.127² - (3.527 x 4.254)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.254 = 2 x 2.127. Using this fact, we can rewrite the given expression as 3.527² + 2.127² - (2 x 3.527 x 2.127). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.527² + 2.127² - (3.527 x 4.254) = 3.527² + 2.127² - (2 x 3.527 x 2.127) = (3.527 - 2.127)² = 1.4² = 1.96. ---411. What is the value of [420.1² + 295² - 125.1²] / [840.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 420.1, 295, and 125.1. We note that 125.1 = 420.1 - 295. We also note, in the denominator, that 840.2 = 2 x 420.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 420.1, and b = 295, to get [(420.1² + 295² - (420.1 - 295)²] / 840.2 = [420.1² + 295² - 125.1²] / 840.2 = 295. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 295. ---412. What is the positive square root of [3202² + 70² - 448280]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3202, and b = 70, we see that 2ab = 448280, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3202² + 70² - 448280] = [3202 - 70]². Therefore, the positive square root of the expression given is equal to [3202 70] = 3132. ---413. Please find the value of 69999², by using algebraic identities to simplify computation.
-Solution: Since 69999 is only a little lower than 70000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 70000 and b = 1. After the substitution, we get 69999² = (70000 - 1)² = 70000² + 1² - (2 x 70000 x 1) = 4900000000 + 1 - 140000 = 4899860001. ---414. What is the value of 3.362² + 2.562² - (3.362 x 5.124)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.124 = 2 x 2.562. Using this fact, we can rewrite the given expression as 3.362² + 2.562² - (2 x 3.362 x 2.562). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.362² + 2.562² - (3.362 x 5.124) = 3.362² + 2.562² - (2 x 3.362 x 2.562) = (3.362 - 2.562)² = 0.8² = 0.64. ---415. What is the value of [554.4² + 201.6² - 352.8²] / [1108.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 554.4, 201.6, and 352.8. We note that 352.8 = 554.4 - 201.6. We also note, in the denominator, that 1108.8 = 2 x 554.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 554.4, and b = 201.6, to get [(554.4² + 201.6² - (554.4 - 201.6)²] / 1108.8 = [554.4² + 201.6² - 352.8²] / 1108.8 = 201.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 201.6. ---416. What is the positive square root of [3023² + 80² - 483680]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3023, and b = 80, we see that 2ab = 483680, which is exactly equal to the absolute value of the third term in the expression.
Hence, we can say that [3023² + 80² - 483680] = [3023 - 80]². Therefore, the positive square root of the expression given is equal to [3023 80] = 2943. ---417. Please find the value of 3997², by using algebraic identities to simplify computation. -Solution: Since 3997 is only a little lower than 4000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 4000 and b = 3. After the substitution, we get 3997² = (4000 - 3)² = 4000² + 3² - (2 x 4000 x 3) = 16000000 + 9 - 24000 = 15976009. ---418. What is the value of 1.462² + 0.462² - (1.462 x 0.924)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.924 = 2 x 0.462. Using this fact, we can rewrite the given expression as 1.462² + 0.462² - (2 x 1.462 x 0.462). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.462² + 0.462² - (1.462 x 0.924) = 1.462² + 0.462² - (2 x 1.462 x 0.462) = (1.462 - 0.462)² = 1² = 1. ---419. What is the value of [541.7² + 145.6² - 396.1²] / [1083.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 541.7, 145.6, and 396.1. We note that 396.1 = 541.7 - 145.6. We also note, in the denominator, that 1083.4 = 2 x 541.7. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 541.7, and b = 145.6, to get [(541.7² + 145.6² - (541.7 - 145.6)²] / 1083.4 = [541.7² + 145.6² - 396.1²] / 1083.4 = 145.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 145.6.
---420. What is the positive square root of [2403² + 20² - 96120]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2403, and b = 20, we see that 2ab = 96120, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2403² + 20² - 96120] = [2403 - 20]². Therefore, the positive square root of the expression given is equal to [2403 20] = 2383. ---421. Please find the value of 49997², by using algebraic identities to simplify computation. -Solution: Since 49997 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 3. After the substitution, we get 49997² = (50000 - 3)² = 50000² + 3² - (2 x 50000 x 3) = 2500000000 + 9 - 300000 = 2499700009. ---422. What is the value of 1.924² + 0.724² - (1.924 x 1.448)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.448 = 2 x 0.724. Using this fact, we can rewrite the given expression as 1.924² + 0.724² - (2 x 1.924 x 0.724). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.924² + 0.724² - (1.924 x 1.448) = 1.924² + 0.724² - (2 x 1.924 x 0.724) = (1.924 - 0.724)² = 1.2² = 1.44. ---423. What is the value of [506.3² + 172.3² - 334²] / [1012.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 506.3, 172.3, and 334. We note that 334 = 506.3 - 172.3. We also note, in the denominator, that 1012.6 = 2 x 506.3. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to
be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 506.3, and b = 172.3, to get [(506.3² + 172.3² - (506.3 - 172.3)²] / 1012.6 = [506.3² + 172.3² - 334²] / 1012.6 = 172.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 172.3. ---424. What is the positive square root of [3957² + 50² - 395700]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3957, and b = 50, we see that 2ab = 395700, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3957² + 50² - 395700] = [3957 - 50]². Therefore, the positive square root of the expression given is equal to [3957 50] = 3907. ---425. Please find the value of 89999², by using algebraic identities to simplify computation. -Solution: Since 89999 is only a little lower than 90000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 90000 and b = 1. After the substitution, we get 89999² = (90000 - 1)² = 90000² + 1² - (2 x 90000 x 1) = 8100000000 + 1 - 180000 = 8099820001. ---426. What is the value of 3.413² + 3.013² - (3.413 x 6.026)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.026 = 2 x 3.013. Using this fact, we can rewrite the given expression as 3.413² + 3.013² - (2 x 3.413 x 3.013). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.413² + 3.013² - (3.413 x 6.026) = 3.413² + 3.013² - (2 x 3.413 x 3.013) = (3.413 - 3.013)² = 0.4² = 0.16.
---427. What is the value of [531.2² + 179.9² - 351.3²] / [1062.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 531.2, 179.9, and 351.3. We note that 351.3 = 531.2 - 179.9. We also note, in the denominator, that 1062.4 = 2 x 531.2. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 531.2, and b = 179.9, to get [(531.2² + 179.9² - (531.2 - 179.9)²] / 1062.4 = [531.2² + 179.9² - 351.3²] / 1062.4 = 179.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 179.9. ---428. What is the positive square root of [3927² + 30² - 235620]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3927, and b = 30, we see that 2ab = 235620, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3927² + 30² - 235620] = [3927 - 30]². Therefore, the positive square root of the expression given is equal to [3927 30] = 3897. ---429. Please find the value of 199², by using algebraic identities to simplify computation. -Solution: Since 199 is only a little lower than 200, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 200 and b = 1. After the substitution, we get 199² = (200 - 1)² = 200² + 1² - (2 x 200 x 1) = 40000 + 1 - 400 = 39601. ---430. What is the value of 2.435² + 1.235² - (2.435 x 2.47)? Please use
algebraic identities to make the computation simple. -Solution: First, we note that 2.47 = 2 x 1.235. Using this fact, we can rewrite the given expression as 2.435² + 1.235² - (2 x 2.435 x 1.235). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.435² + 1.235² - (2.435 x 2.47) = 2.435² + 1.235² - (2 x 2.435 x 1.235) = (2.435 - 1.235)² = 1.2² = 1.44. ---431. What is the value of [400.5² + 142.4² - 258.1²] / [801]? -Solution: First, we look at the three numbers under the square signs in the numerator: 400.5, 142.4, and 258.1. We note that 258.1 = 400.5 - 142.4. We also note, in the denominator, that 801 = 2 x 400.5. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 400.5, and b = 142.4, to get [(400.5² + 142.4² - (400.5 - 142.4)²] / 801 = [400.5² + 142.4² - 258.1²] / 801 = 142.4. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 142.4. ---432. What is the positive square root of [2632² + 40² - 210560]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2632, and b = 40, we see that 2ab = 210560, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2632² + 40² - 210560] = [2632 - 40]². Therefore, the positive square root of the expression given is equal to [2632 40] = 2592. ---433. Please find the value of 497², by using algebraic identities to simplify computation. -Solution: Since 497 is only a little lower than 500, we can deduce that the
identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 500 and b = 3. After the substitution, we get 497² = (500 - 3)² = 500² + 3² - (2 x 500 x 3) = 250000 + 9 - 3000 = 247009. ---434. What is the value of 2.988² + 2.588² - (2.988 x 5.176)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.176 = 2 x 2.588. Using this fact, we can rewrite the given expression as 2.988² + 2.588² - (2 x 2.988 x 2.588). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.988² + 2.588² - (2.988 x 5.176) = 2.988² + 2.588² - (2 x 2.988 x 2.588) = (2.988 - 2.588)² = 0.4² = 0.16. ---435. What is the value of [562.5² + 144.8² - 417.7²] / [1125]? -Solution: First, we look at the three numbers under the square signs in the numerator: 562.5, 144.8, and 417.7. We note that 417.7 = 562.5 - 144.8. We also note, in the denominator, that 1125 = 2 x 562.5. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 562.5, and b = 144.8, to get [(562.5² + 144.8² - (562.5 - 144.8)²] / 1125 = [562.5² + 144.8² - 417.7²] / 1125 = 144.8. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 144.8. ---436. What is the positive square root of [3812² + 80² - 609920]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3812, and b = 80, we see that 2ab = 609920, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3812² + 80² - 609920] = [3812 - 80]².
Therefore, the positive square root of the expression given is equal to [3812 80] = 3732. ---437. Please find the value of 197², by using algebraic identities to simplify computation. -Solution: Since 197 is only a little lower than 200, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 200 and b = 3. After the substitution, we get 197² = (200 - 3)² = 200² + 3² - (2 x 200 x 3) = 40000 + 9 - 1200 = 38809. ---438. What is the value of 1.596² + 0.396² - (1.596 x 0.792)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.792 = 2 x 0.396. Using this fact, we can rewrite the given expression as 1.596² + 0.396² - (2 x 1.596 x 0.396). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.596² + 0.396² - (1.596 x 0.792) = 1.596² + 0.396² - (2 x 1.596 x 0.396) = (1.596 - 0.396)² = 1.2² = 1.44. ---439. What is the value of [465.9² + 199.3² - 266.6²] / [931.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 465.9, 199.3, and 266.6. We note that 266.6 = 465.9 - 199.3. We also note, in the denominator, that 931.8 = 2 x 465.9. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 465.9, and b = 199.3, to get [(465.9² + 199.3² - (465.9 - 199.3)²] / 931.8 = [465.9² + 199.3² - 266.6²] / 931.8 = 199.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 199.3. ----
440. What is the positive square root of [3203² + 30² - 192180]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3203, and b = 30, we see that 2ab = 192180, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3203² + 30² - 192180] = [3203 - 30]². Therefore, the positive square root of the expression given is equal to [3203 30] = 3173. ---441. Please find the value of 49998², by using algebraic identities to simplify computation. -Solution: Since 49998 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 2. After the substitution, we get 49998² = (50000 - 2)² = 50000² + 2² - (2 x 50000 x 2) = 2500000000 + 4 - 200000 = 2499800004. ---442. What is the value of 3.61² + 3.01² - (3.61 x 6.02)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.02 = 2 x 3.01. Using this fact, we can rewrite the given expression as 3.61² + 3.01² - (2 x 3.61 x 3.01). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.61² + 3.01² - (3.61 x 6.02) = 3.61² + 3.01² - (2 x 3.61 x 3.01) = (3.61 - 3.01)² = 0.6² = 0.36. ---443. What is the value of [430² + 106.6² - 323.4²] / [860]? -Solution: First, we look at the three numbers under the square signs in the numerator: 430, 106.6, and 323.4. We note that 323.4 = 430 - 106.6. We also note, in the denominator, that 860 = 2 x 430. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression.
After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 430, and b = 106.6, to get [(430² + 106.6² - (430 - 106.6)²] / 860 = [430² + 106.6² - 323.4²] / 860 = 106.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 106.6. ---444. What is the positive square root of [2720² + 50² - 272000]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2720, and b = 50, we see that 2ab = 272000, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2720² + 50² - 272000] = [2720 - 50]². Therefore, the positive square root of the expression given is equal to [2720 50] = 2670. ---445. Please find the value of 299², by using algebraic identities to simplify computation. -Solution: Since 299 is only a little lower than 300, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 1. After the substitution, we get 299² = (300 - 1)² = 300² + 1² - (2 x 300 x 1) = 90000 + 1 - 600 = 89401. ---446. What is the value of 1.661² + 0.261² - (1.661 x 0.522)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.522 = 2 x 0.261. Using this fact, we can rewrite the given expression as 1.661² + 0.261² - (2 x 1.661 x 0.261). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.661² + 0.261² - (1.661 x 0.522) = 1.661² + 0.261² - (2 x 1.661 x 0.261) = (1.661 - 0.261)² = 1.4² = 1.96. ---447. What is the value of [588.9² + 185² - 403.9²] / [1177.8]?
-Solution: First, we look at the three numbers under the square signs in the numerator: 588.9, 185, and 403.9. We note that 403.9 = 588.9 - 185. We also note, in the denominator, that 1177.8 = 2 x 588.9. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 588.9, and b = 185, to get [(588.9² + 185² - (588.9 - 185)²] / 1177.8 = [588.9² + 185² - 403.9²] / 1177.8 = 185. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 185. ---448. What is the positive square root of [2343² + 30² - 140580]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2343, and b = 30, we see that 2ab = 140580, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2343² + 30² - 140580] = [2343 - 30]². Therefore, the positive square root of the expression given is equal to [2343 30] = 2313. ---449. Please find the value of 69997², by using algebraic identities to simplify computation. -Solution: Since 69997 is only a little lower than 70000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 70000 and b = 3. After the substitution, we get 69997² = (70000 - 3)² = 70000² + 3² - (2 x 70000 x 3) = 4900000000 + 9 - 420000 = 4899580009. ---450. What is the value of 3.69² + 2.89² - (3.69 x 5.78)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.78 = 2 x 2.89. Using this fact, we can rewrite
the given expression as 3.69² + 2.89² - (2 x 3.69 x 2.89). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.69² + 2.89² - (3.69 x 5.78) = 3.69² + 2.89² - (2 x 3.69 x 2.89) = (3.69 - 2.89)² = 0.8² = 0.640000000000001. ---451. What is the value of [585.1² + 139.4² - 445.7²] / [1170.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 585.1, 139.4, and 445.7. We note that 445.7 = 585.1 - 139.4. We also note, in the denominator, that 1170.2 = 2 x 585.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 585.1, and b = 139.4, to get [(585.1² + 139.4² - (585.1 - 139.4)²] / 1170.2 = [585.1² + 139.4² - 445.7²] / 1170.2 = 139.4. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 139.4. ---452. What is the positive square root of [3035² + 50² - 303500]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3035, and b = 50, we see that 2ab = 303500, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3035² + 50² - 303500] = [3035 - 50]². Therefore, the positive square root of the expression given is equal to [3035 50] = 2985. ---453. Please find the value of 198², by using algebraic identities to simplify computation. -Solution: Since 198 is only a little lower than 200, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 200
and b = 2. After the substitution, we get 198² = (200 - 2)² = 200² + 2² - (2 x 200 x 2) = 40000 + 4 - 800 = 39204. ---454. What is the value of 1.55² + 0.35² - (1.55 x 0.7)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.7 = 2 x 0.35. Using this fact, we can rewrite the given expression as 1.55² + 0.35² - (2 x 1.55 x 0.35). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.55² + 0.35² - (1.55 x 0.7) = 1.55² + 0.35² - (2 x 1.55 x 0.35) = (1.55 - 0.35)² = 1.2² = 1.44. ---455. What is the value of [516.7² + 250.7² - 266.²] / [1033.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 516.7, 250.7, and 266.. We note that 266. = 516.7 - 250.7. We also note, in the denominator, that 1033.4 = 2 x 516.7. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 516.7, and b = 250.7, to get [(516.7² + 250.7² - (516.7 - 250.7)²] / 1033.4 = [516.7² + 250.7² - 266.²] / 1033.4 = 250.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 250.7. ---456. What is the positive square root of [3104² + 30² - 186240]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3104, and b = 30, we see that 2ab = 186240, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3104² + 30² - 186240] = [3104 - 30]². Therefore, the positive square root of the expression given is equal to [3104 30] = 3074.
---457. Please find the value of 89997², by using algebraic identities to simplify computation. -Solution: Since 89997 is only a little lower than 90000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 90000 and b = 3. After the substitution, we get 89997² = (90000 - 3)² = 90000² + 3² - (2 x 90000 x 3) = 8100000000 + 9 - 540000 = 8099460009. ---458. What is the value of 1.765² + 1.565² - (1.765 x 3.13)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.13 = 2 x 1.565. Using this fact, we can rewrite the given expression as 1.765² + 1.565² - (2 x 1.765 x 1.565). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.765² + 1.565² - (1.765 x 3.13) = 1.765² + 1.565² - (2 x 1.765 x 1.565) = (1.765 - 1.565)² = 0.2² = 0.04. ---459. What is the value of [590.1² + 141² - 449.1²] / [1180.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 590.1, 141, and 449.1. We note that 449.1 = 590.1 - 141. We also note, in the denominator, that 1180.2 = 2 x 590.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 590.1, and b = 141, to get [(590.1² + 141² - (590.1 - 141)²] / 1180.2 = [590.1² + 141² - 449.1²] / 1180.2 = 141. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 141. ---460. What is the positive square root of [2219² + 90² - 399420]? --
Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2219, and b = 90, we see that 2ab = 399420, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2219² + 90² - 399420] = [2219 - 90]². Therefore, the positive square root of the expression given is equal to [2219 90] = 2129. ---461. Please find the value of 1997², by using algebraic identities to simplify computation. -Solution: Since 1997 is only a little lower than 2000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 3. After the substitution, we get 1997² = (2000 - 3)² = 2000² + 3² - (2 x 2000 x 3) = 4000000 + 9 - 12000 = 3988009. ---462. What is the value of 1.978² + 0.978² - (1.978 x 1.956)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.956 = 2 x 0.978. Using this fact, we can rewrite the given expression as 1.978² + 0.978² - (2 x 1.978 x 0.978). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.978² + 0.978² - (1.978 x 1.956) = 1.978² + 0.978² - (2 x 1.978 x 0.978) = (1.978 - 0.978)² = 1² = 1. ---463. What is the value of [525.6² + 102.2² - 423.4²] / [1051.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 525.6, 102.2, and 423.4. We note that 423.4 = 525.6 - 102.2. We also note, in the denominator, that 1051.2 = 2 x 525.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b.
In this rearranged equation, we put a = 525.6, and b = 102.2, to get [(525.6² + 102.2² - (525.6 - 102.2)²] / 1051.2 = [525.6² + 102.2² - 423.4²] / 1051.2 = 102.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 102.2. ---464. What is the positive square root of [2989² + 90² - 538020]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2989, and b = 90, we see that 2ab = 538020, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2989² + 90² - 538020] = [2989 - 90]². Therefore, the positive square root of the expression given is equal to [2989 90] = 2899. ---465. Please find the value of 8998², by using algebraic identities to simplify computation. -Solution: Since 8998 is only a little lower than 9000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 9000 and b = 2. After the substitution, we get 8998² = (9000 - 2)² = 9000² + 2² - (2 x 9000 x 2) = 81000000 + 4 - 36000 = 80964004. ---466. What is the value of 3.456² + 2.656² - (3.456 x 5.312)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.312 = 2 x 2.656. Using this fact, we can rewrite the given expression as 3.456² + 2.656² - (2 x 3.456 x 2.656). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.456² + 2.656² - (3.456 x 5.312) = 3.456² + 2.656² - (2 x 3.456 x 2.656) = (3.456 - 2.656)² = 0.8² = 0.640000000000001. ---467. What is the value of [463.9² + 148² - 315.9²] / [927.8]?
-Solution: First, we look at the three numbers under the square signs in the numerator: 463.9, 148, and 315.9. We note that 315.9 = 463.9 - 148. We also note, in the denominator, that 927.8 = 2 x 463.9. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 463.9, and b = 148, to get [(463.9² + 148² - (463.9 - 148)²] / 927.8 = [463.9² + 148² - 315.9²] / 927.8 = 148. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 148. ---468. What is the positive square root of [2121² + 60² - 254520]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2121, and b = 60, we see that 2ab = 254520, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2121² + 60² - 254520] = [2121 - 60]². Therefore, the positive square root of the expression given is equal to [2121 60] = 2061. ---469. Please find the value of 79999², by using algebraic identities to simplify computation. -Solution: Since 79999 is only a little lower than 80000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 1. After the substitution, we get 79999² = (80000 - 1)² = 80000² + 1² - (2 x 80000 x 1) = 6400000000 + 1 - 160000 = 6399840001. ---470. What is the value of 1.826² + 1.626² - (1.826 x 3.252)? Please use algebraic identities to make the computation simple. --
Solution: First, we note that 3.252 = 2 x 1.626. Using this fact, we can rewrite the given expression as 1.826² + 1.626² - (2 x 1.826 x 1.626). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.826² + 1.626² - (1.826 x 3.252) = 1.826² + 1.626² - (2 x 1.826 x 1.626) = (1.826 - 1.626)² = 0.2² = 0.04. ---471. What is the value of [405.1² + 212.2² - 192.9²] / [810.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 405.1, 212.2, and 192.9. We note that 192.9 = 405.1 - 212.2. We also note, in the denominator, that 810.2 = 2 x 405.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 405.1, and b = 212.2, to get [(405.1² + 212.2² - (405.1 - 212.2)²] / 810.2 = [405.1² + 212.2² - 192.9²] / 810.2 = 212.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 212.2. ---472. What is the positive square root of [3695² + 30² - 221700]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3695, and b = 30, we see that 2ab = 221700, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3695² + 30² - 221700] = [3695 - 30]². Therefore, the positive square root of the expression given is equal to [3695 30] = 3665. ---473. Please find the value of 59997², by using algebraic identities to simplify computation. -Solution: Since 59997 is only a little lower than 60000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a =
60000 and b = 3. After the substitution, we get 59997² = (60000 - 3)² = 60000² + 3² - (2 x 60000 x 3) = 3600000000 + 9 - 360000 = 3599640009. ---474. What is the value of 2.494² + 2.294² - (2.494 x 4.588)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.588 = 2 x 2.294. Using this fact, we can rewrite the given expression as 2.494² + 2.294² - (2 x 2.494 x 2.294). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.494² + 2.294² - (2.494 x 4.588) = 2.494² + 2.294² - (2 x 2.494 x 2.294) = (2.494 - 2.294)² = 0.2² = 0.0400000000000001. ---475. What is the value of [450.3² + 142² - 308.3²] / [900.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 450.3, 142, and 308.3. We note that 308.3 = 450.3 - 142. We also note, in the denominator, that 900.6 = 2 x 450.3. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 450.3, and b = 142, to get [(450.3² + 142² - (450.3 - 142)²] / 900.6 = [450.3² + 142² - 308.3²] / 900.6 = 142. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 142. ---476. What is the positive square root of [2628² + 70² - 367920]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2628, and b = 70, we see that 2ab = 367920, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2628² + 70² - 367920] = [2628 - 70]². Therefore, the positive square root of the expression given is equal to [2628 70] = 2558.
---477. Please find the value of 49998², by using algebraic identities to simplify computation. -Solution: Since 49998 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 2. After the substitution, we get 49998² = (50000 - 2)² = 50000² + 2² - (2 x 50000 x 2) = 2500000000 + 4 - 200000 = 2499800004. ---478. What is the value of 1.664² + 1.464² - (1.664 x 2.928)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.928 = 2 x 1.464. Using this fact, we can rewrite the given expression as 1.664² + 1.464² - (2 x 1.664 x 1.464). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.664² + 1.464² - (1.664 x 2.928) = 1.664² + 1.464² - (2 x 1.664 x 1.464) = (1.664 - 1.464)² = 0.2² = 0.04. ---479. What is the value of [502² + 267.1² - 234.9²] / [1004]? -Solution: First, we look at the three numbers under the square signs in the numerator: 502, 267.1, and 234.9. We note that 234.9 = 502 - 267.1. We also note, in the denominator, that 1004 = 2 x 502. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 502, and b = 267.1, to get [(502² + 267.1² - (502 - 267.1)²] / 1004 = [502² + 267.1² - 234.9²] / 1004 = 267.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 267.1. ---480. What is the positive square root of [2444² + 70² - 342160]? --
Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2444, and b = 70, we see that 2ab = 342160, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2444² + 70² - 342160] = [2444 - 70]². Therefore, the positive square root of the expression given is equal to [2444 70] = 2374. ---481. Please find the value of 49997², by using algebraic identities to simplify computation. -Solution: Since 49997 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 3. After the substitution, we get 49997² = (50000 - 3)² = 50000² + 3² - (2 x 50000 x 3) = 2500000000 + 9 - 300000 = 2499700009. ---482. What is the value of 3.368² + 2.968² - (3.368 x 5.936)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.936 = 2 x 2.968. Using this fact, we can rewrite the given expression as 3.368² + 2.968² - (2 x 3.368 x 2.968). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.368² + 2.968² - (3.368 x 5.936) = 3.368² + 2.968² - (2 x 3.368 x 2.968) = (3.368 - 2.968)² = 0.4² = 0.16. ---483. What is the value of [488.7² + 265.9² - 222.8²] / [977.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 488.7, 265.9, and 222.8. We note that 222.8 = 488.7 - 265.9. We also note, in the denominator, that 977.4 = 2 x 488.7. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b.
In this rearranged equation, we put a = 488.7, and b = 265.9, to get [(488.7² + 265.9² - (488.7 - 265.9)²] / 977.4 = [488.7² + 265.9² - 222.8²] / 977.4 = 265.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 265.9. ---484. What is the positive square root of [1908² + 20² - 76320]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1908, and b = 20, we see that 2ab = 76320, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1908² + 20² - 76320] = [1908 - 20]². Therefore, the positive square root of the expression given is equal to [1908 20] = 1888. ---485. Please find the value of 499², by using algebraic identities to simplify computation. -Solution: Since 499 is only a little lower than 500, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 500 and b = 1. After the substitution, we get 499² = (500 - 1)² = 500² + 1² - (2 x 500 x 1) = 250000 + 1 - 1000 = 249001. ---486. What is the value of 2.583² + 1.183² - (2.583 x 2.366)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.366 = 2 x 1.183. Using this fact, we can rewrite the given expression as 2.583² + 1.183² - (2 x 2.583 x 1.183). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.583² + 1.183² - (2.583 x 2.366) = 2.583² + 1.183² - (2 x 2.583 x 1.183) = (2.583 - 1.183)² = 1.4² = 1.96. ---487. What is the value of [512.1² + 196.3² - 315.8²] / [1024.2]? --
Solution: First, we look at the three numbers under the square signs in the numerator: 512.1, 196.3, and 315.8. We note that 315.8 = 512.1 - 196.3. We also note, in the denominator, that 1024.2 = 2 x 512.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 512.1, and b = 196.3, to get [(512.1² + 196.3² - (512.1 - 196.3)²] / 1024.2 = [512.1² + 196.3² - 315.8²] / 1024.2 = 196.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 196.3. ---488. What is the positive square root of [2779² + 70² - 389060]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2779, and b = 70, we see that 2ab = 389060, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2779² + 70² - 389060] = [2779 - 70]². Therefore, the positive square root of the expression given is equal to [2779 70] = 2709. ---489. Please find the value of 699², by using algebraic identities to simplify computation. -Solution: Since 699 is only a little lower than 700, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 1. After the substitution, we get 699² = (700 - 1)² = 700² + 1² - (2 x 700 x 1) = 490000 + 1 - 1400 = 488601. ---490. What is the value of 3.186² + 2.986² - (3.186 x 5.972)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.972 = 2 x 2.986. Using this fact, we can
rewrite the given expression as 3.186² + 2.986² - (2 x 3.186 x 2.986). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.186² + 2.986² - (3.186 x 5.972) = 3.186² + 2.986² - (2 x 3.186 x 2.986) = (3.186 - 2.986)² = 0.2² = 0.0400000000000001. ---491. What is the value of [440.1² + 107.5² - 332.6²] / [880.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 440.1, 107.5, and 332.6. We note that 332.6 = 440.1 - 107.5. We also note, in the denominator, that 880.2 = 2 x 440.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 440.1, and b = 107.5, to get [(440.1² + 107.5² - (440.1 - 107.5)²] / 880.2 = [440.1² + 107.5² - 332.6²] / 880.2 = 107.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 107.5. ---492. What is the positive square root of [1797² + 50² - 179700]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1797, and b = 50, we see that 2ab = 179700, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1797² + 50² - 179700] = [1797 - 50]². Therefore, the positive square root of the expression given is equal to [1797 50] = 1747. ---493. Please find the value of 699², by using algebraic identities to simplify computation. -Solution: Since 699 is only a little lower than 700, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 1. After the substitution, we get 699² = (700 - 1)² = 700² + 1² - (2 x
700 x 1) = 490000 + 1 - 1400 = 488601. ---494. What is the value of 2.11² + 1.11² - (2.11 x 2.22)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.22 = 2 x 1.11. Using this fact, we can rewrite the given expression as 2.11² + 1.11² - (2 x 2.11 x 1.11). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.11² + 1.11² - (2.11 x 2.22) = 2.11² + 1.11² - (2 x 2.11 x 1.11) = (2.11 - 1.11)² = 1² = 1. ---495. What is the value of [573.7² + 211.3² - 362.4²] / [1147.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 573.7, 211.3, and 362.4. We note that 362.4 = 573.7 - 211.3. We also note, in the denominator, that 1147.4 = 2 x 573.7. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 573.7, and b = 211.3, to get [(573.7² + 211.3² - (573.7 - 211.3)²] / 1147.4 = [573.7² + 211.3² - 362.4²] / 1147.4 = 211.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 211.3. ---496. What is the positive square root of [3997² + 40² - 319760]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3997, and b = 40, we see that 2ab = 319760, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3997² + 40² - 319760] = [3997 - 40]². Therefore, the positive square root of the expression given is equal to [3997 40] = 3957. ----
497. Please find the value of 897², by using algebraic identities to simplify computation. -Solution: Since 897 is only a little lower than 900, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 900 and b = 3. After the substitution, we get 897² = (900 - 3)² = 900² + 3² - (2 x 900 x 3) = 810000 + 9 - 5400 = 804609. ---498. What is the value of 1.559² + 0.759² - (1.559 x 1.518)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.518 = 2 x 0.759. Using this fact, we can rewrite the given expression as 1.559² + 0.759² - (2 x 1.559 x 0.759). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.559² + 0.759² - (1.559 x 1.518) = 1.559² + 0.759² - (2 x 1.559 x 0.759) = (1.559 - 0.759)² = 0.8² = 0.64. ---499. What is the value of [431.6² + 150.9² - 280.7²] / [863.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 431.6, 150.9, and 280.7. We note that 280.7 = 431.6 - 150.9. We also note, in the denominator, that 863.2 = 2 x 431.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 431.6, and b = 150.9, to get [(431.6² + 150.9² - (431.6 - 150.9)²] / 863.2 = [431.6² + 150.9² - 280.7²] / 863.2 = 150.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 150.9. ---500. What is the positive square root of [2682² + 40² - 214560]? -Solution: By observation, we can deduce that the given expression looks a
little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2682, and b = 40, we see that 2ab = 214560, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2682² + 40² - 214560] = [2682 - 40]². Therefore, the positive square root of the expression given is equal to [2682 40] = 2642. ---501. Please find the value of 19998², by using algebraic identities to simplify computation. -Solution: Since 19998 is only a little lower than 20000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 20000 and b = 2. After the substitution, we get 19998² = (20000 - 2)² = 20000² + 2² - (2 x 20000 x 2) = 400000000 + 4 - 80000 = 399920004. ---502. What is the value of 3.635² + 3.035² - (3.635 x 6.07)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.07 = 2 x 3.035. Using this fact, we can rewrite the given expression as 3.635² + 3.035² - (2 x 3.635 x 3.035). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.635² + 3.035² - (3.635 x 6.07) = 3.635² + 3.035² - (2 x 3.635 x 3.035) = (3.635 - 3.035)² = 0.6² = 0.36. ---503. What is the value of [520.1² + 281.2² - 238.9²] / [1040.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 520.1, 281.2, and 238.9. We note that 238.9 = 520.1 - 281.2. We also note, in the denominator, that 1040.2 = 2 x 520.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 520.1, and b = 281.2, to get [(520.1²
+ 281.2² - (520.1 - 281.2)²] / 1040.2 = [520.1² + 281.2² - 238.9²] / 1040.2 = 281.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 281.2. ---504. What is the positive square root of [2915² + 20² - 116600]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2915, and b = 20, we see that 2ab = 116600, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2915² + 20² - 116600] = [2915 - 20]². Therefore, the positive square root of the expression given is equal to [2915 20] = 2895. ---505. Please find the value of 599², by using algebraic identities to simplify computation. -Solution: Since 599 is only a little lower than 600, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 600 and b = 1. After the substitution, we get 599² = (600 - 1)² = 600² + 1² - (2 x 600 x 1) = 360000 + 1 - 1200 = 358801. ---506. What is the value of 2.91² + 1.91² - (2.91 x 3.82)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.82 = 2 x 1.91. Using this fact, we can rewrite the given expression as 2.91² + 1.91² - (2 x 2.91 x 1.91). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.91² + 1.91² - (2.91 x 3.82) = 2.91² + 1.91² - (2 x 2.91 x 1.91) = (2.91 - 1.91)² = 1² = 1. ---507. What is the value of [562.6² + 245.1² - 317.5²] / [1125.2]? -Solution: First, we look at the three numbers under the square signs in the
numerator: 562.6, 245.1, and 317.5. We note that 317.5 = 562.6 - 245.1. We also note, in the denominator, that 1125.2 = 2 x 562.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 562.6, and b = 245.1, to get [(562.6² + 245.1² - (562.6 - 245.1)²] / 1125.2 = [562.6² + 245.1² - 317.5²] / 1125.2 = 245.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 245.1. ---508. What is the positive square root of [2443² + 50² - 244300]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2443, and b = 50, we see that 2ab = 244300, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2443² + 50² - 244300] = [2443 - 50]². Therefore, the positive square root of the expression given is equal to [2443 50] = 2393. ---509. Please find the value of 2999², by using algebraic identities to simplify computation. -Solution: Since 2999 is only a little lower than 3000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 3000 and b = 1. After the substitution, we get 2999² = (3000 - 1)² = 3000² + 1² - (2 x 3000 x 1) = 9000000 + 1 - 6000 = 8994001. ---510. What is the value of 3.217² + 2.417² - (3.217 x 4.834)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.834 = 2 x 2.417. Using this fact, we can rewrite the given expression as 3.217² + 2.417² - (2 x 3.217 x 2.417).
Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.217² + 2.417² - (3.217 x 4.834) = 3.217² + 2.417² - (2 x 3.217 x 2.417) = (3.217 - 2.417)² = 0.8² = 0.640000000000001. ---511. What is the value of [488.8² + 119.8² - 369²] / [977.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 488.8, 119.8, and 369. We note that 369 = 488.8 - 119.8. We also note, in the denominator, that 977.6 = 2 x 488.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 488.8, and b = 119.8, to get [(488.8² + 119.8² - (488.8 - 119.8)²] / 977.6 = [488.8² + 119.8² - 369²] / 977.6 = 119.8. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 119.8. ---512. What is the positive square root of [2138² + 90² - 384840]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2138, and b = 90, we see that 2ab = 384840, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2138² + 90² - 384840] = [2138 - 90]². Therefore, the positive square root of the expression given is equal to [2138 90] = 2048. ---513. Please find the value of 299², by using algebraic identities to simplify computation. -Solution: Since 299 is only a little lower than 300, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 1. After the substitution, we get 299² = (300 - 1)² = 300² + 1² - (2 x
300 x 1) = 90000 + 1 - 600 = 89401. ---514. What is the value of 2.601² + 2.401² - (2.601 x 4.802)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.802 = 2 x 2.401. Using this fact, we can rewrite the given expression as 2.601² + 2.401² - (2 x 2.601 x 2.401). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.601² + 2.401² - (2.601 x 4.802) = 2.601² + 2.401² - (2 x 2.601 x 2.401) = (2.601 - 2.401)² = 0.2² = 0.0400000000000001. ---515. What is the value of [563.9² + 191.2² - 372.7²] / [1127.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 563.9, 191.2, and 372.7. We note that 372.7 = 563.9 - 191.2. We also note, in the denominator, that 1127.8 = 2 x 563.9. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 563.9, and b = 191.2, to get [(563.9² + 191.2² - (563.9 - 191.2)²] / 1127.8 = [563.9² + 191.2² - 372.7²] / 1127.8 = 191.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 191.2. ---516. What is the positive square root of [3525² + 40² - 282000]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3525, and b = 40, we see that 2ab = 282000, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3525² + 40² - 282000] = [3525 - 40]². Therefore, the positive square root of the expression given is equal to [3525 40] = 3485. ----
517. Please find the value of 1997², by using algebraic identities to simplify computation. -Solution: Since 1997 is only a little lower than 2000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 3. After the substitution, we get 1997² = (2000 - 3)² = 2000² + 3² - (2 x 2000 x 3) = 4000000 + 9 - 12000 = 3988009. ---518. What is the value of 3.321² + 2.521² - (3.321 x 5.042)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.042 = 2 x 2.521. Using this fact, we can rewrite the given expression as 3.321² + 2.521² - (2 x 3.321 x 2.521). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.321² + 2.521² - (3.321 x 5.042) = 3.321² + 2.521² - (2 x 3.321 x 2.521) = (3.321 - 2.521)² = 0.8² = 0.640000000000001. ---519. What is the value of [424² + 211.6² - 212.4²] / [848]? -Solution: First, we look at the three numbers under the square signs in the numerator: 424, 211.6, and 212.4. We note that 212.4 = 424 - 211.6. We also note, in the denominator, that 848 = 2 x 424. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 424, and b = 211.6, to get [(424² + 211.6² - (424 - 211.6)²] / 848 = [424² + 211.6² - 212.4²] / 848 = 211.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 211.6. ---520. What is the positive square root of [3775² + 40² - 302000]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a
= 3775, and b = 40, we see that 2ab = 302000, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3775² + 40² - 302000] = [3775 - 40]². Therefore, the positive square root of the expression given is equal to [3775 40] = 3735. ---521. Please find the value of 5999², by using algebraic identities to simplify computation. -Solution: Since 5999 is only a little lower than 6000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 1. After the substitution, we get 5999² = (6000 - 1)² = 6000² + 1² - (2 x 6000 x 1) = 36000000 + 1 - 12000 = 35988001. ---522. What is the value of 3.177² + 1.777² - (3.177 x 3.554)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.554 = 2 x 1.777. Using this fact, we can rewrite the given expression as 3.177² + 1.777² - (2 x 3.177 x 1.777). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.177² + 1.777² - (3.177 x 3.554) = 3.177² + 1.777² - (2 x 3.177 x 1.777) = (3.177 - 1.777)² = 1.4² = 1.96. ---523. What is the value of [508.1² + 200.3² - 307.8²] / [1016.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 508.1, 200.3, and 307.8. We note that 307.8 = 508.1 - 200.3. We also note, in the denominator, that 1016.2 = 2 x 508.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 508.1, and b = 200.3, to get [(508.1² + 200.3² - (508.1 - 200.3)²] / 1016.2 = [508.1² + 200.3² - 307.8²] / 1016.2 = 200.3. The Left Hand Side of this equation is exactly the expression in our
question, so the value of that expression is equal to 200.3. ---524. What is the positive square root of [3708² + 50² - 370800]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3708, and b = 50, we see that 2ab = 370800, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3708² + 50² - 370800] = [3708 - 50]². Therefore, the positive square root of the expression given is equal to [3708 50] = 3658. ---525. Please find the value of 5999², by using algebraic identities to simplify computation. -Solution: Since 5999 is only a little lower than 6000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 6000 and b = 1. After the substitution, we get 5999² = (6000 - 1)² = 6000² + 1² - (2 x 6000 x 1) = 36000000 + 1 - 12000 = 35988001. ---526. What is the value of 2.127² + 1.727² - (2.127 x 3.454)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.454 = 2 x 1.727. Using this fact, we can rewrite the given expression as 2.127² + 1.727² - (2 x 2.127 x 1.727). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.127² + 1.727² - (2.127 x 3.454) = 2.127² + 1.727² - (2 x 2.127 x 1.727) = (2.127 - 1.727)² = 0.4² = 0.16. ---527. What is the value of [452.1² + 250² - 202.1²] / [904.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 452.1, 250, and 202.1. We note that 202.1 = 452.1 - 250. We also note, in the denominator, that 904.2 = 2 x 452.1.
Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 452.1, and b = 250, to get [(452.1² + 250² - (452.1 - 250)²] / 904.2 = [452.1² + 250² - 202.1²] / 904.2 = 250. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 250. ---528. What is the positive square root of [3646² + 90² - 656280]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3646, and b = 90, we see that 2ab = 656280, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3646² + 90² - 656280] = [3646 - 90]². Therefore, the positive square root of the expression given is equal to [3646 90] = 3556. ---529. Please find the value of 79997², by using algebraic identities to simplify computation. -Solution: Since 79997 is only a little lower than 80000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 3. After the substitution, we get 79997² = (80000 - 3)² = 80000² + 3² - (2 x 80000 x 3) = 6400000000 + 9 - 480000 = 6399520009. ---530. What is the value of 3.545² + 2.345² - (3.545 x 4.69)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.69 = 2 x 2.345. Using this fact, we can rewrite the given expression as 3.545² + 2.345² - (2 x 3.545 x 2.345). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.545² + 2.345² - (3.545 x 4.69) = 3.545² + 2.345² - (2 x 3.545 x 2.345) = (3.545 - 2.345)² = 1.2² = 1.44.
---531. What is the value of [439.1² + 219.9² - 219.2²] / [878.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 439.1, 219.9, and 219.2. We note that 219.2 = 439.1 - 219.9. We also note, in the denominator, that 878.2 = 2 x 439.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 439.1, and b = 219.9, to get [(439.1² + 219.9² - (439.1 - 219.9)²] / 878.2 = [439.1² + 219.9² - 219.2²] / 878.2 = 219.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 219.9. ---532. What is the positive square root of [3439² + 30² - 206340]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3439, and b = 30, we see that 2ab = 206340, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3439² + 30² - 206340] = [3439 - 30]². Therefore, the positive square root of the expression given is equal to [3439 30] = 3409. ---533. Please find the value of 598², by using algebraic identities to simplify computation. -Solution: Since 598 is only a little lower than 600, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 600 and b = 2. After the substitution, we get 598² = (600 - 2)² = 600² + 2² - (2 x 600 x 2) = 360000 + 4 - 2400 = 357604. ----
534. What is the value of 1.832² + 1.632² - (1.832 x 3.264)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.264 = 2 x 1.632. Using this fact, we can rewrite the given expression as 1.832² + 1.632² - (2 x 1.832 x 1.632). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.832² + 1.632² - (1.832 x 3.264) = 1.832² + 1.632² - (2 x 1.832 x 1.632) = (1.832 - 1.632)² = 0.2² = 0.04. ---535. What is the value of [475.5² + 215.7² - 259.8²] / [951]? -Solution: First, we look at the three numbers under the square signs in the numerator: 475.5, 215.7, and 259.8. We note that 259.8 = 475.5 - 215.7. We also note, in the denominator, that 951 = 2 x 475.5. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 475.5, and b = 215.7, to get [(475.5² + 215.7² - (475.5 - 215.7)²] / 951 = [475.5² + 215.7² - 259.8²] / 951 = 215.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 215.7. ---536. What is the positive square root of [2164² + 60² - 259680]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2164, and b = 60, we see that 2ab = 259680, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2164² + 60² - 259680] = [2164 - 60]². Therefore, the positive square root of the expression given is equal to [2164 60] = 2104. ---537. Please find the value of 897², by using algebraic identities to simplify computation. --
Solution: Since 897 is only a little lower than 900, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 900 and b = 3. After the substitution, we get 897² = (900 - 3)² = 900² + 3² - (2 x 900 x 3) = 810000 + 9 - 5400 = 804609. ---538. What is the value of 3.392² + 2.992² - (3.392 x 5.984)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.984 = 2 x 2.992. Using this fact, we can rewrite the given expression as 3.392² + 2.992² - (2 x 3.392 x 2.992). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.392² + 2.992² - (3.392 x 5.984) = 3.392² + 2.992² - (2 x 3.392 x 2.992) = (3.392 - 2.992)² = 0.4² = 0.16. ---539. What is the value of [582² + 182.4² - 399.6²] / [1164]? -Solution: First, we look at the three numbers under the square signs in the numerator: 582, 182.4, and 399.6. We note that 399.6 = 582 - 182.4. We also note, in the denominator, that 1164 = 2 x 582. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 582, and b = 182.4, to get [(582² + 182.4² - (582 - 182.4)²] / 1164 = [582² + 182.4² - 399.6²] / 1164 = 182.4. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 182.4. ---540. What is the positive square root of [2188² + 30² - 131280]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2188, and b = 30, we see that 2ab = 131280, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2188² + 30² - 131280] = [2188 - 30]².
Therefore, the positive square root of the expression given is equal to [2188 30] = 2158. ---541. Please find the value of 298², by using algebraic identities to simplify computation. -Solution: Since 298 is only a little lower than 300, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 2. After the substitution, we get 298² = (300 - 2)² = 300² + 2² - (2 x 300 x 2) = 90000 + 4 - 1200 = 88804. ---542. What is the value of 2.263² + 2.063² - (2.263 x 4.126)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.126 = 2 x 2.063. Using this fact, we can rewrite the given expression as 2.263² + 2.063² - (2 x 2.263 x 2.063). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.263² + 2.063² - (2.263 x 4.126) = 2.263² + 2.063² - (2 x 2.263 x 2.063) = (2.263 - 2.063)² = 0.2² = 0.0400000000000001. ---543. What is the value of [440.6² + 286.9² - 153.7²] / [881.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 440.6, 286.9, and 153.7. We note that 153.7 = 440.6 - 286.9. We also note, in the denominator, that 881.2 = 2 x 440.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 440.6, and b = 286.9, to get [(440.6² + 286.9² - (440.6 - 286.9)²] / 881.2 = [440.6² + 286.9² - 153.7²] / 881.2 = 286.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 286.9. ----
544. What is the positive square root of [2028² + 40² - 162240]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2028, and b = 40, we see that 2ab = 162240, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2028² + 40² - 162240] = [2028 - 40]². Therefore, the positive square root of the expression given is equal to [2028 40] = 1988. ---545. Please find the value of 598², by using algebraic identities to simplify computation. -Solution: Since 598 is only a little lower than 600, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 600 and b = 2. After the substitution, we get 598² = (600 - 2)² = 600² + 2² - (2 x 600 x 2) = 360000 + 4 - 2400 = 357604. ---546. What is the value of 3.423² + 2.023² - (3.423 x 4.046)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.046 = 2 x 2.023. Using this fact, we can rewrite the given expression as 3.423² + 2.023² - (2 x 3.423 x 2.023). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.423² + 2.023² - (3.423 x 4.046) = 3.423² + 2.023² - (2 x 3.423 x 2.023) = (3.423 - 2.023)² = 1.4² = 1.96. ---547. What is the value of [493.4² + 226.2² - 267.2²] / [986.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 493.4, 226.2, and 267.2. We note that 267.2 = 493.4 - 226.2. We also note, in the denominator, that 986.8 = 2 x 493.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression.
After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 493.4, and b = 226.2, to get [(493.4² + 226.2² - (493.4 - 226.2)²] / 986.8 = [493.4² + 226.2² - 267.2²] / 986.8 = 226.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 226.2. ---548. What is the positive square root of [1932² + 90² - 347760]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1932, and b = 90, we see that 2ab = 347760, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1932² + 90² - 347760] = [1932 - 90]². Therefore, the positive square root of the expression given is equal to [1932 90] = 1842. ---549. Please find the value of 49998², by using algebraic identities to simplify computation. -Solution: Since 49998 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 2. After the substitution, we get 49998² = (50000 - 2)² = 50000² + 2² - (2 x 50000 x 2) = 2500000000 + 4 - 200000 = 2499800004. ---550. What is the value of 3.287² + 2.487² - (3.287 x 4.974)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.974 = 2 x 2.487. Using this fact, we can rewrite the given expression as 3.287² + 2.487² - (2 x 3.287 x 2.487). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.287² + 2.487² - (3.287 x 4.974) = 3.287² + 2.487² - (2 x 3.287 x 2.487) = (3.287 - 2.487)² = 0.8² = 0.64. ---551. What is the value of [442.6² + 115.3² - 327.3²] / [885.2]?
-Solution: First, we look at the three numbers under the square signs in the numerator: 442.6, 115.3, and 327.3. We note that 327.3 = 442.6 - 115.3. We also note, in the denominator, that 885.2 = 2 x 442.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 442.6, and b = 115.3, to get [(442.6² + 115.3² - (442.6 - 115.3)²] / 885.2 = [442.6² + 115.3² - 327.3²] / 885.2 = 115.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 115.3. ---552. What is the positive square root of [2864² + 90² - 515520]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2864, and b = 90, we see that 2ab = 515520, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2864² + 90² - 515520] = [2864 - 90]². Therefore, the positive square root of the expression given is equal to [2864 90] = 2774. ---553. Please find the value of 7999², by using algebraic identities to simplify computation. -Solution: Since 7999 is only a little lower than 8000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 8000 and b = 1. After the substitution, we get 7999² = (8000 - 1)² = 8000² + 1² - (2 x 8000 x 1) = 64000000 + 1 - 16000 = 63984001. ---554. What is the value of 3.567² + 2.967² - (3.567 x 5.934)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.934 = 2 x 2.967. Using this fact, we can
rewrite the given expression as 3.567² + 2.967² - (2 x 3.567 x 2.967). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.567² + 2.967² - (3.567 x 5.934) = 3.567² + 2.967² - (2 x 3.567 x 2.967) = (3.567 - 2.967)² = 0.6² = 0.36. ---555. What is the value of [569.7² + 178.6² - 391.1²] / [1139.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 569.7, 178.6, and 391.1. We note that 391.1 = 569.7 - 178.6. We also note, in the denominator, that 1139.4 = 2 x 569.7. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 569.7, and b = 178.6, to get [(569.7² + 178.6² - (569.7 - 178.6)²] / 1139.4 = [569.7² + 178.6² - 391.1²] / 1139.4 = 178.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 178.6. ---556. What is the positive square root of [1726² + 50² - 172600]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1726, and b = 50, we see that 2ab = 172600, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1726² + 50² - 172600] = [1726 - 50]². Therefore, the positive square root of the expression given is equal to [1726 50] = 1676. ---557. Please find the value of 6999², by using algebraic identities to simplify computation. -Solution: Since 6999 is only a little lower than 7000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 1. After the substitution, we get 6999² = (7000 - 1)² = 7000² + 1² - (2
x 7000 x 1) = 49000000 + 1 - 14000 = 48986001. ---558. What is the value of 2.318² + 1.718² - (2.318 x 3.436)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.436 = 2 x 1.718. Using this fact, we can rewrite the given expression as 2.318² + 1.718² - (2 x 2.318 x 1.718). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.318² + 1.718² - (2.318 x 3.436) = 2.318² + 1.718² - (2 x 2.318 x 1.718) = (2.318 - 1.718)² = 0.6² = 0.36. ---559. What is the value of [489² + 144.2² - 344.8²] / [978]? -Solution: First, we look at the three numbers under the square signs in the numerator: 489, 144.2, and 344.8. We note that 344.8 = 489 - 144.2. We also note, in the denominator, that 978 = 2 x 489. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 489, and b = 144.2, to get [(489² + 144.2² - (489 - 144.2)²] / 978 = [489² + 144.2² - 344.8²] / 978 = 144.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 144.2. ---560. What is the positive square root of [3716² + 20² - 148640]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3716, and b = 20, we see that 2ab = 148640, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3716² + 20² - 148640] = [3716 - 20]². Therefore, the positive square root of the expression given is equal to [3716 20] = 3696. ----
561. Please find the value of 3999², by using algebraic identities to simplify computation. -Solution: Since 3999 is only a little lower than 4000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 4000 and b = 1. After the substitution, we get 3999² = (4000 - 1)² = 4000² + 1² - (2 x 4000 x 1) = 16000000 + 1 - 8000 = 15992001. ---562. What is the value of 3.277² + 2.077² - (3.277 x 4.154)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.154 = 2 x 2.077. Using this fact, we can rewrite the given expression as 3.277² + 2.077² - (2 x 3.277 x 2.077). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.277² + 2.077² - (3.277 x 4.154) = 3.277² + 2.077² - (2 x 3.277 x 2.077) = (3.277 - 2.077)² = 1.2² = 1.44. ---563. What is the value of [595.8² + 222² - 373.8²] / [1191.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 595.8, 222, and 373.8. We note that 373.8 = 595.8 - 222. We also note, in the denominator, that 1191.6 = 2 x 595.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 595.8, and b = 222, to get [(595.8² + 222² - (595.8 - 222)²] / 1191.6 = [595.8² + 222² - 373.8²] / 1191.6 = 222. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 222. ---564. What is the positive square root of [3895² + 20² - 155800]? -Solution: By observation, we can deduce that the given expression looks a
little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3895, and b = 20, we see that 2ab = 155800, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3895² + 20² - 155800] = [3895 - 20]². Therefore, the positive square root of the expression given is equal to [3895 20] = 3875. ---565. Please find the value of 49997², by using algebraic identities to simplify computation. -Solution: Since 49997 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 3. After the substitution, we get 49997² = (50000 - 3)² = 50000² + 3² - (2 x 50000 x 3) = 2500000000 + 9 - 300000 = 2499700009. ---566. What is the value of 2.71² + 1.31² - (2.71 x 2.62)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.62 = 2 x 1.31. Using this fact, we can rewrite the given expression as 2.71² + 1.31² - (2 x 2.71 x 1.31). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.71² + 1.31² - (2.71 x 2.62) = 2.71² + 1.31² - (2 x 2.71 x 1.31) = (2.71 - 1.31)² = 1.4² = 1.96. ---567. What is the value of [439.6² + 168.5² - 271.1²] / [879.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 439.6, 168.5, and 271.1. We note that 271.1 = 439.6 - 168.5. We also note, in the denominator, that 879.2 = 2 x 439.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 439.6, and b = 168.5, to get [(439.6² + 168.5² - (439.6 - 168.5)²] / 879.2 = [439.6² + 168.5² - 271.1²] / 879.2 =
168.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 168.5. ---568. What is the positive square root of [2802² + 80² - 448320]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2802, and b = 80, we see that 2ab = 448320, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2802² + 80² - 448320] = [2802 - 80]². Therefore, the positive square root of the expression given is equal to [2802 80] = 2722. ---569. Please find the value of 6999², by using algebraic identities to simplify computation. -Solution: Since 6999 is only a little lower than 7000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 1. After the substitution, we get 6999² = (7000 - 1)² = 7000² + 1² - (2 x 7000 x 1) = 49000000 + 1 - 14000 = 48986001. ---570. What is the value of 3.049² + 1.649² - (3.049 x 3.298)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.298 = 2 x 1.649. Using this fact, we can rewrite the given expression as 3.049² + 1.649² - (2 x 3.049 x 1.649). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.049² + 1.649² - (3.049 x 3.298) = 3.049² + 1.649² - (2 x 3.049 x 1.649) = (3.049 - 1.649)² = 1.4² = 1.96. ---571. What is the value of [439.4² + 278.1² - 161.3²] / [878.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 439.4, 278.1, and 161.3. We note that 161.3 = 439.4 - 278.1. We
also note, in the denominator, that 878.8 = 2 x 439.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 439.4, and b = 278.1, to get [(439.4² + 278.1² - (439.4 - 278.1)²] / 878.8 = [439.4² + 278.1² - 161.3²] / 878.8 = 278.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 278.1. ---572. What is the positive square root of [2817² + 50² - 281700]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2817, and b = 50, we see that 2ab = 281700, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2817² + 50² - 281700] = [2817 - 50]². Therefore, the positive square root of the expression given is equal to [2817 50] = 2767. ---573. Please find the value of 8997², by using algebraic identities to simplify computation. -Solution: Since 8997 is only a little lower than 9000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 9000 and b = 3. After the substitution, we get 8997² = (9000 - 3)² = 9000² + 3² - (2 x 9000 x 3) = 81000000 + 9 - 54000 = 80946009. ---574. What is the value of 2.403² + 2.203² - (2.403 x 4.406)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.406 = 2 x 2.203. Using this fact, we can rewrite the given expression as 2.403² + 2.203² - (2 x 2.403 x 2.203). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.403² + 2.203² - (2.403 x 4.406) = 2.403² + 2.203² - (2 x 2.403
x 2.203) = (2.403 - 2.203)² = 0.2² = 0.0400000000000001. ---575. What is the value of [454.4² + 199.4² - 255.²] / [908.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 454.4, 199.4, and 255.. We note that 255. = 454.4 - 199.4. We also note, in the denominator, that 908.8 = 2 x 454.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 454.4, and b = 199.4, to get [(454.4² + 199.4² - (454.4 - 199.4)²] / 908.8 = [454.4² + 199.4² - 255.²] / 908.8 = 199.4. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 199.4. ---576. What is the positive square root of [3201² + 50² - 320100]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3201, and b = 50, we see that 2ab = 320100, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3201² + 50² - 320100] = [3201 - 50]². Therefore, the positive square root of the expression given is equal to [3201 50] = 3151. ---577. Please find the value of 8998², by using algebraic identities to simplify computation. -Solution: Since 8998 is only a little lower than 9000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 9000 and b = 2. After the substitution, we get 8998² = (9000 - 2)² = 9000² + 2² - (2 x 9000 x 2) = 81000000 + 4 - 36000 = 80964004. ----
578. What is the value of 3.381² + 2.581² - (3.381 x 5.162)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.162 = 2 x 2.581. Using this fact, we can rewrite the given expression as 3.381² + 2.581² - (2 x 3.381 x 2.581). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.381² + 2.581² - (3.381 x 5.162) = 3.381² + 2.581² - (2 x 3.381 x 2.581) = (3.381 - 2.581)² = 0.8² = 0.640000000000001. ---579. What is the value of [595² + 100.4² - 494.6²] / [1190]? -Solution: First, we look at the three numbers under the square signs in the numerator: 595, 100.4, and 494.6. We note that 494.6 = 595 - 100.4. We also note, in the denominator, that 1190 = 2 x 595. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 595, and b = 100.4, to get [(595² + 100.4² - (595 - 100.4)²] / 1190 = [595² + 100.4² - 494.6²] / 1190 = 100.4. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 100.4. ---580. What is the positive square root of [2446² + 20² - 97840]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2446, and b = 20, we see that 2ab = 97840, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2446² + 20² - 97840] = [2446 - 20]². Therefore, the positive square root of the expression given is equal to [2446 20] = 2426. ---581. Please find the value of 59998², by using algebraic identities to simplify computation. --
Solution: Since 59998 is only a little lower than 60000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 60000 and b = 2. After the substitution, we get 59998² = (60000 - 2)² = 60000² + 2² - (2 x 60000 x 2) = 3600000000 + 4 - 240000 = 3599760004. ---582. What is the value of 2.871² + 2.471² - (2.871 x 4.942)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.942 = 2 x 2.471. Using this fact, we can rewrite the given expression as 2.871² + 2.471² - (2 x 2.871 x 2.471). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.871² + 2.471² - (2.871 x 4.942) = 2.871² + 2.471² - (2 x 2.871 x 2.471) = (2.871 - 2.471)² = 0.4² = 0.16. ---583. What is the value of [533.6² + 273.7² - 259.9²] / [1067.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 533.6, 273.7, and 259.9. We note that 259.9 = 533.6 - 273.7. We also note, in the denominator, that 1067.2 = 2 x 533.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 533.6, and b = 273.7, to get [(533.6² + 273.7² - (533.6 - 273.7)²] / 1067.2 = [533.6² + 273.7² - 259.9²] / 1067.2 = 273.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 273.7. ---584. What is the positive square root of [3090² + 90² - 556200]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3090, and b = 90, we see that 2ab = 556200, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3090² + 90² - 556200] = [3090 - 90]².
Therefore, the positive square root of the expression given is equal to [3090 90] = 3000. ---585. Please find the value of 6998², by using algebraic identities to simplify computation. -Solution: Since 6998 is only a little lower than 7000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 2. After the substitution, we get 6998² = (7000 - 2)² = 7000² + 2² - (2 x 7000 x 2) = 49000000 + 4 - 28000 = 48972004. ---586. What is the value of 2.54² + 2.14² - (2.54 x 4.28)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.28 = 2 x 2.14. Using this fact, we can rewrite the given expression as 2.54² + 2.14² - (2 x 2.54 x 2.14). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.54² + 2.14² - (2.54 x 4.28) = 2.54² + 2.14² - (2 x 2.54 x 2.14) = (2.54 - 2.14)² = 0.4² = 0.16. ---587. What is the value of [485.1² + 210.6² - 274.5²] / [970.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 485.1, 210.6, and 274.5. We note that 274.5 = 485.1 - 210.6. We also note, in the denominator, that 970.2 = 2 x 485.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 485.1, and b = 210.6, to get [(485.1² + 210.6² - (485.1 - 210.6)²] / 970.2 = [485.1² + 210.6² - 274.5²] / 970.2 = 210.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 210.6. ----
588. What is the positive square root of [3625² + 90² - 652500]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3625, and b = 90, we see that 2ab = 652500, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3625² + 90² - 652500] = [3625 - 90]². Therefore, the positive square root of the expression given is equal to [3625 90] = 3535. ---589. Please find the value of 698², by using algebraic identities to simplify computation. -Solution: Since 698 is only a little lower than 700, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 2. After the substitution, we get 698² = (700 - 2)² = 700² + 2² - (2 x 700 x 2) = 490000 + 4 - 2800 = 487204. ---590. What is the value of 2.605² + 1.205² - (2.605 x 2.41)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.41 = 2 x 1.205. Using this fact, we can rewrite the given expression as 2.605² + 1.205² - (2 x 2.605 x 1.205). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.605² + 1.205² - (2.605 x 2.41) = 2.605² + 1.205² - (2 x 2.605 x 1.205) = (2.605 - 1.205)² = 1.4² = 1.96. ---591. What is the value of [436.4² + 258.7² - 177.7²] / [872.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 436.4, 258.7, and 177.7. We note that 177.7 = 436.4 - 258.7. We also note, in the denominator, that 872.8 = 2 x 436.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression.
After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 436.4, and b = 258.7, to get [(436.4² + 258.7² - (436.4 - 258.7)²] / 872.8 = [436.4² + 258.7² - 177.7²] / 872.8 = 258.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 258.7. ---592. What is the positive square root of [3747² + 50² - 374700]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3747, and b = 50, we see that 2ab = 374700, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3747² + 50² - 374700] = [3747 - 50]². Therefore, the positive square root of the expression given is equal to [3747 50] = 3697. ---593. Please find the value of 899², by using algebraic identities to simplify computation. -Solution: Since 899 is only a little lower than 900, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 900 and b = 1. After the substitution, we get 899² = (900 - 1)² = 900² + 1² - (2 x 900 x 1) = 810000 + 1 - 1800 = 808201. ---594. What is the value of 2.308² + 1.708² - (2.308 x 3.416)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.416 = 2 x 1.708. Using this fact, we can rewrite the given expression as 2.308² + 1.708² - (2 x 2.308 x 1.708). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.308² + 1.708² - (2.308 x 3.416) = 2.308² + 1.708² - (2 x 2.308 x 1.708) = (2.308 - 1.708)² = 0.6² = 0.36. ---595. What is the value of [457.4² + 219.6² - 237.8²] / [914.8]?
-Solution: First, we look at the three numbers under the square signs in the numerator: 457.4, 219.6, and 237.8. We note that 237.8 = 457.4 - 219.6. We also note, in the denominator, that 914.8 = 2 x 457.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 457.4, and b = 219.6, to get [(457.4² + 219.6² - (457.4 - 219.6)²] / 914.8 = [457.4² + 219.6² - 237.8²] / 914.8 = 219.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 219.6. ---596. What is the positive square root of [2937² + 70² - 411180]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2937, and b = 70, we see that 2ab = 411180, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2937² + 70² - 411180] = [2937 - 70]². Therefore, the positive square root of the expression given is equal to [2937 70] = 2867. ---597. Please find the value of 497², by using algebraic identities to simplify computation. -Solution: Since 497 is only a little lower than 500, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 500 and b = 3. After the substitution, we get 497² = (500 - 3)² = 500² + 3² - (2 x 500 x 3) = 250000 + 9 - 3000 = 247009. ---598. What is the value of 1.993² + 1.193² - (1.993 x 2.386)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.386 = 2 x 1.193. Using this fact, we can
rewrite the given expression as 1.993² + 1.193² - (2 x 1.993 x 1.193). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.993² + 1.193² - (1.993 x 2.386) = 1.993² + 1.193² - (2 x 1.993 x 1.193) = (1.993 - 1.193)² = 0.8² = 0.64. ---599. What is the value of [569.4² + 246.5² - 322.9²] / [1138.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 569.4, 246.5, and 322.9. We note that 322.9 = 569.4 - 246.5. We also note, in the denominator, that 1138.8 = 2 x 569.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 569.4, and b = 246.5, to get [(569.4² + 246.5² - (569.4 - 246.5)²] / 1138.8 = [569.4² + 246.5² - 322.9²] / 1138.8 = 246.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 246.5. ---600. What is the positive square root of [3710² + 60² - 445200]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3710, and b = 60, we see that 2ab = 445200, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3710² + 60² - 445200] = [3710 - 60]². Therefore, the positive square root of the expression given is equal to [3710 60] = 3650. ---601. Please find the value of 89997², by using algebraic identities to simplify computation. -Solution: Since 89997 is only a little lower than 90000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 90000 and b = 3. After the substitution, we get 89997² = (90000 - 3)² =
90000² + 3² - (2 x 90000 x 3) = 8100000000 + 9 - 540000 = 8099460009. ---602. What is the value of 1.601² + 1.401² - (1.601 x 2.802)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.802 = 2 x 1.401. Using this fact, we can rewrite the given expression as 1.601² + 1.401² - (2 x 1.601 x 1.401). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.601² + 1.401² - (1.601 x 2.802) = 1.601² + 1.401² - (2 x 1.601 x 1.401) = (1.601 - 1.401)² = 0.2² = 0.04. ---603. What is the value of [549.8² + 156.2² - 393.6²] / [1099.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 549.8, 156.2, and 393.6. We note that 393.6 = 549.8 - 156.2. We also note, in the denominator, that 1099.6 = 2 x 549.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 549.8, and b = 156.2, to get [(549.8² + 156.2² - (549.8 - 156.2)²] / 1099.6 = [549.8² + 156.2² - 393.6²] / 1099.6 = 156.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 156.2. ---604. What is the positive square root of [1968² + 50² - 196800]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1968, and b = 50, we see that 2ab = 196800, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1968² + 50² - 196800] = [1968 - 50]². Therefore, the positive square root of the expression given is equal to [1968 50] = 1918. ----
605. Please find the value of 7997², by using algebraic identities to simplify computation. -Solution: Since 7997 is only a little lower than 8000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 8000 and b = 3. After the substitution, we get 7997² = (8000 - 3)² = 8000² + 3² - (2 x 8000 x 3) = 64000000 + 9 - 48000 = 63952009. ---606. What is the value of 3.063² + 1.663² - (3.063 x 3.326)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.326 = 2 x 1.663. Using this fact, we can rewrite the given expression as 3.063² + 1.663² - (2 x 3.063 x 1.663). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.063² + 1.663² - (3.063 x 3.326) = 3.063² + 1.663² - (2 x 3.063 x 1.663) = (3.063 - 1.663)² = 1.4² = 1.96. ---607. What is the value of [480.2² + 182.8² - 297.4²] / [960.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 480.2, 182.8, and 297.4. We note that 297.4 = 480.2 - 182.8. We also note, in the denominator, that 960.4 = 2 x 480.2. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 480.2, and b = 182.8, to get [(480.2² + 182.8² - (480.2 - 182.8)²] / 960.4 = [480.2² + 182.8² - 297.4²] / 960.4 = 182.8. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 182.8. ---608. What is the positive square root of [2834² + 80² - 453440]? -Solution: By observation, we can deduce that the given expression looks a
little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2834, and b = 80, we see that 2ab = 453440, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2834² + 80² - 453440] = [2834 - 80]². Therefore, the positive square root of the expression given is equal to [2834 80] = 2754. ---609. Please find the value of 59997², by using algebraic identities to simplify computation. -Solution: Since 59997 is only a little lower than 60000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 60000 and b = 3. After the substitution, we get 59997² = (60000 - 3)² = 60000² + 3² - (2 x 60000 x 3) = 3600000000 + 9 - 360000 = 3599640009. ---610. What is the value of 2.547² + 2.347² - (2.547 x 4.694)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.694 = 2 x 2.347. Using this fact, we can rewrite the given expression as 2.547² + 2.347² - (2 x 2.547 x 2.347). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.547² + 2.347² - (2.547 x 4.694) = 2.547² + 2.347² - (2 x 2.547 x 2.347) = (2.547 - 2.347)² = 0.2² = 0.0400000000000001. ---611. What is the value of [563.7² + 270.1² - 293.6²] / [1127.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 563.7, 270.1, and 293.6. We note that 293.6 = 563.7 - 270.1. We also note, in the denominator, that 1127.4 = 2 x 563.7. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 563.7, and b = 270.1, to get [(563.7²
+ 270.1² - (563.7 - 270.1)²] / 1127.4 = [563.7² + 270.1² - 293.6²] / 1127.4 = 270.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 270.1. ---612. What is the positive square root of [2449² + 40² - 195920]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2449, and b = 40, we see that 2ab = 195920, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2449² + 40² - 195920] = [2449 - 40]². Therefore, the positive square root of the expression given is equal to [2449 40] = 2409. ---613. Please find the value of 8997², by using algebraic identities to simplify computation. -Solution: Since 8997 is only a little lower than 9000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 9000 and b = 3. After the substitution, we get 8997² = (9000 - 3)² = 9000² + 3² - (2 x 9000 x 3) = 81000000 + 9 - 54000 = 80946009. ---614. What is the value of 3.444² + 2.444² - (3.444 x 4.888)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.888 = 2 x 2.444. Using this fact, we can rewrite the given expression as 3.444² + 2.444² - (2 x 3.444 x 2.444). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.444² + 2.444² - (3.444 x 4.888) = 3.444² + 2.444² - (2 x 3.444 x 2.444) = (3.444 - 2.444)² = 1² = 1. ---615. What is the value of [571.6² + 224.2² - 347.4²] / [1143.2]? --
Solution: First, we look at the three numbers under the square signs in the numerator: 571.6, 224.2, and 347.4. We note that 347.4 = 571.6 - 224.2. We also note, in the denominator, that 1143.2 = 2 x 571.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 571.6, and b = 224.2, to get [(571.6² + 224.2² - (571.6 - 224.2)²] / 1143.2 = [571.6² + 224.2² - 347.4²] / 1143.2 = 224.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 224.2. ---616. What is the positive square root of [3068² + 80² - 490880]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3068, and b = 80, we see that 2ab = 490880, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3068² + 80² - 490880] = [3068 - 80]². Therefore, the positive square root of the expression given is equal to [3068 80] = 2988. ---617. Please find the value of 49999², by using algebraic identities to simplify computation. -Solution: Since 49999 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 1. After the substitution, we get 49999² = (50000 - 1)² = 50000² + 1² - (2 x 50000 x 1) = 2500000000 + 1 - 100000 = 2499900001. ---618. What is the value of 1.942² + 0.942² - (1.942 x 1.884)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.884 = 2 x 0.942. Using this fact, we can rewrite the given expression as 1.942² + 0.942² - (2 x 1.942 x 0.942).
Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.942² + 0.942² - (1.942 x 1.884) = 1.942² + 0.942² - (2 x 1.942 x 0.942) = (1.942 - 0.942)² = 1² = 1. ---619. What is the value of [536.7² + 229.3² - 307.4²] / [1073.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 536.7, 229.3, and 307.4. We note that 307.4 = 536.7 - 229.3. We also note, in the denominator, that 1073.4 = 2 x 536.7. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 536.7, and b = 229.3, to get [(536.7² + 229.3² - (536.7 - 229.3)²] / 1073.4 = [536.7² + 229.3² - 307.4²] / 1073.4 = 229.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 229.3. ---620. What is the positive square root of [2287² + 50² - 228700]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2287, and b = 50, we see that 2ab = 228700, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2287² + 50² - 228700] = [2287 - 50]². Therefore, the positive square root of the expression given is equal to [2287 50] = 2237. ---621. Please find the value of 298², by using algebraic identities to simplify computation. -Solution: Since 298 is only a little lower than 300, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 2. After the substitution, we get 298² = (300 - 2)² = 300² + 2² - (2 x
300 x 2) = 90000 + 4 - 1200 = 88804. ---622. What is the value of 1.695² + 1.295² - (1.695 x 2.59)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.59 = 2 x 1.295. Using this fact, we can rewrite the given expression as 1.695² + 1.295² - (2 x 1.695 x 1.295). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.695² + 1.295² - (1.695 x 2.59) = 1.695² + 1.295² - (2 x 1.695 x 1.295) = (1.695 - 1.295)² = 0.4² = 0.16. ---623. What is the value of [420.1² + 168.4² - 251.7²] / [840.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 420.1, 168.4, and 251.7. We note that 251.7 = 420.1 - 168.4. We also note, in the denominator, that 840.2 = 2 x 420.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 420.1, and b = 168.4, to get [(420.1² + 168.4² - (420.1 - 168.4)²] / 840.2 = [420.1² + 168.4² - 251.7²] / 840.2 = 168.4. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 168.4. ---624. What is the positive square root of [1845² + 40² - 147600]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1845, and b = 40, we see that 2ab = 147600, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1845² + 40² - 147600] = [1845 - 40]². Therefore, the positive square root of the expression given is equal to [1845 40] = 1805. ----
625. Please find the value of 397², by using algebraic identities to simplify computation. -Solution: Since 397 is only a little lower than 400, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 400 and b = 3. After the substitution, we get 397² = (400 - 3)² = 400² + 3² - (2 x 400 x 3) = 160000 + 9 - 2400 = 157609. ---626. What is the value of 1.987² + 1.587² - (1.987 x 3.174)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.174 = 2 x 1.587. Using this fact, we can rewrite the given expression as 1.987² + 1.587² - (2 x 1.987 x 1.587). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.987² + 1.587² - (1.987 x 3.174) = 1.987² + 1.587² - (2 x 1.987 x 1.587) = (1.987 - 1.587)² = 0.4² = 0.16. ---627. What is the value of [542.1² + 282.3² - 259.8²] / [1084.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 542.1, 282.3, and 259.8. We note that 259.8 = 542.1 - 282.3. We also note, in the denominator, that 1084.2 = 2 x 542.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 542.1, and b = 282.3, to get [(542.1² + 282.3² - (542.1 - 282.3)²] / 1084.2 = [542.1² + 282.3² - 259.8²] / 1084.2 = 282.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 282.3. ---628. What is the positive square root of [1749² + 80² - 279840]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a
= 1749, and b = 80, we see that 2ab = 279840, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1749² + 80² - 279840] = [1749 - 80]². Therefore, the positive square root of the expression given is equal to [1749 80] = 1669. ---629. Please find the value of 4997², by using algebraic identities to simplify computation. -Solution: Since 4997 is only a little lower than 5000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 5000 and b = 3. After the substitution, we get 4997² = (5000 - 3)² = 5000² + 3² - (2 x 5000 x 3) = 25000000 + 9 - 30000 = 24970009. ---630. What is the value of 2.174² + 0.974² - (2.174 x 1.948)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.948 = 2 x 0.974. Using this fact, we can rewrite the given expression as 2.174² + 0.974² - (2 x 2.174 x 0.974). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.174² + 0.974² - (2.174 x 1.948) = 2.174² + 0.974² - (2 x 2.174 x 0.974) = (2.174 - 0.974)² = 1.2² = 1.44. ---631. What is the value of [476.9² + 153.2² - 323.7²] / [953.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 476.9, 153.2, and 323.7. We note that 323.7 = 476.9 - 153.2. We also note, in the denominator, that 953.8 = 2 x 476.9. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 476.9, and b = 153.2, to get [(476.9² + 153.2² - (476.9 - 153.2)²] / 953.8 = [476.9² + 153.2² - 323.7²] / 953.8 =
153.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 153.2. ---632. What is the positive square root of [3840² + 50² - 384000]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3840, and b = 50, we see that 2ab = 384000, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3840² + 50² - 384000] = [3840 - 50]². Therefore, the positive square root of the expression given is equal to [3840 50] = 3790. ---633. Please find the value of 298², by using algebraic identities to simplify computation. -Solution: Since 298 is only a little lower than 300, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 2. After the substitution, we get 298² = (300 - 2)² = 300² + 2² - (2 x 300 x 2) = 90000 + 4 - 1200 = 88804. ---634. What is the value of 2.084² + 1.484² - (2.084 x 2.968)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.968 = 2 x 1.484. Using this fact, we can rewrite the given expression as 2.084² + 1.484² - (2 x 2.084 x 1.484). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.084² + 1.484² - (2.084 x 2.968) = 2.084² + 1.484² - (2 x 2.084 x 1.484) = (2.084 - 1.484)² = 0.6² = 0.36. ---635. What is the value of [512.8² + 203.9² - 308.9²] / [1025.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 512.8, 203.9, and 308.9. We note that 308.9 = 512.8 - 203.9. We
also note, in the denominator, that 1025.6 = 2 x 512.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 512.8, and b = 203.9, to get [(512.8² + 203.9² - (512.8 - 203.9)²] / 1025.6 = [512.8² + 203.9² - 308.9²] / 1025.6 = 203.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 203.9. ---636. What is the positive square root of [2359² + 60² - 283080]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2359, and b = 60, we see that 2ab = 283080, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2359² + 60² - 283080] = [2359 - 60]². Therefore, the positive square root of the expression given is equal to [2359 60] = 2299. ---637. Please find the value of 698², by using algebraic identities to simplify computation. -Solution: Since 698 is only a little lower than 700, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 2. After the substitution, we get 698² = (700 - 2)² = 700² + 2² - (2 x 700 x 2) = 490000 + 4 - 2800 = 487204. ---638. What is the value of 1.861² + 0.861² - (1.861 x 1.722)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.722 = 2 x 0.861. Using this fact, we can rewrite the given expression as 1.861² + 0.861² - (2 x 1.861 x 0.861). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² -
2ab), we get 1.861² + 0.861² - (1.861 x 1.722) = 1.861² + 0.861² - (2 x 1.861 x 0.861) = (1.861 - 0.861)² = 1² = 1. ---639. What is the value of [536.5² + 266.7² - 269.8²] / [1073]? -Solution: First, we look at the three numbers under the square signs in the numerator: 536.5, 266.7, and 269.8. We note that 269.8 = 536.5 - 266.7. We also note, in the denominator, that 1073 = 2 x 536.5. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 536.5, and b = 266.7, to get [(536.5² + 266.7² - (536.5 - 266.7)²] / 1073 = [536.5² + 266.7² - 269.8²] / 1073 = 266.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 266.7. ---640. What is the positive square root of [3250² + 70² - 455000]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3250, and b = 70, we see that 2ab = 455000, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3250² + 70² - 455000] = [3250 - 70]². Therefore, the positive square root of the expression given is equal to [3250 70] = 3180. ---641. Please find the value of 798², by using algebraic identities to simplify computation. -Solution: Since 798 is only a little lower than 800, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 800 and b = 2. After the substitution, we get 798² = (800 - 2)² = 800² + 2² - (2 x 800 x 2) = 640000 + 4 - 3200 = 636804. ----
642. What is the value of 3.057² + 2.057² - (3.057 x 4.114)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.114 = 2 x 2.057. Using this fact, we can rewrite the given expression as 3.057² + 2.057² - (2 x 3.057 x 2.057). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.057² + 2.057² - (3.057 x 4.114) = 3.057² + 2.057² - (2 x 3.057 x 2.057) = (3.057 - 2.057)² = 1² = 1. ---643. What is the value of [456.4² + 294.7² - 161.7²] / [912.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 456.4, 294.7, and 161.7. We note that 161.7 = 456.4 - 294.7. We also note, in the denominator, that 912.8 = 2 x 456.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 456.4, and b = 294.7, to get [(456.4² + 294.7² - (456.4 - 294.7)²] / 912.8 = [456.4² + 294.7² - 161.7²] / 912.8 = 294.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 294.7. ---644. What is the positive square root of [2740² + 70² - 383600]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2740, and b = 70, we see that 2ab = 383600, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2740² + 70² - 383600] = [2740 - 70]². Therefore, the positive square root of the expression given is equal to [2740 70] = 2670. ---645. Please find the value of 4998², by using algebraic identities to simplify computation.
-Solution: Since 4998 is only a little lower than 5000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 5000 and b = 2. After the substitution, we get 4998² = (5000 - 2)² = 5000² + 2² - (2 x 5000 x 2) = 25000000 + 4 - 20000 = 24980004. ---646. What is the value of 2.005² + 1.605² - (2.005 x 3.21)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.21 = 2 x 1.605. Using this fact, we can rewrite the given expression as 2.005² + 1.605² - (2 x 2.005 x 1.605). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.005² + 1.605² - (2.005 x 3.21) = 2.005² + 1.605² - (2 x 2.005 x 1.605) = (2.005 - 1.605)² = 0.4² = 0.16. ---647. What is the value of [431.2² + 165² - 266.2²] / [862.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 431.2, 165, and 266.2. We note that 266.2 = 431.2 - 165. We also note, in the denominator, that 862.4 = 2 x 431.2. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 431.2, and b = 165, to get [(431.2² + 165² - (431.2 - 165)²] / 862.4 = [431.2² + 165² - 266.2²] / 862.4 = 165. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 165. ---648. What is the positive square root of [2636² + 20² - 105440]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2636, and b = 20, we see that 2ab = 105440, which is exactly equal to the
absolute value of the third term in the expression. Hence, we can say that [2636² + 20² - 105440] = [2636 - 20]². Therefore, the positive square root of the expression given is equal to [2636 20] = 2616. ---649. Please find the value of 2997², by using algebraic identities to simplify computation. -Solution: Since 2997 is only a little lower than 3000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 3000 and b = 3. After the substitution, we get 2997² = (3000 - 3)² = 3000² + 3² - (2 x 3000 x 3) = 9000000 + 9 - 18000 = 8982009. ---650. What is the value of 2.758² + 2.158² - (2.758 x 4.316)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.316 = 2 x 2.158. Using this fact, we can rewrite the given expression as 2.758² + 2.158² - (2 x 2.758 x 2.158). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.758² + 2.158² - (2.758 x 4.316) = 2.758² + 2.158² - (2 x 2.758 x 2.158) = (2.758 - 2.158)² = 0.6² = 0.36. ---651. What is the value of [542.4² + 152.4² - 390²] / [1084.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 542.4, 152.4, and 390. We note that 390 = 542.4 - 152.4. We also note, in the denominator, that 1084.8 = 2 x 542.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 542.4, and b = 152.4, to get [(542.4² + 152.4² - (542.4 - 152.4)²] / 1084.8 = [542.4² + 152.4² - 390²] / 1084.8 = 152.4. The Left Hand Side of this equation is exactly the expression in our
question, so the value of that expression is equal to 152.4. ---652. What is the positive square root of [3991² + 50² - 399100]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3991, and b = 50, we see that 2ab = 399100, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3991² + 50² - 399100] = [3991 - 50]². Therefore, the positive square root of the expression given is equal to [3991 50] = 3941. ---653. Please find the value of 397², by using algebraic identities to simplify computation. -Solution: Since 397 is only a little lower than 400, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 400 and b = 3. After the substitution, we get 397² = (400 - 3)² = 400² + 3² - (2 x 400 x 3) = 160000 + 9 - 2400 = 157609. ---654. What is the value of 2.454² + 2.254² - (2.454 x 4.508)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.508 = 2 x 2.254. Using this fact, we can rewrite the given expression as 2.454² + 2.254² - (2 x 2.454 x 2.254). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.454² + 2.254² - (2.454 x 4.508) = 2.454² + 2.254² - (2 x 2.454 x 2.254) = (2.454 - 2.254)² = 0.2² = 0.0400000000000001. ---655. What is the value of [489.5² + 241.2² - 248.3²] / [979]? -Solution: First, we look at the three numbers under the square signs in the numerator: 489.5, 241.2, and 248.3. We note that 248.3 = 489.5 - 241.2. We also note, in the denominator, that 979 = 2 x 489.5.
Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 489.5, and b = 241.2, to get [(489.5² + 241.2² - (489.5 - 241.2)²] / 979 = [489.5² + 241.2² - 248.3²] / 979 = 241.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 241.2. ---656. What is the positive square root of [3019² + 50² - 301900]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3019, and b = 50, we see that 2ab = 301900, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3019² + 50² - 301900] = [3019 - 50]². Therefore, the positive square root of the expression given is equal to [3019 50] = 2969. ---657. Please find the value of 7997², by using algebraic identities to simplify computation. -Solution: Since 7997 is only a little lower than 8000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 8000 and b = 3. After the substitution, we get 7997² = (8000 - 3)² = 8000² + 3² - (2 x 8000 x 3) = 64000000 + 9 - 48000 = 63952009. ---658. What is the value of 1.814² + 1.014² - (1.814 x 2.028)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.028 = 2 x 1.014. Using this fact, we can rewrite the given expression as 1.814² + 1.014² - (2 x 1.814 x 1.014). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.814² + 1.014² - (1.814 x 2.028) = 1.814² + 1.014² - (2 x 1.814
x 1.014) = (1.814 - 1.014)² = 0.8² = 0.64. ---659. What is the value of [434.2² + 187.1² - 247.1²] / [868.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 434.2, 187.1, and 247.1. We note that 247.1 = 434.2 - 187.1. We also note, in the denominator, that 868.4 = 2 x 434.2. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 434.2, and b = 187.1, to get [(434.2² + 187.1² - (434.2 - 187.1)²] / 868.4 = [434.2² + 187.1² - 247.1²] / 868.4 = 187.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 187.1. ---660. What is the positive square root of [3091² + 60² - 370920]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3091, and b = 60, we see that 2ab = 370920, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3091² + 60² - 370920] = [3091 - 60]². Therefore, the positive square root of the expression given is equal to [3091 60] = 3031. ---661. Please find the value of 297², by using algebraic identities to simplify computation. -Solution: Since 297 is only a little lower than 300, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 300 and b = 3. After the substitution, we get 297² = (300 - 3)² = 300² + 3² - (2 x 300 x 3) = 90000 + 9 - 1800 = 88209. ----
662. What is the value of 1.534² + 1.334² - (1.534 x 2.668)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.668 = 2 x 1.334. Using this fact, we can rewrite the given expression as 1.534² + 1.334² - (2 x 1.534 x 1.334). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.534² + 1.334² - (1.534 x 2.668) = 1.534² + 1.334² - (2 x 1.534 x 1.334) = (1.534 - 1.334)² = 0.2² = 0.04. ---663. What is the value of [442.9² + 140.1² - 302.8²] / [885.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 442.9, 140.1, and 302.8. We note that 302.8 = 442.9 - 140.1. We also note, in the denominator, that 885.8 = 2 x 442.9. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 442.9, and b = 140.1, to get [(442.9² + 140.1² - (442.9 - 140.1)²] / 885.8 = [442.9² + 140.1² - 302.8²] / 885.8 = 140.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 140.1. ---664. What is the positive square root of [3779² + 90² - 680220]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3779, and b = 90, we see that 2ab = 680220, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3779² + 90² - 680220] = [3779 - 90]². Therefore, the positive square root of the expression given is equal to [3779 90] = 3689. ---665. Please find the value of 4998², by using algebraic identities to simplify computation.
-Solution: Since 4998 is only a little lower than 5000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 5000 and b = 2. After the substitution, we get 4998² = (5000 - 2)² = 5000² + 2² - (2 x 5000 x 2) = 25000000 + 4 - 20000 = 24980004. ---666. What is the value of 3.457² + 3.057² - (3.457 x 6.114)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.114 = 2 x 3.057. Using this fact, we can rewrite the given expression as 3.457² + 3.057² - (2 x 3.457 x 3.057). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.457² + 3.057² - (3.457 x 6.114) = 3.457² + 3.057² - (2 x 3.457 x 3.057) = (3.457 - 3.057)² = 0.4² = 0.16. ---667. What is the value of [544.8² + 297.1² - 247.7²] / [1089.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 544.8, 297.1, and 247.7. We note that 247.7 = 544.8 - 297.1. We also note, in the denominator, that 1089.6 = 2 x 544.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 544.8, and b = 297.1, to get [(544.8² + 297.1² - (544.8 - 297.1)²] / 1089.6 = [544.8² + 297.1² - 247.7²] / 1089.6 = 297.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 297.1. ---668. What is the positive square root of [2373² + 20² - 94920]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2373, and b = 20, we see that 2ab = 94920, which is exactly equal to the absolute value of the third term in the expression.
Hence, we can say that [2373² + 20² - 94920] = [2373 - 20]². Therefore, the positive square root of the expression given is equal to [2373 20] = 2353. ---669. Please find the value of 39998², by using algebraic identities to simplify computation. -Solution: Since 39998 is only a little lower than 40000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 40000 and b = 2. After the substitution, we get 39998² = (40000 - 2)² = 40000² + 2² - (2 x 40000 x 2) = 1600000000 + 4 - 160000 = 1599840004. ---670. What is the value of 3.413² + 2.613² - (3.413 x 5.226)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.226 = 2 x 2.613. Using this fact, we can rewrite the given expression as 3.413² + 2.613² - (2 x 3.413 x 2.613). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.413² + 2.613² - (3.413 x 5.226) = 3.413² + 2.613² - (2 x 3.413 x 2.613) = (3.413 - 2.613)² = 0.8² = 0.640000000000001. ---671. What is the value of [571.5² + 263.6² - 307.9²] / [1143]? -Solution: First, we look at the three numbers under the square signs in the numerator: 571.5, 263.6, and 307.9. We note that 307.9 = 571.5 - 263.6. We also note, in the denominator, that 1143 = 2 x 571.5. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 571.5, and b = 263.6, to get [(571.5² + 263.6² - (571.5 - 263.6)²] / 1143 = [571.5² + 263.6² - 307.9²] / 1143 = 263.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 263.6. ----
672. What is the positive square root of [3167² + 60² - 380040]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3167, and b = 60, we see that 2ab = 380040, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3167² + 60² - 380040] = [3167 - 60]². Therefore, the positive square root of the expression given is equal to [3167 60] = 3107. ---673. Please find the value of 49997², by using algebraic identities to simplify computation. -Solution: Since 49997 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 3. After the substitution, we get 49997² = (50000 - 3)² = 50000² + 3² - (2 x 50000 x 3) = 2500000000 + 9 - 300000 = 2499700009. ---674. What is the value of 3.506² + 3.306² - (3.506 x 6.612)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 6.612 = 2 x 3.306. Using this fact, we can rewrite the given expression as 3.506² + 3.306² - (2 x 3.506 x 3.306). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.506² + 3.306² - (3.506 x 6.612) = 3.506² + 3.306² - (2 x 3.506 x 3.306) = (3.506 - 3.306)² = 0.2² = 0.0400000000000001. ---675. What is the value of [525.3² + 176.7² - 348.6²] / [1050.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 525.3, 176.7, and 348.6. We note that 348.6 = 525.3 - 176.7. We also note, in the denominator, that 1050.6 = 2 x 525.3. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get
something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 525.3, and b = 176.7, to get [(525.3² + 176.7² - (525.3 - 176.7)²] / 1050.6 = [525.3² + 176.7² - 348.6²] / 1050.6 = 176.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 176.7. ---676. What is the positive square root of [1906² + 20² - 76240]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1906, and b = 20, we see that 2ab = 76240, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1906² + 20² - 76240] = [1906 - 20]². Therefore, the positive square root of the expression given is equal to [1906 20] = 1886. ---677. Please find the value of 798², by using algebraic identities to simplify computation. -Solution: Since 798 is only a little lower than 800, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 800 and b = 2. After the substitution, we get 798² = (800 - 2)² = 800² + 2² - (2 x 800 x 2) = 640000 + 4 - 3200 = 636804. ---678. What is the value of 2.612² + 2.012² - (2.612 x 4.024)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.024 = 2 x 2.012. Using this fact, we can rewrite the given expression as 2.612² + 2.012² - (2 x 2.612 x 2.012). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.612² + 2.012² - (2.612 x 4.024) = 2.612² + 2.012² - (2 x 2.612 x 2.012) = (2.612 - 2.012)² = 0.6² = 0.36. ----
679. What is the value of [426.6² + 160.7² - 265.9²] / [853.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 426.6, 160.7, and 265.9. We note that 265.9 = 426.6 - 160.7. We also note, in the denominator, that 853.2 = 2 x 426.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 426.6, and b = 160.7, to get [(426.6² + 160.7² - (426.6 - 160.7)²] / 853.2 = [426.6² + 160.7² - 265.9²] / 853.2 = 160.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 160.7. ---680. What is the positive square root of [2986² + 20² - 119440]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2986, and b = 20, we see that 2ab = 119440, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2986² + 20² - 119440] = [2986 - 20]². Therefore, the positive square root of the expression given is equal to [2986 20] = 2966. ---681. Please find the value of 399², by using algebraic identities to simplify computation. -Solution: Since 399 is only a little lower than 400, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 400 and b = 1. After the substitution, we get 399² = (400 - 1)² = 400² + 1² - (2 x 400 x 1) = 160000 + 1 - 800 = 159201. ---682. What is the value of 1.592² + 1.392² - (1.592 x 2.784)? Please use algebraic identities to make the computation simple. --
Solution: First, we note that 2.784 = 2 x 1.392. Using this fact, we can rewrite the given expression as 1.592² + 1.392² - (2 x 1.592 x 1.392). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.592² + 1.392² - (1.592 x 2.784) = 1.592² + 1.392² - (2 x 1.592 x 1.392) = (1.592 - 1.392)² = 0.2² = 0.04. ---683. What is the value of [567.8² + 199.4² - 368.4²] / [1135.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 567.8, 199.4, and 368.4. We note that 368.4 = 567.8 - 199.4. We also note, in the denominator, that 1135.6 = 2 x 567.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 567.8, and b = 199.4, to get [(567.8² + 199.4² - (567.8 - 199.4)²] / 1135.6 = [567.8² + 199.4² - 368.4²] / 1135.6 = 199.4. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 199.4. ---684. What is the positive square root of [3168² + 50² - 316800]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3168, and b = 50, we see that 2ab = 316800, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3168² + 50² - 316800] = [3168 - 50]². Therefore, the positive square root of the expression given is equal to [3168 50] = 3118. ---685. Please find the value of 6997², by using algebraic identities to simplify computation. -Solution: Since 6997 is only a little lower than 7000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately
chosen values of a and b. To do this particular computation, we put a = 7000 and b = 3. After the substitution, we get 6997² = (7000 - 3)² = 7000² + 3² - (2 x 7000 x 3) = 49000000 + 9 - 42000 = 48958009. ---686. What is the value of 2.219² + 1.419² - (2.219 x 2.838)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.838 = 2 x 1.419. Using this fact, we can rewrite the given expression as 2.219² + 1.419² - (2 x 2.219 x 1.419). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.219² + 1.419² - (2.219 x 2.838) = 2.219² + 1.419² - (2 x 2.219 x 1.419) = (2.219 - 1.419)² = 0.8² = 0.64. ---687. What is the value of [473.2² + 292.2² - 181²] / [946.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 473.2, 292.2, and 181. We note that 181 = 473.2 - 292.2. We also note, in the denominator, that 946.4 = 2 x 473.2. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 473.2, and b = 292.2, to get [(473.2² + 292.2² - (473.2 - 292.2)²] / 946.4 = [473.2² + 292.2² - 181²] / 946.4 = 292.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 292.2. ---688. What is the positive square root of [1795² + 30² - 107700]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1795, and b = 30, we see that 2ab = 107700, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1795² + 30² - 107700] = [1795 - 30]². Therefore, the positive square root of the expression given is equal to [1795 30] = 1765.
---689. Please find the value of 6999², by using algebraic identities to simplify computation. -Solution: Since 6999 is only a little lower than 7000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 1. After the substitution, we get 6999² = (7000 - 1)² = 7000² + 1² - (2 x 7000 x 1) = 49000000 + 1 - 14000 = 48986001. ---690. What is the value of 2.105² + 1.505² - (2.105 x 3.01)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.01 = 2 x 1.505. Using this fact, we can rewrite the given expression as 2.105² + 1.505² - (2 x 2.105 x 1.505). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.105² + 1.505² - (2.105 x 3.01) = 2.105² + 1.505² - (2 x 2.105 x 1.505) = (2.105 - 1.505)² = 0.6² = 0.36. ---691. What is the value of [512.7² + 158.3² - 354.4²] / [1025.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 512.7, 158.3, and 354.4. We note that 354.4 = 512.7 - 158.3. We also note, in the denominator, that 1025.4 = 2 x 512.7. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 512.7, and b = 158.3, to get [(512.7² + 158.3² - (512.7 - 158.3)²] / 1025.4 = [512.7² + 158.3² - 354.4²] / 1025.4 = 158.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 158.3. ---692. What is the positive square root of [3399² + 90² - 611820]? --
Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3399, and b = 90, we see that 2ab = 611820, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3399² + 90² - 611820] = [3399 - 90]². Therefore, the positive square root of the expression given is equal to [3399 90] = 3309. ---693. Please find the value of 497², by using algebraic identities to simplify computation. -Solution: Since 497 is only a little lower than 500, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 500 and b = 3. After the substitution, we get 497² = (500 - 3)² = 500² + 3² - (2 x 500 x 3) = 250000 + 9 - 3000 = 247009. ---694. What is the value of 2.656² + 1.656² - (2.656 x 3.312)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.312 = 2 x 1.656. Using this fact, we can rewrite the given expression as 2.656² + 1.656² - (2 x 2.656 x 1.656). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.656² + 1.656² - (2.656 x 3.312) = 2.656² + 1.656² - (2 x 2.656 x 1.656) = (2.656 - 1.656)² = 1² = 1. ---695. What is the value of [592.8² + 263.6² - 329.2²] / [1185.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 592.8, 263.6, and 329.2. We note that 329.2 = 592.8 - 263.6. We also note, in the denominator, that 1185.6 = 2 x 592.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 592.8, and b = 263.6, to get [(592.8²
+ 263.6² - (592.8 - 263.6)²] / 1185.6 = [592.8² + 263.6² - 329.2²] / 1185.6 = 263.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 263.6. ---696. What is the positive square root of [3142² + 40² - 251360]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3142, and b = 40, we see that 2ab = 251360, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3142² + 40² - 251360] = [3142 - 40]². Therefore, the positive square root of the expression given is equal to [3142 40] = 3102. ---697. Please find the value of 798², by using algebraic identities to simplify computation. -Solution: Since 798 is only a little lower than 800, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 800 and b = 2. After the substitution, we get 798² = (800 - 2)² = 800² + 2² - (2 x 800 x 2) = 640000 + 4 - 3200 = 636804. ---698. What is the value of 3.149² + 2.549² - (3.149 x 5.098)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.098 = 2 x 2.549. Using this fact, we can rewrite the given expression as 3.149² + 2.549² - (2 x 3.149 x 2.549). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.149² + 2.549² - (3.149 x 5.098) = 3.149² + 2.549² - (2 x 3.149 x 2.549) = (3.149 - 2.549)² = 0.6² = 0.36. ---699. What is the value of [488.4² + 294.5² - 193.9²] / [976.8]? -Solution: First, we look at the three numbers under the square signs in the
numerator: 488.4, 294.5, and 193.9. We note that 193.9 = 488.4 - 294.5. We also note, in the denominator, that 976.8 = 2 x 488.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 488.4, and b = 294.5, to get [(488.4² + 294.5² - (488.4 - 294.5)²] / 976.8 = [488.4² + 294.5² - 193.9²] / 976.8 = 294.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 294.5. ---700. What is the positive square root of [3394² + 60² - 407280]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3394, and b = 60, we see that 2ab = 407280, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3394² + 60² - 407280] = [3394 - 60]². Therefore, the positive square root of the expression given is equal to [3394 60] = 3334. ---701. Please find the value of 7998², by using algebraic identities to simplify computation. -Solution: Since 7998 is only a little lower than 8000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 8000 and b = 2. After the substitution, we get 7998² = (8000 - 2)² = 8000² + 2² - (2 x 8000 x 2) = 64000000 + 4 - 32000 = 63968004. ---702. What is the value of 2.059² + 0.859² - (2.059 x 1.718)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.718 = 2 x 0.859. Using this fact, we can rewrite the given expression as 2.059² + 0.859² - (2 x 2.059 x 0.859). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² -
2ab), we get 2.059² + 0.859² - (2.059 x 1.718) = 2.059² + 0.859² - (2 x 2.059 x 0.859) = (2.059 - 0.859)² = 1.2² = 1.44. ---703. What is the value of [500.4² + 157.8² - 342.6²] / [1000.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 500.4, 157.8, and 342.6. We note that 342.6 = 500.4 - 157.8. We also note, in the denominator, that 1000.8 = 2 x 500.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 500.4, and b = 157.8, to get [(500.4² + 157.8² - (500.4 - 157.8)²] / 1000.8 = [500.4² + 157.8² - 342.6²] / 1000.8 = 157.8. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 157.8. ---704. What is the positive square root of [2626² + 60² - 315120]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2626, and b = 60, we see that 2ab = 315120, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2626² + 60² - 315120] = [2626 - 60]². Therefore, the positive square root of the expression given is equal to [2626 60] = 2566. ---705. Please find the value of 697², by using algebraic identities to simplify computation. -Solution: Since 697 is only a little lower than 700, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 3. After the substitution, we get 697² = (700 - 3)² = 700² + 3² - (2 x 700 x 3) = 490000 + 9 - 4200 = 485809. ----
706. What is the value of 2.625² + 1.825² - (2.625 x 3.65)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.65 = 2 x 1.825. Using this fact, we can rewrite the given expression as 2.625² + 1.825² - (2 x 2.625 x 1.825). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.625² + 1.825² - (2.625 x 3.65) = 2.625² + 1.825² - (2 x 2.625 x 1.825) = (2.625 - 1.825)² = 0.8² = 0.64. ---707. What is the value of [561.8² + 218² - 343.8²] / [1123.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 561.8, 218, and 343.8. We note that 343.8 = 561.8 - 218. We also note, in the denominator, that 1123.6 = 2 x 561.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 561.8, and b = 218, to get [(561.8² + 218² - (561.8 - 218)²] / 1123.6 = [561.8² + 218² - 343.8²] / 1123.6 = 218. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 218. ---708. What is the positive square root of [2658² + 60² - 318960]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2658, and b = 60, we see that 2ab = 318960, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2658² + 60² - 318960] = [2658 - 60]². Therefore, the positive square root of the expression given is equal to [2658 60] = 2598. ---709. Please find the value of 49998², by using algebraic identities to simplify
computation. -Solution: Since 49998 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 2. After the substitution, we get 49998² = (50000 - 2)² = 50000² + 2² - (2 x 50000 x 2) = 2500000000 + 4 - 200000 = 2499800004. ---710. What is the value of 3.051² + 2.451² - (3.051 x 4.902)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.902 = 2 x 2.451. Using this fact, we can rewrite the given expression as 3.051² + 2.451² - (2 x 3.051 x 2.451). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.051² + 2.451² - (3.051 x 4.902) = 3.051² + 2.451² - (2 x 3.051 x 2.451) = (3.051 - 2.451)² = 0.6² = 0.36. ---711. What is the value of [541.5² + 149.7² - 391.8²] / [1083]? -Solution: First, we look at the three numbers under the square signs in the numerator: 541.5, 149.7, and 391.8. We note that 391.8 = 541.5 - 149.7. We also note, in the denominator, that 1083 = 2 x 541.5. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 541.5, and b = 149.7, to get [(541.5² + 149.7² - (541.5 - 149.7)²] / 1083 = [541.5² + 149.7² - 391.8²] / 1083 = 149.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 149.7. ---712. What is the positive square root of [2967² + 60² - 356040]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2967, and b = 60, we see that 2ab = 356040, which is exactly equal to the
absolute value of the third term in the expression. Hence, we can say that [2967² + 60² - 356040] = [2967 - 60]². Therefore, the positive square root of the expression given is equal to [2967 60] = 2907. ---713. Please find the value of 6997², by using algebraic identities to simplify computation. -Solution: Since 6997 is only a little lower than 7000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 7000 and b = 3. After the substitution, we get 6997² = (7000 - 3)² = 7000² + 3² - (2 x 7000 x 3) = 49000000 + 9 - 42000 = 48958009. ---714. What is the value of 1.541² + 0.341² - (1.541 x 0.682)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.682 = 2 x 0.341. Using this fact, we can rewrite the given expression as 1.541² + 0.341² - (2 x 1.541 x 0.341). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.541² + 0.341² - (1.541 x 0.682) = 1.541² + 0.341² - (2 x 1.541 x 0.341) = (1.541 - 0.341)² = 1.2² = 1.44. ---715. What is the value of [508.4² + 160.2² - 348.2²] / [1016.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 508.4, 160.2, and 348.2. We note that 348.2 = 508.4 - 160.2. We also note, in the denominator, that 1016.8 = 2 x 508.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 508.4, and b = 160.2, to get [(508.4² + 160.2² - (508.4 - 160.2)²] / 1016.8 = [508.4² + 160.2² - 348.2²] / 1016.8 = 160.2. The Left Hand Side of this equation is exactly the expression in our
question, so the value of that expression is equal to 160.2. ---716. What is the positive square root of [2921² + 70² - 408940]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2921, and b = 70, we see that 2ab = 408940, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2921² + 70² - 408940] = [2921 - 70]². Therefore, the positive square root of the expression given is equal to [2921 70] = 2851. ---717. Please find the value of 79998², by using algebraic identities to simplify computation. -Solution: Since 79998 is only a little lower than 80000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 2. After the substitution, we get 79998² = (80000 - 2)² = 80000² + 2² - (2 x 80000 x 2) = 6400000000 + 4 - 320000 = 6399680004. ---718. What is the value of 2.496² + 1.896² - (2.496 x 3.792)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.792 = 2 x 1.896. Using this fact, we can rewrite the given expression as 2.496² + 1.896² - (2 x 2.496 x 1.896). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.496² + 1.896² - (2.496 x 3.792) = 2.496² + 1.896² - (2 x 2.496 x 1.896) = (2.496 - 1.896)² = 0.6² = 0.36. ---719. What is the value of [492.1² + 211.8² - 280.3²] / [984.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 492.1, 211.8, and 280.3. We note that 280.3 = 492.1 - 211.8. We also note, in the denominator, that 984.2 = 2 x 492.1.
Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 492.1, and b = 211.8, to get [(492.1² + 211.8² - (492.1 - 211.8)²] / 984.2 = [492.1² + 211.8² - 280.3²] / 984.2 = 211.8. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 211.8. ---720. What is the positive square root of [3079² + 30² - 184740]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3079, and b = 30, we see that 2ab = 184740, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3079² + 30² - 184740] = [3079 - 30]². Therefore, the positive square root of the expression given is equal to [3079 30] = 3049. ---721. Please find the value of 3999², by using algebraic identities to simplify computation. -Solution: Since 3999 is only a little lower than 4000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 4000 and b = 1. After the substitution, we get 3999² = (4000 - 1)² = 4000² + 1² - (2 x 4000 x 1) = 16000000 + 1 - 8000 = 15992001. ---722. What is the value of 2.653² + 1.653² - (2.653 x 3.306)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.306 = 2 x 1.653. Using this fact, we can rewrite the given expression as 2.653² + 1.653² - (2 x 2.653 x 1.653). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.653² + 1.653² - (2.653 x 3.306) = 2.653² + 1.653² - (2 x 2.653 x 1.653) = (2.653 - 1.653)² = 1² = 1.
---723. What is the value of [589.5² + 265.4² - 324.1²] / [1179]? -Solution: First, we look at the three numbers under the square signs in the numerator: 589.5, 265.4, and 324.1. We note that 324.1 = 589.5 - 265.4. We also note, in the denominator, that 1179 = 2 x 589.5. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 589.5, and b = 265.4, to get [(589.5² + 265.4² - (589.5 - 265.4)²] / 1179 = [589.5² + 265.4² - 324.1²] / 1179 = 265.4. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 265.4. ---724. What is the positive square root of [2326² + 30² - 139560]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2326, and b = 30, we see that 2ab = 139560, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2326² + 30² - 139560] = [2326 - 30]². Therefore, the positive square root of the expression given is equal to [2326 30] = 2296. ---725. Please find the value of 29999², by using algebraic identities to simplify computation. -Solution: Since 29999 is only a little lower than 30000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 30000 and b = 1. After the substitution, we get 29999² = (30000 - 1)² = 30000² + 1² - (2 x 30000 x 1) = 900000000 + 1 - 60000 = 899940001. ---726. What is the value of 1.48² + 0.28² - (1.48 x 0.56)? Please use algebraic
identities to make the computation simple. -Solution: First, we note that 0.56 = 2 x 0.28. Using this fact, we can rewrite the given expression as 1.48² + 0.28² - (2 x 1.48 x 0.28). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.48² + 0.28² - (1.48 x 0.56) = 1.48² + 0.28² - (2 x 1.48 x 0.28) = (1.48 - 0.28)² = 1.2² = 1.44. ---727. What is the value of [503.8² + 253.3² - 250.5²] / [1007.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 503.8, 253.3, and 250.5. We note that 250.5 = 503.8 - 253.3. We also note, in the denominator, that 1007.6 = 2 x 503.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 503.8, and b = 253.3, to get [(503.8² + 253.3² - (503.8 - 253.3)²] / 1007.6 = [503.8² + 253.3² - 250.5²] / 1007.6 = 253.3. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 253.3. ---728. What is the positive square root of [2682² + 20² - 107280]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2682, and b = 20, we see that 2ab = 107280, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2682² + 20² - 107280] = [2682 - 20]². Therefore, the positive square root of the expression given is equal to [2682 20] = 2662. ---729. Please find the value of 29998², by using algebraic identities to simplify computation. -Solution: Since 29998 is only a little lower than 30000, we can deduce that
the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 30000 and b = 2. After the substitution, we get 29998² = (30000 - 2)² = 30000² + 2² - (2 x 30000 x 2) = 900000000 + 4 - 120000 = 899880004. ---730. What is the value of 2.526² + 1.726² - (2.526 x 3.452)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.452 = 2 x 1.726. Using this fact, we can rewrite the given expression as 2.526² + 1.726² - (2 x 2.526 x 1.726). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.526² + 1.726² - (2.526 x 3.452) = 2.526² + 1.726² - (2 x 2.526 x 1.726) = (2.526 - 1.726)² = 0.8² = 0.64. ---731. What is the value of [499² + 293.5² - 205.5²] / [998]? -Solution: First, we look at the three numbers under the square signs in the numerator: 499, 293.5, and 205.5. We note that 205.5 = 499 - 293.5. We also note, in the denominator, that 998 = 2 x 499. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 499, and b = 293.5, to get [(499² + 293.5² - (499 - 293.5)²] / 998 = [499² + 293.5² - 205.5²] / 998 = 293.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 293.5. ---732. What is the positive square root of [2310² + 40² - 184800]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2310, and b = 40, we see that 2ab = 184800, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2310² + 40² - 184800] = [2310 - 40]². Therefore, the positive square root of the expression given is equal to [2310 -
40] = 2270. ---733. Please find the value of 7999², by using algebraic identities to simplify computation. -Solution: Since 7999 is only a little lower than 8000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 8000 and b = 1. After the substitution, we get 7999² = (8000 - 1)² = 8000² + 1² - (2 x 8000 x 1) = 64000000 + 1 - 16000 = 63984001. ---734. What is the value of 3.159² + 1.959² - (3.159 x 3.918)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.918 = 2 x 1.959. Using this fact, we can rewrite the given expression as 3.159² + 1.959² - (2 x 3.159 x 1.959). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.159² + 1.959² - (3.159 x 3.918) = 3.159² + 1.959² - (2 x 3.159 x 1.959) = (3.159 - 1.959)² = 1.2² = 1.44. ---735. What is the value of [596.1² + 157.5² - 438.6²] / [1192.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 596.1, 157.5, and 438.6. We note that 438.6 = 596.1 - 157.5. We also note, in the denominator, that 1192.2 = 2 x 596.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 596.1, and b = 157.5, to get [(596.1² + 157.5² - (596.1 - 157.5)²] / 1192.2 = [596.1² + 157.5² - 438.6²] / 1192.2 = 157.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 157.5. ---736. What is the positive square root of [3006² + 20² - 120240]?
-Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3006, and b = 20, we see that 2ab = 120240, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3006² + 20² - 120240] = [3006 - 20]². Therefore, the positive square root of the expression given is equal to [3006 20] = 2986. ---737. Please find the value of 4998², by using algebraic identities to simplify computation. -Solution: Since 4998 is only a little lower than 5000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 5000 and b = 2. After the substitution, we get 4998² = (5000 - 2)² = 5000² + 2² - (2 x 5000 x 2) = 25000000 + 4 - 20000 = 24980004. ---738. What is the value of 1.929² + 1.529² - (1.929 x 3.058)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.058 = 2 x 1.529. Using this fact, we can rewrite the given expression as 1.929² + 1.529² - (2 x 1.929 x 1.529). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.929² + 1.529² - (1.929 x 3.058) = 1.929² + 1.529² - (2 x 1.929 x 1.529) = (1.929 - 1.529)² = 0.4² = 0.16. ---739. What is the value of [462.6² + 219.9² - 242.7²] / [925.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 462.6, 219.9, and 242.7. We note that 242.7 = 462.6 - 219.9. We also note, in the denominator, that 925.2 = 2 x 462.6. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b.
In this rearranged equation, we put a = 462.6, and b = 219.9, to get [(462.6² + 219.9² - (462.6 - 219.9)²] / 925.2 = [462.6² + 219.9² - 242.7²] / 925.2 = 219.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 219.9. ---740. What is the positive square root of [2128² + 30² - 127680]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2128, and b = 30, we see that 2ab = 127680, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2128² + 30² - 127680] = [2128 - 30]². Therefore, the positive square root of the expression given is equal to [2128 30] = 2098. ---741. Please find the value of 7998², by using algebraic identities to simplify computation. -Solution: Since 7998 is only a little lower than 8000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 8000 and b = 2. After the substitution, we get 7998² = (8000 - 2)² = 8000² + 2² - (2 x 8000 x 2) = 64000000 + 4 - 32000 = 63968004. ---742. What is the value of 3.115² + 1.915² - (3.115 x 3.83)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.83 = 2 x 1.915. Using this fact, we can rewrite the given expression as 3.115² + 1.915² - (2 x 3.115 x 1.915). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.115² + 1.915² - (3.115 x 3.83) = 3.115² + 1.915² - (2 x 3.115 x 1.915) = (3.115 - 1.915)² = 1.2² = 1.44. ---743. What is the value of [541.3² + 229.8² - 311.5²] / [1082.6]? --
Solution: First, we look at the three numbers under the square signs in the numerator: 541.3, 229.8, and 311.5. We note that 311.5 = 541.3 - 229.8. We also note, in the denominator, that 1082.6 = 2 x 541.3. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 541.3, and b = 229.8, to get [(541.3² + 229.8² - (541.3 - 229.8)²] / 1082.6 = [541.3² + 229.8² - 311.5²] / 1082.6 = 229.8. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 229.8. ---744. What is the positive square root of [3039² + 30² - 182340]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3039, and b = 30, we see that 2ab = 182340, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3039² + 30² - 182340] = [3039 - 30]². Therefore, the positive square root of the expression given is equal to [3039 30] = 3009. ---745. Please find the value of 49997², by using algebraic identities to simplify computation. -Solution: Since 49997 is only a little lower than 50000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 50000 and b = 3. After the substitution, we get 49997² = (50000 - 3)² = 50000² + 3² - (2 x 50000 x 3) = 2500000000 + 9 - 300000 = 2499700009. ---746. What is the value of 3.188² + 2.188² - (3.188 x 4.376)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.376 = 2 x 2.188. Using this fact, we can rewrite the given expression as 3.188² + 2.188² - (2 x 3.188 x 2.188).
Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.188² + 2.188² - (3.188 x 4.376) = 3.188² + 2.188² - (2 x 3.188 x 2.188) = (3.188 - 2.188)² = 1² = 1. ---747. What is the value of [466² + 272.9² - 193.1²] / [932]? -Solution: First, we look at the three numbers under the square signs in the numerator: 466, 272.9, and 193.1. We note that 193.1 = 466 - 272.9. We also note, in the denominator, that 932 = 2 x 466. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 466, and b = 272.9, to get [(466² + 272.9² - (466 - 272.9)²] / 932 = [466² + 272.9² - 193.1²] / 932 = 272.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 272.9. ---748. What is the positive square root of [3959² + 40² - 316720]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3959, and b = 40, we see that 2ab = 316720, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3959² + 40² - 316720] = [3959 - 40]². Therefore, the positive square root of the expression given is equal to [3959 40] = 3919. ---749. Please find the value of 8997², by using algebraic identities to simplify computation. -Solution: Since 8997 is only a little lower than 9000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 9000 and b = 3. After the substitution, we get 8997² = (9000 - 3)² = 9000² + 3² - (2
x 9000 x 3) = 81000000 + 9 - 54000 = 80946009. ---750. What is the value of 1.817² + 0.417² - (1.817 x 0.834)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 0.834 = 2 x 0.417. Using this fact, we can rewrite the given expression as 1.817² + 0.417² - (2 x 1.817 x 0.417). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.817² + 0.417² - (1.817 x 0.834) = 1.817² + 0.417² - (2 x 1.817 x 0.417) = (1.817 - 0.417)² = 1.4² = 1.96. ---751. What is the value of [535.2² + 220.1² - 315.1²] / [1070.4]? -Solution: First, we look at the three numbers under the square signs in the numerator: 535.2, 220.1, and 315.1. We note that 315.1 = 535.2 - 220.1. We also note, in the denominator, that 1070.4 = 2 x 535.2. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 535.2, and b = 220.1, to get [(535.2² + 220.1² - (535.2 - 220.1)²] / 1070.4 = [535.2² + 220.1² - 315.1²] / 1070.4 = 220.1. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 220.1. ---752. What is the positive square root of [3157² + 40² - 252560]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3157, and b = 40, we see that 2ab = 252560, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3157² + 40² - 252560] = [3157 - 40]². Therefore, the positive square root of the expression given is equal to [3157 40] = 3117. ----
753. Please find the value of 89997², by using algebraic identities to simplify computation. -Solution: Since 89997 is only a little lower than 90000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 90000 and b = 3. After the substitution, we get 89997² = (90000 - 3)² = 90000² + 3² - (2 x 90000 x 3) = 8100000000 + 9 - 540000 = 8099460009. ---754. What is the value of 3.481² + 2.481² - (3.481 x 4.962)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.962 = 2 x 2.481. Using this fact, we can rewrite the given expression as 3.481² + 2.481² - (2 x 3.481 x 2.481). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.481² + 2.481² - (3.481 x 4.962) = 3.481² + 2.481² - (2 x 3.481 x 2.481) = (3.481 - 2.481)² = 1² = 1. ---755. What is the value of [591.5² + 208.8² - 382.7²] / [1183]? -Solution: First, we look at the three numbers under the square signs in the numerator: 591.5, 208.8, and 382.7. We note that 382.7 = 591.5 - 208.8. We also note, in the denominator, that 1183 = 2 x 591.5. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 591.5, and b = 208.8, to get [(591.5² + 208.8² - (591.5 - 208.8)²] / 1183 = [591.5² + 208.8² - 382.7²] / 1183 = 208.8. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 208.8. ---756. What is the positive square root of [2037² + 50² - 203700]? -Solution: By observation, we can deduce that the given expression looks a
little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2037, and b = 50, we see that 2ab = 203700, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2037² + 50² - 203700] = [2037 - 50]². Therefore, the positive square root of the expression given is equal to [2037 50] = 1987. ---757. Please find the value of 79999², by using algebraic identities to simplify computation. -Solution: Since 79999 is only a little lower than 80000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 1. After the substitution, we get 79999² = (80000 - 1)² = 80000² + 1² - (2 x 80000 x 1) = 6400000000 + 1 - 160000 = 6399840001. ---758. What is the value of 3.561² + 2.761² - (3.561 x 5.522)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 5.522 = 2 x 2.761. Using this fact, we can rewrite the given expression as 3.561² + 2.761² - (2 x 3.561 x 2.761). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.561² + 2.761² - (3.561 x 5.522) = 3.561² + 2.761² - (2 x 3.561 x 2.761) = (3.561 - 2.761)² = 0.8² = 0.64. ---759. What is the value of [557.3² + 175.9² - 381.4²] / [1114.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 557.3, 175.9, and 381.4. We note that 381.4 = 557.3 - 175.9. We also note, in the denominator, that 1114.6 = 2 x 557.3. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 557.3, and b = 175.9, to get [(557.3² + 175.9² - (557.3 - 175.9)²] / 1114.6 = [557.3² + 175.9² - 381.4²] / 1114.6 =
175.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 175.9. ---760. What is the positive square root of [2171² + 30² - 130260]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2171, and b = 30, we see that 2ab = 130260, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2171² + 30² - 130260] = [2171 - 30]². Therefore, the positive square root of the expression given is equal to [2171 30] = 2141. ---761. Please find the value of 697², by using algebraic identities to simplify computation. -Solution: Since 697 is only a little lower than 700, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 700 and b = 3. After the substitution, we get 697² = (700 - 3)² = 700² + 3² - (2 x 700 x 3) = 490000 + 9 - 4200 = 485809. ---762. What is the value of 1.985² + 0.785² - (1.985 x 1.57)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.57 = 2 x 0.785. Using this fact, we can rewrite the given expression as 1.985² + 0.785² - (2 x 1.985 x 0.785). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.985² + 0.785² - (1.985 x 1.57) = 1.985² + 0.785² - (2 x 1.985 x 0.785) = (1.985 - 0.785)² = 1.2² = 1.44. ---763. What is the value of [554.1² + 175.5² - 378.6²] / [1108.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 554.1, 175.5, and 378.6. We note that 378.6 = 554.1 - 175.5. We
also note, in the denominator, that 1108.2 = 2 x 554.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 554.1, and b = 175.5, to get [(554.1² + 175.5² - (554.1 - 175.5)²] / 1108.2 = [554.1² + 175.5² - 378.6²] / 1108.2 = 175.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 175.5. ---764. What is the positive square root of [2576² + 20² - 103040]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2576, and b = 20, we see that 2ab = 103040, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2576² + 20² - 103040] = [2576 - 20]². Therefore, the positive square root of the expression given is equal to [2576 20] = 2556. ---765. Please find the value of 1999², by using algebraic identities to simplify computation. -Solution: Since 1999 is only a little lower than 2000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 1. After the substitution, we get 1999² = (2000 - 1)² = 2000² + 1² - (2 x 2000 x 1) = 4000000 + 1 - 4000 = 3996001. ---766. What is the value of 3.307² + 2.307² - (3.307 x 4.614)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.614 = 2 x 2.307. Using this fact, we can rewrite the given expression as 3.307² + 2.307² - (2 x 3.307 x 2.307). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.307² + 2.307² - (3.307 x 4.614) = 3.307² + 2.307² - (2 x 3.307
x 2.307) = (3.307 - 2.307)² = 1² = 1. ---767. What is the value of [463.3² + 278.6² - 184.7²] / [926.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 463.3, 278.6, and 184.7. We note that 184.7 = 463.3 - 278.6. We also note, in the denominator, that 926.6 = 2 x 463.3. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 463.3, and b = 278.6, to get [(463.3² + 278.6² - (463.3 - 278.6)²] / 926.6 = [463.3² + 278.6² - 184.7²] / 926.6 = 278.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 278.6. ---768. What is the positive square root of [2541² + 90² - 457380]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2541, and b = 90, we see that 2ab = 457380, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2541² + 90² - 457380] = [2541 - 90]². Therefore, the positive square root of the expression given is equal to [2541 90] = 2451. ---769. Please find the value of 39997², by using algebraic identities to simplify computation. -Solution: Since 39997 is only a little lower than 40000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 40000 and b = 3. After the substitution, we get 39997² = (40000 - 3)² = 40000² + 3² - (2 x 40000 x 3) = 1600000000 + 9 - 240000 = 1599760009. ----
770. What is the value of 2.212² + 1.012² - (2.212 x 2.024)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.024 = 2 x 1.012. Using this fact, we can rewrite the given expression as 2.212² + 1.012² - (2 x 2.212 x 1.012). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.212² + 1.012² - (2.212 x 2.024) = 2.212² + 1.012² - (2 x 2.212 x 1.012) = (2.212 - 1.012)² = 1.2² = 1.44. ---771. What is the value of [575.1² + 259² - 316.1²] / [1150.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 575.1, 259, and 316.1. We note that 316.1 = 575.1 - 259. We also note, in the denominator, that 1150.2 = 2 x 575.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 575.1, and b = 259, to get [(575.1² + 259² - (575.1 - 259)²] / 1150.2 = [575.1² + 259² - 316.1²] / 1150.2 = 259. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 259. ---772. What is the positive square root of [3147² + 20² - 125880]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3147, and b = 20, we see that 2ab = 125880, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3147² + 20² - 125880] = [3147 - 20]². Therefore, the positive square root of the expression given is equal to [3147 20] = 3127. ---773. Please find the value of 29997², by using algebraic identities to simplify computation. --
Solution: Since 29997 is only a little lower than 30000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 30000 and b = 3. After the substitution, we get 29997² = (30000 - 3)² = 30000² + 3² - (2 x 30000 x 3) = 900000000 + 9 - 180000 = 899820009. ---774. What is the value of 3.291² + 2.491² - (3.291 x 4.982)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.982 = 2 x 2.491. Using this fact, we can rewrite the given expression as 3.291² + 2.491² - (2 x 3.291 x 2.491). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 3.291² + 2.491² - (3.291 x 4.982) = 3.291² + 2.491² - (2 x 3.291 x 2.491) = (3.291 - 2.491)² = 0.8² = 0.640000000000001. ---775. What is the value of [507.9² + 162.5² - 345.4²] / [1015.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 507.9, 162.5, and 345.4. We note that 345.4 = 507.9 - 162.5. We also note, in the denominator, that 1015.8 = 2 x 507.9. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 507.9, and b = 162.5, to get [(507.9² + 162.5² - (507.9 - 162.5)²] / 1015.8 = [507.9² + 162.5² - 345.4²] / 1015.8 = 162.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 162.5. ---776. What is the positive square root of [3621² + 40² - 289680]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3621, and b = 40, we see that 2ab = 289680, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3621² + 40² - 289680] = [3621 - 40]².
Therefore, the positive square root of the expression given is equal to [3621 40] = 3581. ---777. Please find the value of 79999², by using algebraic identities to simplify computation. -Solution: Since 79999 is only a little lower than 80000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 80000 and b = 1. After the substitution, we get 79999² = (80000 - 1)² = 80000² + 1² - (2 x 80000 x 1) = 6400000000 + 1 - 160000 = 6399840001. ---778. What is the value of 1.756² + 0.756² - (1.756 x 1.512)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.512 = 2 x 0.756. Using this fact, we can rewrite the given expression as 1.756² + 0.756² - (2 x 1.756 x 0.756). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 1.756² + 0.756² - (1.756 x 1.512) = 1.756² + 0.756² - (2 x 1.756 x 0.756) = (1.756 - 0.756)² = 1² = 1. ---779. What is the value of [491.3² + 107.5² - 383.8²] / [982.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 491.3, 107.5, and 383.8. We note that 383.8 = 491.3 - 107.5. We also note, in the denominator, that 982.6 = 2 x 491.3. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 491.3, and b = 107.5, to get [(491.3² + 107.5² - (491.3 - 107.5)²] / 982.6 = [491.3² + 107.5² - 383.8²] / 982.6 = 107.5. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 107.5. ----
780. What is the positive square root of [2575² + 20² - 103000]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2575, and b = 20, we see that 2ab = 103000, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2575² + 20² - 103000] = [2575 - 20]². Therefore, the positive square root of the expression given is equal to [2575 20] = 2555. ---781. Please find the value of 1999², by using algebraic identities to simplify computation. -Solution: Since 1999 is only a little lower than 2000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 2000 and b = 1. After the substitution, we get 1999² = (2000 - 1)² = 2000² + 1² - (2 x 2000 x 1) = 4000000 + 1 - 4000 = 3996001. ---782. What is the value of 2.759² + 1.959² - (2.759 x 3.918)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.918 = 2 x 1.959. Using this fact, we can rewrite the given expression as 2.759² + 1.959² - (2 x 2.759 x 1.959). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.759² + 1.959² - (2.759 x 3.918) = 2.759² + 1.959² - (2 x 2.759 x 1.959) = (2.759 - 1.959)² = 0.8² = 0.64. ---783. What is the value of [552.1² + 124.2² - 427.9²] / [1104.2]? -Solution: First, we look at the three numbers under the square signs in the numerator: 552.1, 124.2, and 427.9. We note that 427.9 = 552.1 - 124.2. We also note, in the denominator, that 1104.2 = 2 x 552.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression.
After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 552.1, and b = 124.2, to get [(552.1² + 124.2² - (552.1 - 124.2)²] / 1104.2 = [552.1² + 124.2² - 427.9²] / 1104.2 = 124.2. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 124.2. ---784. What is the positive square root of [2843² + 20² - 113720]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2843, and b = 20, we see that 2ab = 113720, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2843² + 20² - 113720] = [2843 - 20]². Therefore, the positive square root of the expression given is equal to [2843 20] = 2823. ---785. Please find the value of 69997², by using algebraic identities to simplify computation. -Solution: Since 69997 is only a little lower than 70000, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 70000 and b = 3. After the substitution, we get 69997² = (70000 - 3)² = 70000² + 3² - (2 x 70000 x 3) = 4900000000 + 9 - 420000 = 4899580009. ---786. What is the value of 2.845² + 1.445² - (2.845 x 2.89)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 2.89 = 2 x 1.445. Using this fact, we can rewrite the given expression as 2.845² + 1.445² - (2 x 2.845 x 1.445). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.845² + 1.445² - (2.845 x 2.89) = 2.845² + 1.445² - (2 x 2.845 x 1.445) = (2.845 - 1.445)² = 1.4² = 1.96. ---787. What is the value of [473.1² + 238.9² - 234.2²] / [946.2]?
-Solution: First, we look at the three numbers under the square signs in the numerator: 473.1, 238.9, and 234.2. We note that 234.2 = 473.1 - 238.9. We also note, in the denominator, that 946.2 = 2 x 473.1. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 473.1, and b = 238.9, to get [(473.1² + 238.9² - (473.1 - 238.9)²] / 946.2 = [473.1² + 238.9² - 234.2²] / 946.2 = 238.9. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 238.9. ---788. What is the positive square root of [3615² + 70² - 506100]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 3615, and b = 70, we see that 2ab = 506100, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [3615² + 70² - 506100] = [3615 - 70]². Therefore, the positive square root of the expression given is equal to [3615 70] = 3545. ---789. Please find the value of 899², by using algebraic identities to simplify computation. -Solution: Since 899 is only a little lower than 900, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 900 and b = 1. After the substitution, we get 899² = (900 - 1)² = 900² + 1² - (2 x 900 x 1) = 810000 + 1 - 1800 = 808201. ---790. What is the value of 2.131² + 0.931² - (2.131 x 1.862)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 1.862 = 2 x 0.931. Using this fact, we can
rewrite the given expression as 2.131² + 0.931² - (2 x 2.131 x 0.931). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.131² + 0.931² - (2.131 x 1.862) = 2.131² + 0.931² - (2 x 2.131 x 0.931) = (2.131 - 0.931)² = 1.2² = 1.44. ---791. What is the value of [403.8² + 194.6² - 209.2²] / [807.6]? -Solution: First, we look at the three numbers under the square signs in the numerator: 403.8, 194.6, and 209.2. We note that 209.2 = 403.8 - 194.6. We also note, in the denominator, that 807.6 = 2 x 403.8. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 403.8, and b = 194.6, to get [(403.8² + 194.6² - (403.8 - 194.6)²] / 807.6 = [403.8² + 194.6² - 209.2²] / 807.6 = 194.6. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 194.6. ---792. What is the positive square root of [1799² + 90² - 323820]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1799, and b = 90, we see that 2ab = 323820, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1799² + 90² - 323820] = [1799 - 90]². Therefore, the positive square root of the expression given is equal to [1799 90] = 1709. ---793. Please find the value of 198², by using algebraic identities to simplify computation. -Solution: Since 198 is only a little lower than 200, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 200
and b = 2. After the substitution, we get 198² = (200 - 2)² = 200² + 2² - (2 x 200 x 2) = 40000 + 4 - 800 = 39204. ---794. What is the value of 2.802² + 2.202² - (2.802 x 4.404)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 4.404 = 2 x 2.202. Using this fact, we can rewrite the given expression as 2.802² + 2.202² - (2 x 2.802 x 2.202). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.802² + 2.202² - (2.802 x 4.404) = 2.802² + 2.202² - (2 x 2.802 x 2.202) = (2.802 - 2.202)² = 0.6² = 0.36. ---795. What is the value of [584² + 222.4² - 361.6²] / [1168]? -Solution: First, we look at the three numbers under the square signs in the numerator: 584, 222.4, and 361.6. We note that 361.6 = 584 - 222.4. We also note, in the denominator, that 1168 = 2 x 584. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 584, and b = 222.4, to get [(584² + 222.4² - (584 - 222.4)²] / 1168 = [584² + 222.4² - 361.6²] / 1168 = 222.4. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 222.4. ---796. What is the positive square root of [1906² + 50² - 190600]? -Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 1906, and b = 50, we see that 2ab = 190600, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [1906² + 50² - 190600] = [1906 - 50]². Therefore, the positive square root of the expression given is equal to [1906 50] = 1856.
---797. Please find the value of 198², by using algebraic identities to simplify computation. -Solution: Since 198 is only a little lower than 200, we can deduce that the identity (a - b)² = (a² + b² - 2ab) is to be used, with some appropriately chosen values of a and b. To do this particular computation, we put a = 200 and b = 2. After the substitution, we get 198² = (200 - 2)² = 200² + 2² - (2 x 200 x 2) = 40000 + 4 - 800 = 39204. ---798. What is the value of 2.162² + 1.762² - (2.162 x 3.524)? Please use algebraic identities to make the computation simple. -Solution: First, we note that 3.524 = 2 x 1.762. Using this fact, we can rewrite the given expression as 2.162² + 1.762² - (2 x 2.162 x 1.762). Comparing this with the Right Hand Side of the identity (a - b)² = (a² + b² 2ab), we get 2.162² + 1.762² - (2.162 x 3.524) = 2.162² + 1.762² - (2 x 2.162 x 1.762) = (2.162 - 1.762)² = 0.4² = 0.16. ---799. What is the value of [475.4² + 299.7² - 175.7²] / [950.8]? -Solution: First, we look at the three numbers under the square signs in the numerator: 475.4, 299.7, and 175.7. We note that 175.7 = 475.4 - 299.7. We also note, in the denominator, that 950.8 = 2 x 475.4. Hence, we can guess that the algebraic identity (a - b)² = (a² + b² - 2ab) is to be used, but we will need to rearrange the terms of this identity to get something that will help us with our expression. After some trial and error, we note that [a² + b² - (a - b)²] / 2a = b. In this rearranged equation, we put a = 475.4, and b = 299.7, to get [(475.4² + 299.7² - (475.4 - 299.7)²] / 950.8 = [475.4² + 299.7² - 175.7²] / 950.8 = 299.7. The Left Hand Side of this equation is exactly the expression in our question, so the value of that expression is equal to 299.7. ---800. What is the positive square root of [2115² + 60² - 253800]? --
Solution: By observation, we can deduce that the given expression looks a little like the LHS of the identity (a - b)² = (a² + b² - 2ab). Indeed, if we set a = 2115, and b = 60, we see that 2ab = 253800, which is exactly equal to the absolute value of the third term in the expression. Hence, we can say that [2115² + 60² - 253800] = [2115 - 60]². Therefore, the positive square root of the expression given is equal to [2115 60] = 2055. ----
801. Evaluate the expression (8134² - 1866²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8134 and b = 1866 in the difference-of-squares identity. We then get (8134² - 1866²) = (8134 + 1866)(8134 - 1866). The Right Hand Side of this equation can be simplified to 10000 x 6268 = 62680000. ---802. Evaluate the expression (90007² - 89993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 90007 and b = 89993 in the difference-of-squares identity. We then get (90007² - 89993²) = (90007 + 89993)(90007 - 89993). The Right Hand Side of this equation can be simplified to 180000 x 14 = 2520000. ---803. Find the value of (50006 x 49994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 50006 and 49994 are close in value to 50000. We can rewrite the expression (50006 x 49994) as (50000 + 6)(50000 - 6).
Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (50000 + 6)(50000 - 6) = 50000² - 6². This is relatively easy to compute; we have 50000² - 6² = 2500000000 - 36 = 2499999964. ---804. What is the value of the positive square root of [(340 + 213)(340 - 213) + 45369]? -Solution: First, we note that the number 45369 looks approximately like the square of 213. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(340 + 213)(340 - 213) + 45369] = [(340 + 213)(340 - 213) + 213²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(340 + 213)(340 - 213) + 213²] = [340² - 213² + 213²]. The second and third terms cancel, so the expression is just equal to 340². Clearly, the square root of the expression is equal to 340. ---805. Evaluate the expression (9986² - 14²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9986 and b = 14 in the difference-of-squares identity. We then get (9986² - 14²) = (9986 + 14)(9986 - 14). The Right Hand Side of this equation can be simplified to 10000 x 9972 = 99720000. ---806. Evaluate the expression (7008² - 6992²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7008 and b = 6992 in the difference-of-squares identity. We then get (7008² - 6992²) = (7008 + 6992)(7008 - 6992). The Right Hand Side of this equation can be simplified to 14000 x 16 = 224000.
---807. Find the value of (7006 x 6994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7006 and 6994 are close in value to 7000. We can rewrite the expression (7006 x 6994) as (7000 + 6)(7000 - 6). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 6)(7000 - 6) = 7000² - 6². This is relatively easy to compute; we have 7000² - 6² = 49000000 - 36 = 48999964. ---808. What is the value of the positive square root of [(660 + 139)(660 - 139) + 19321]? -Solution: First, we note that the number 19321 looks approximately like the square of 139. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(660 + 139)(660 - 139) + 19321] = [(660 + 139)(660 - 139) + 139²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(660 + 139)(660 - 139) + 139²] = [660² - 139² + 139²]. The second and third terms cancel, so the expression is just equal to 660². Clearly, the square root of the expression is equal to 660. ---809. Evaluate the expression (6459² - 3541²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6459 and b = 3541 in the difference-of-squares identity. We then get (6459² - 3541²) = (6459 + 3541)(6459 - 3541). The Right Hand Side of this equation can be simplified to 10000 x 2918 = 29180000. ---810. Evaluate the expression (7004² - 6996²), by using algebraic identities to simplify computation.
-Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7004 and b = 6996 in the difference-of-squares identity. We then get (7004² - 6996²) = (7004 + 6996)(7004 - 6996). The Right Hand Side of this equation can be simplified to 14000 x 8 = 112000. ---811. Find the value of (40009 x 39991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 40009 and 39991 are close in value to 40000. We can rewrite the expression (40009 x 39991) as (40000 + 9)(40000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (40000 + 9)(40000 - 9) = 40000² - 9². This is relatively easy to compute; we have 40000² - 9² = 1600000000 - 81 = 1599999919. ---812. What is the value of the positive square root of [(340 + 255)(340 - 255) + 65025]? -Solution: First, we note that the number 65025 looks approximately like the square of 255. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(340 + 255)(340 - 255) + 65025] = [(340 + 255)(340 - 255) + 255²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(340 + 255)(340 - 255) + 255²] = [340² - 255² + 255²]. The second and third terms cancel, so the expression is just equal to 340². Clearly, the square root of the expression is equal to 340. ---813. Evaluate the expression (7254² - 2746²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7254 and b =
2746 in the difference-of-squares identity. We then get (7254² - 2746²) = (7254 + 2746)(7254 - 2746). The Right Hand Side of this equation can be simplified to 10000 x 4508 = 45080000. ---814. Evaluate the expression (30007² - 29993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 30007 and b = 29993 in the difference-of-squares identity. We then get (30007² - 29993²) = (30007 + 29993)(30007 - 29993). The Right Hand Side of this equation can be simplified to 60000 x 14 = 840000. ---815. Find the value of (70009 x 69991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 70009 and 69991 are close in value to 70000. We can rewrite the expression (70009 x 69991) as (70000 + 9)(70000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (70000 + 9)(70000 - 9) = 70000² - 9². This is relatively easy to compute; we have 70000² - 9² = 4900000000 - 81 = 4899999919. ---816. What is the value of the positive square root of [(730 + 222)(730 - 222) + 49284]? -Solution: First, we note that the number 49284 looks approximately like the square of 222. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(730 + 222)(730 - 222) + 49284] = [(730 + 222)(730 - 222) + 222²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(730 + 222)(730 - 222) + 222²] = [730² - 222² + 222²]. The second and third terms cancel, so the expression is just equal to 730². Clearly, the square root of the expression is equal to 730.
---817. Evaluate the expression (6684² - 3316²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6684 and b = 3316 in the difference-of-squares identity. We then get (6684² - 3316²) = (6684 + 3316)(6684 - 3316). The Right Hand Side of this equation can be simplified to 10000 x 3368 = 33680000. ---818. Evaluate the expression (8003² - 7997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8003 and b = 7997 in the difference-of-squares identity. We then get (8003² - 7997²) = (8003 + 7997)(8003 - 7997). The Right Hand Side of this equation can be simplified to 16000 x 6 = 96000. ---819. Find the value of (2004 x 1996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2004 and 1996 are close in value to 2000. We can rewrite the expression (2004 x 1996) as (2000 + 4)(2000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 + 4)(2000 - 4) = 2000² - 4². This is relatively easy to compute; we have 2000² - 4² = 4000000 - 16 = 3999984. ---820. What is the value of the positive square root of [(600 + 175)(600 - 175) + 30625]? -Solution: First, we note that the number 30625 looks approximately like the
square of 175. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(600 + 175)(600 - 175) + 30625] = [(600 + 175)(600 - 175) + 175²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(600 + 175)(600 - 175) + 175²] = [600² - 175² + 175²]. The second and third terms cancel, so the expression is just equal to 600². Clearly, the square root of the expression is equal to 600. ---821. Evaluate the expression (9662² - 338²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9662 and b = 338 in the difference-of-squares identity. We then get (9662² - 338²) = (9662 + 338)(9662 - 338). The Right Hand Side of this equation can be simplified to 10000 x 9324 = 93240000. ---822. Evaluate the expression (60004² - 59996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 60004 and b = 59996 in the difference-of-squares identity. We then get (60004² - 59996²) = (60004 + 59996)(60004 - 59996). The Right Hand Side of this equation can be simplified to 120000 x 8 = 960000. ---823. Find the value of (40005 x 39995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 40005 and 39995 are close in value to 40000. We can rewrite the expression (40005 x 39995) as (40000 + 5)(40000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (40000 + 5)(40000 - 5) = 40000² - 5². This is relatively easy to compute; we have 40000² - 5² = 1600000000 - 25 =
1599999975. ---824. What is the value of the positive square root of [(520 + 247)(520 - 247) + 61009]? -Solution: First, we note that the number 61009 looks approximately like the square of 247. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(520 + 247)(520 - 247) + 61009] = [(520 + 247)(520 - 247) + 247²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(520 + 247)(520 - 247) + 247²] = [520² - 247² + 247²]. The second and third terms cancel, so the expression is just equal to 520². Clearly, the square root of the expression is equal to 520. ---825. Evaluate the expression (9865² - 135²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9865 and b = 135 in the difference-of-squares identity. We then get (9865² - 135²) = (9865 + 135)(9865 - 135). The Right Hand Side of this equation can be simplified to 10000 x 9730 = 97300000. ---826. Evaluate the expression (5008² - 4992²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5008 and b = 4992 in the difference-of-squares identity. We then get (5008² - 4992²) = (5008 + 4992)(5008 - 4992). The Right Hand Side of this equation can be simplified to 10000 x 16 = 160000. ---827. Find the value of (40008 x 39992), by using algebraic identities to
simplify the calculation. -Solution: We observe that both 40008 and 39992 are close in value to 40000. We can rewrite the expression (40008 x 39992) as (40000 + 8)(40000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (40000 + 8)(40000 - 8) = 40000² - 8². This is relatively easy to compute; we have 40000² - 8² = 1600000000 - 64 = 1599999936. ---828. What is the value of the positive square root of [(480 + 245)(480 - 245) + 60025]? -Solution: First, we note that the number 60025 looks approximately like the square of 245. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(480 + 245)(480 - 245) + 60025] = [(480 + 245)(480 - 245) + 245²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(480 + 245)(480 - 245) + 245²] = [480² - 245² + 245²]. The second and third terms cancel, so the expression is just equal to 480². Clearly, the square root of the expression is equal to 480. ---829. Evaluate the expression (5597² - 4403²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5597 and b = 4403 in the difference-of-squares identity. We then get (5597² - 4403²) = (5597 + 4403)(5597 - 4403). The Right Hand Side of this equation can be simplified to 10000 x 1194 = 11940000. ---830. Evaluate the expression (2002² - 1998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 2002 and b =
1998 in the difference-of-squares identity. We then get (2002² - 1998²) = (2002 + 1998)(2002 - 1998). The Right Hand Side of this equation can be simplified to 4000 x 4 = 16000. ---831. Find the value of (30009 x 29991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 30009 and 29991 are close in value to 30000. We can rewrite the expression (30009 x 29991) as (30000 + 9)(30000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (30000 + 9)(30000 - 9) = 30000² - 9². This is relatively easy to compute; we have 30000² - 9² = 900000000 - 81 = 899999919. ---832. What is the value of the positive square root of [(750 + 201)(750 - 201) + 40401]? -Solution: First, we note that the number 40401 looks approximately like the square of 201. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(750 + 201)(750 - 201) + 40401] = [(750 + 201)(750 - 201) + 201²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(750 + 201)(750 - 201) + 201²] = [750² - 201² + 201²]. The second and third terms cancel, so the expression is just equal to 750². Clearly, the square root of the expression is equal to 750. ---833. Evaluate the expression (7906² - 2094²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7906 and b = 2094 in the difference-of-squares identity. We then get (7906² - 2094²) = (7906 + 2094)(7906 - 2094). The Right Hand Side of this equation can be simplified to 10000 x 5812 = 58120000. ----
834. Evaluate the expression (5009² - 4991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5009 and b = 4991 in the difference-of-squares identity. We then get (5009² - 4991²) = (5009 + 4991)(5009 - 4991). The Right Hand Side of this equation can be simplified to 10000 x 18 = 180000. ---835. Find the value of (90007 x 89993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 90007 and 89993 are close in value to 90000. We can rewrite the expression (90007 x 89993) as (90000 + 7)(90000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (90000 + 7)(90000 - 7) = 90000² - 7². This is relatively easy to compute; we have 90000² - 7² = 8100000000 - 49 = 8099999951. ---836. What is the value of the positive square root of [(440 + 293)(440 - 293) + 85849]? -Solution: First, we note that the number 85849 looks approximately like the square of 293. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(440 + 293)(440 - 293) + 85849] = [(440 + 293)(440 - 293) + 293²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(440 + 293)(440 - 293) + 293²] = [440² - 293² + 293²]. The second and third terms cancel, so the expression is just equal to 440². Clearly, the square root of the expression is equal to 440. ---837. Evaluate the expression (7038² - 2962²), by using algebraic identities to simplify computation. --
Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7038 and b = 2962 in the difference-of-squares identity. We then get (7038² - 2962²) = (7038 + 2962)(7038 - 2962). The Right Hand Side of this equation can be simplified to 10000 x 4076 = 40760000. ---838. Evaluate the expression (8003² - 7997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8003 and b = 7997 in the difference-of-squares identity. We then get (8003² - 7997²) = (8003 + 7997)(8003 - 7997). The Right Hand Side of this equation can be simplified to 16000 x 6 = 96000. ---839. Find the value of (2008 x 1992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2008 and 1992 are close in value to 2000. We can rewrite the expression (2008 x 1992) as (2000 + 8)(2000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 + 8)(2000 - 8) = 2000² - 8². This is relatively easy to compute; we have 2000² - 8² = 4000000 - 64 = 3999936. ---840. What is the value of the positive square root of [(810 + 281)(810 - 281) + 78961]? -Solution: First, we note that the number 78961 looks approximately like the square of 281. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(810 + 281)(810 - 281) + 78961] = [(810 + 281)(810 - 281) + 281²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as
[(810 + 281)(810 - 281) + 281²] = [810² - 281² + 281²]. The second and third terms cancel, so the expression is just equal to 810². Clearly, the square root of the expression is equal to 810. ---841. Evaluate the expression (9108² - 892²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9108 and b = 892 in the difference-of-squares identity. We then get (9108² - 892²) = (9108 + 892)(9108 - 892). The Right Hand Side of this equation can be simplified to 10000 x 8216 = 82160000. ---842. Evaluate the expression (3009² - 2991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 3009 and b = 2991 in the difference-of-squares identity. We then get (3009² - 2991²) = (3009 + 2991)(3009 - 2991). The Right Hand Side of this equation can be simplified to 6000 x 18 = 108000. ---843. Find the value of (8009 x 7991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 8009 and 7991 are close in value to 8000. We can rewrite the expression (8009 x 7991) as (8000 + 9)(8000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (8000 + 9)(8000 - 9) = 8000² - 9². This is relatively easy to compute; we have 8000² - 9² = 64000000 - 81 = 63999919. ---844. What is the value of the positive square root of [(620 + 130)(620 - 130)
+ 16900]? -Solution: First, we note that the number 16900 looks approximately like the square of 130. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(620 + 130)(620 - 130) + 16900] = [(620 + 130)(620 - 130) + 130²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(620 + 130)(620 - 130) + 130²] = [620² - 130² + 130²]. The second and third terms cancel, so the expression is just equal to 620². Clearly, the square root of the expression is equal to 620. ---845. Evaluate the expression (6281² - 3719²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6281 and b = 3719 in the difference-of-squares identity. We then get (6281² - 3719²) = (6281 + 3719)(6281 - 3719). The Right Hand Side of this equation can be simplified to 10000 x 2562 = 25620000. ---846. Evaluate the expression (40003² - 39997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 40003 and b = 39997 in the difference-of-squares identity. We then get (40003² - 39997²) = (40003 + 39997)(40003 - 39997). The Right Hand Side of this equation can be simplified to 80000 x 6 = 480000. ---847. Find the value of (9007 x 8993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 9007 and 8993 are close in value to 9000. We can rewrite the expression (9007 x 8993) as (9000 + 7)(9000 - 7).
Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (9000 + 7)(9000 - 7) = 9000² - 7². This is relatively easy to compute; we have 9000² - 7² = 81000000 - 49 = 80999951. ---848. What is the value of the positive square root of [(470 + 165)(470 - 165) + 27225]? -Solution: First, we note that the number 27225 looks approximately like the square of 165. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(470 + 165)(470 - 165) + 27225] = [(470 + 165)(470 - 165) + 165²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(470 + 165)(470 - 165) + 165²] = [470² - 165² + 165²]. The second and third terms cancel, so the expression is just equal to 470². Clearly, the square root of the expression is equal to 470. ---849. Evaluate the expression (9966² - 34²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9966 and b = 34 in the difference-of-squares identity. We then get (9966² - 34²) = (9966 + 34)(9966 - 34). The Right Hand Side of this equation can be simplified to 10000 x 9932 = 99320000. ---850. Evaluate the expression (4007² - 3993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 4007 and b = 3993 in the difference-of-squares identity. We then get (4007² - 3993²) = (4007 + 3993)(4007 - 3993). The Right Hand Side of this equation can be simplified to 8000 x 14 =
112000. ---851. Find the value of (30005 x 29995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 30005 and 29995 are close in value to 30000. We can rewrite the expression (30005 x 29995) as (30000 + 5)(30000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (30000 + 5)(30000 - 5) = 30000² - 5². This is relatively easy to compute; we have 30000² - 5² = 900000000 - 25 = 899999975. ---852. What is the value of the positive square root of [(610 + 137)(610 - 137) + 18769]? -Solution: First, we note that the number 18769 looks approximately like the square of 137. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(610 + 137)(610 - 137) + 18769] = [(610 + 137)(610 - 137) + 137²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(610 + 137)(610 - 137) + 137²] = [610² - 137² + 137²]. The second and third terms cancel, so the expression is just equal to 610². Clearly, the square root of the expression is equal to 610. ---853. Evaluate the expression (7615² - 2385²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7615 and b = 2385 in the difference-of-squares identity. We then get (7615² - 2385²) = (7615 + 2385)(7615 - 2385). The Right Hand Side of this equation can be simplified to 10000 x 5230 = 52300000. ---854. Evaluate the expression (7008² - 6992²), by using algebraic identities to
simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7008 and b = 6992 in the difference-of-squares identity. We then get (7008² - 6992²) = (7008 + 6992)(7008 - 6992). The Right Hand Side of this equation can be simplified to 14000 x 16 = 224000. ---855. Find the value of (80008 x 79992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 80008 and 79992 are close in value to 80000. We can rewrite the expression (80008 x 79992) as (80000 + 8)(80000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (80000 + 8)(80000 - 8) = 80000² - 8². This is relatively easy to compute; we have 80000² - 8² = 6400000000 - 64 = 6399999936. ---856. What is the value of the positive square root of [(900 + 275)(900 - 275) + 75625]? -Solution: First, we note that the number 75625 looks approximately like the square of 275. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(900 + 275)(900 - 275) + 75625] = [(900 + 275)(900 - 275) + 275²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(900 + 275)(900 - 275) + 275²] = [900² - 275² + 275²]. The second and third terms cancel, so the expression is just equal to 900². Clearly, the square root of the expression is equal to 900. ---857. Evaluate the expression (5001² - 4999²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5001 and b =
4999 in the difference-of-squares identity. We then get (5001² - 4999²) = (5001 + 4999)(5001 - 4999). The Right Hand Side of this equation can be simplified to 10000 x 2 = 20000. ---858. Evaluate the expression (40004² - 39996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 40004 and b = 39996 in the difference-of-squares identity. We then get (40004² - 39996²) = (40004 + 39996)(40004 - 39996). The Right Hand Side of this equation can be simplified to 80000 x 8 = 640000. ---859. Find the value of (40006 x 39994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 40006 and 39994 are close in value to 40000. We can rewrite the expression (40006 x 39994) as (40000 + 6)(40000 - 6). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (40000 + 6)(40000 - 6) = 40000² - 6². This is relatively easy to compute; we have 40000² - 6² = 1600000000 - 36 = 1599999964. ---860. What is the value of the positive square root of [(840 + 109)(840 - 109) + 11881]? -Solution: First, we note that the number 11881 looks approximately like the square of 109. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(840 + 109)(840 - 109) + 11881] = [(840 + 109)(840 - 109) + 109²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(840 + 109)(840 - 109) + 109²] = [840² - 109² + 109²]. The second and third terms cancel, so the expression is just equal to 840². Clearly, the square root of the expression is equal to 840.
---861. Evaluate the expression (7164² - 2836²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7164 and b = 2836 in the difference-of-squares identity. We then get (7164² - 2836²) = (7164 + 2836)(7164 - 2836). The Right Hand Side of this equation can be simplified to 10000 x 4328 = 43280000. ---862. Evaluate the expression (20009² - 19991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 20009 and b = 19991 in the difference-of-squares identity. We then get (20009² - 19991²) = (20009 + 19991)(20009 - 19991). The Right Hand Side of this equation can be simplified to 40000 x 18 = 720000. ---863. Find the value of (80008 x 79992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 80008 and 79992 are close in value to 80000. We can rewrite the expression (80008 x 79992) as (80000 + 8)(80000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (80000 + 8)(80000 - 8) = 80000² - 8². This is relatively easy to compute; we have 80000² - 8² = 6400000000 - 64 = 6399999936. ---864. What is the value of the positive square root of [(900 + 179)(900 - 179) + 32041]? -Solution: First, we note that the number 32041 looks approximately like the
square of 179. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(900 + 179)(900 - 179) + 32041] = [(900 + 179)(900 - 179) + 179²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(900 + 179)(900 - 179) + 179²] = [900² - 179² + 179²]. The second and third terms cancel, so the expression is just equal to 900². Clearly, the square root of the expression is equal to 900. ---865. Evaluate the expression (6118² - 3882²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6118 and b = 3882 in the difference-of-squares identity. We then get (6118² - 3882²) = (6118 + 3882)(6118 - 3882). The Right Hand Side of this equation can be simplified to 10000 x 2236 = 22360000. ---866. Evaluate the expression (5008² - 4992²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5008 and b = 4992 in the difference-of-squares identity. We then get (5008² - 4992²) = (5008 + 4992)(5008 - 4992). The Right Hand Side of this equation can be simplified to 10000 x 16 = 160000. ---867. Find the value of (8007 x 7993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 8007 and 7993 are close in value to 8000. We can rewrite the expression (8007 x 7993) as (8000 + 7)(8000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (8000 + 7)(8000 - 7) = 8000² - 7². This is relatively easy to compute; we have 8000² - 7² = 64000000 - 49 =
63999951. ---868. What is the value of the positive square root of [(870 + 198)(870 - 198) + 39204]? -Solution: First, we note that the number 39204 looks approximately like the square of 198. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(870 + 198)(870 - 198) + 39204] = [(870 + 198)(870 - 198) + 198²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(870 + 198)(870 - 198) + 198²] = [870² - 198² + 198²]. The second and third terms cancel, so the expression is just equal to 870². Clearly, the square root of the expression is equal to 870. ---869. Evaluate the expression (8317² - 1683²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8317 and b = 1683 in the difference-of-squares identity. We then get (8317² - 1683²) = (8317 + 1683)(8317 - 1683). The Right Hand Side of this equation can be simplified to 10000 x 6634 = 66340000. ---870. Evaluate the expression (90004² - 89996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 90004 and b = 89996 in the difference-of-squares identity. We then get (90004² - 89996²) = (90004 + 89996)(90004 - 89996). The Right Hand Side of this equation can be simplified to 180000 x 8 = 1440000. ---871. Find the value of (50005 x 49995), by using algebraic identities to
simplify the calculation. -Solution: We observe that both 50005 and 49995 are close in value to 50000. We can rewrite the expression (50005 x 49995) as (50000 + 5)(50000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (50000 + 5)(50000 - 5) = 50000² - 5². This is relatively easy to compute; we have 50000² - 5² = 2500000000 - 25 = 2499999975. ---872. What is the value of the positive square root of [(820 + 114)(820 - 114) + 12996]? -Solution: First, we note that the number 12996 looks approximately like the square of 114. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(820 + 114)(820 - 114) + 12996] = [(820 + 114)(820 - 114) + 114²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(820 + 114)(820 - 114) + 114²] = [820² - 114² + 114²]. The second and third terms cancel, so the expression is just equal to 820². Clearly, the square root of the expression is equal to 820. ---873. Evaluate the expression (5614² - 4386²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5614 and b = 4386 in the difference-of-squares identity. We then get (5614² - 4386²) = (5614 + 4386)(5614 - 4386). The Right Hand Side of this equation can be simplified to 10000 x 1228 = 12280000. ---874. Evaluate the expression (20008² - 19992²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 20008 and b
= 19992 in the difference-of-squares identity. We then get (20008² - 19992²) = (20008 + 19992)(20008 - 19992). The Right Hand Side of this equation can be simplified to 40000 x 16 = 640000. ---875. Find the value of (6007 x 5993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 6007 and 5993 are close in value to 6000. We can rewrite the expression (6007 x 5993) as (6000 + 7)(6000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (6000 + 7)(6000 - 7) = 6000² - 7². This is relatively easy to compute; we have 6000² - 7² = 36000000 - 49 = 35999951. ---876. What is the value of the positive square root of [(530 + 182)(530 - 182) + 33124]? -Solution: First, we note that the number 33124 looks approximately like the square of 182. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(530 + 182)(530 - 182) + 33124] = [(530 + 182)(530 - 182) + 182²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(530 + 182)(530 - 182) + 182²] = [530² - 182² + 182²]. The second and third terms cancel, so the expression is just equal to 530². Clearly, the square root of the expression is equal to 530. ---877. Evaluate the expression (8554² - 1446²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8554 and b = 1446 in the difference-of-squares identity. We then get (8554² - 1446²) = (8554 + 1446)(8554 - 1446). The Right Hand Side of this equation can be simplified to 10000 x 7108 =
71080000. ---878. Evaluate the expression (9003² - 8997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9003 and b = 8997 in the difference-of-squares identity. We then get (9003² - 8997²) = (9003 + 8997)(9003 - 8997). The Right Hand Side of this equation can be simplified to 18000 x 6 = 108000. ---879. Find the value of (80005 x 79995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 80005 and 79995 are close in value to 80000. We can rewrite the expression (80005 x 79995) as (80000 + 5)(80000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (80000 + 5)(80000 - 5) = 80000² - 5². This is relatively easy to compute; we have 80000² - 5² = 6400000000 - 25 = 6399999975. ---880. What is the value of the positive square root of [(390 + 287)(390 - 287) + 82369]? -Solution: First, we note that the number 82369 looks approximately like the square of 287. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(390 + 287)(390 - 287) + 82369] = [(390 + 287)(390 - 287) + 287²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(390 + 287)(390 - 287) + 287²] = [390² - 287² + 287²]. The second and third terms cancel, so the expression is just equal to 390². Clearly, the square root of the expression is equal to 390. ---881. Evaluate the expression (9323² - 677²), by using algebraic identities to
simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9323 and b = 677 in the difference-of-squares identity. We then get (9323² - 677²) = (9323 + 677)(9323 - 677). The Right Hand Side of this equation can be simplified to 10000 x 8646 = 86460000. ---882. Evaluate the expression (40007² - 39993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 40007 and b = 39993 in the difference-of-squares identity. We then get (40007² - 39993²) = (40007 + 39993)(40007 - 39993). The Right Hand Side of this equation can be simplified to 80000 x 14 = 1120000. ---883. Find the value of (3002 x 2998), by using algebraic identities to simplify the calculation. -Solution: We observe that both 3002 and 2998 are close in value to 3000. We can rewrite the expression (3002 x 2998) as (3000 + 2)(3000 - 2). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (3000 + 2)(3000 - 2) = 3000² - 2². This is relatively easy to compute; we have 3000² - 2² = 9000000 - 4 = 8999996. ---884. What is the value of the positive square root of [(810 + 203)(810 - 203) + 41209]? -Solution: First, we note that the number 41209 looks approximately like the square of 203. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(810 + 203)(810 - 203) + 41209] = [(810 + 203)(810 - 203) + 203²].
Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(810 + 203)(810 - 203) + 203²] = [810² - 203² + 203²]. The second and third terms cancel, so the expression is just equal to 810². Clearly, the square root of the expression is equal to 810. ---885. Evaluate the expression (6592² - 3408²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6592 and b = 3408 in the difference-of-squares identity. We then get (6592² - 3408²) = (6592 + 3408)(6592 - 3408). The Right Hand Side of this equation can be simplified to 10000 x 3184 = 31840000. ---886. Evaluate the expression (9005² - 8995²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9005 and b = 8995 in the difference-of-squares identity. We then get (9005² - 8995²) = (9005 + 8995)(9005 - 8995). The Right Hand Side of this equation can be simplified to 18000 x 10 = 180000. ---887. Find the value of (4009 x 3991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 4009 and 3991 are close in value to 4000. We can rewrite the expression (4009 x 3991) as (4000 + 9)(4000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (4000 + 9)(4000 - 9) = 4000² - 9². This is relatively easy to compute; we have 4000² - 9² = 16000000 - 81 = 15999919. ----
888. What is the value of the positive square root of [(770 + 178)(770 - 178) + 31684]? -Solution: First, we note that the number 31684 looks approximately like the square of 178. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(770 + 178)(770 - 178) + 31684] = [(770 + 178)(770 - 178) + 178²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(770 + 178)(770 - 178) + 178²] = [770² - 178² + 178²]. The second and third terms cancel, so the expression is just equal to 770². Clearly, the square root of the expression is equal to 770. ---889. Evaluate the expression (6145² - 3855²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6145 and b = 3855 in the difference-of-squares identity. We then get (6145² - 3855²) = (6145 + 3855)(6145 - 3855). The Right Hand Side of this equation can be simplified to 10000 x 2290 = 22900000. ---890. Evaluate the expression (20007² - 19993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 20007 and b = 19993 in the difference-of-squares identity. We then get (20007² - 19993²) = (20007 + 19993)(20007 - 19993). The Right Hand Side of this equation can be simplified to 40000 x 14 = 560000. ---891. Find the value of (7003 x 6997), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7003 and 6997 are close in value to 7000.
We can rewrite the expression (7003 x 6997) as (7000 + 3)(7000 - 3). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 3)(7000 - 3) = 7000² - 3². This is relatively easy to compute; we have 7000² - 3² = 49000000 - 9 = 48999991. ---892. What is the value of the positive square root of [(370 + 126)(370 - 126) + 15876]? -Solution: First, we note that the number 15876 looks approximately like the square of 126. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(370 + 126)(370 - 126) + 15876] = [(370 + 126)(370 - 126) + 126²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(370 + 126)(370 - 126) + 126²] = [370² - 126² + 126²]. The second and third terms cancel, so the expression is just equal to 370². Clearly, the square root of the expression is equal to 370. ---893. Evaluate the expression (8969² - 1031²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8969 and b = 1031 in the difference-of-squares identity. We then get (8969² - 1031²) = (8969 + 1031)(8969 - 1031). The Right Hand Side of this equation can be simplified to 10000 x 7938 = 79380000. ---894. Evaluate the expression (50005² - 49995²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 50005 and b = 49995 in the difference-of-squares identity. We then get (50005² - 49995²) = (50005 + 49995)(50005 - 49995). The Right Hand Side of this equation can be simplified to 100000 x 10 =
1000000. ---895. Find the value of (4008 x 3992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 4008 and 3992 are close in value to 4000. We can rewrite the expression (4008 x 3992) as (4000 + 8)(4000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (4000 + 8)(4000 - 8) = 4000² - 8². This is relatively easy to compute; we have 4000² - 8² = 16000000 - 64 = 15999936. ---896. What is the value of the positive square root of [(530 + 113)(530 - 113) + 12769]? -Solution: First, we note that the number 12769 looks approximately like the square of 113. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(530 + 113)(530 - 113) + 12769] = [(530 + 113)(530 - 113) + 113²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(530 + 113)(530 - 113) + 113²] = [530² - 113² + 113²]. The second and third terms cancel, so the expression is just equal to 530². Clearly, the square root of the expression is equal to 530. ---897. Evaluate the expression (9735² - 265²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9735 and b = 265 in the difference-of-squares identity. We then get (9735² - 265²) = (9735 + 265)(9735 - 265). The Right Hand Side of this equation can be simplified to 10000 x 9470 = 94700000. ----
898. Evaluate the expression (80008² - 79992²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 80008 and b = 79992 in the difference-of-squares identity. We then get (80008² - 79992²) = (80008 + 79992)(80008 - 79992). The Right Hand Side of this equation can be simplified to 160000 x 16 = 2560000. ---899. Find the value of (60007 x 59993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 60007 and 59993 are close in value to 60000. We can rewrite the expression (60007 x 59993) as (60000 + 7)(60000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (60000 + 7)(60000 - 7) = 60000² - 7². This is relatively easy to compute; we have 60000² - 7² = 3600000000 - 49 = 3599999951. ---900. What is the value of the positive square root of [(580 + 191)(580 - 191) + 36481]? -Solution: First, we note that the number 36481 looks approximately like the square of 191. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(580 + 191)(580 - 191) + 36481] = [(580 + 191)(580 - 191) + 191²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(580 + 191)(580 - 191) + 191²] = [580² - 191² + 191²]. The second and third terms cancel, so the expression is just equal to 580². Clearly, the square root of the expression is equal to 580. ---901. Evaluate the expression (8146² - 1854²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) =
(a + b)(a - b). To simplify this particular expression, we set a = 8146 and b = 1854 in the difference-of-squares identity. We then get (8146² - 1854²) = (8146 + 1854)(8146 - 1854). The Right Hand Side of this equation can be simplified to 10000 x 6292 = 62920000. ---902. Evaluate the expression (5004² - 4996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5004 and b = 4996 in the difference-of-squares identity. We then get (5004² - 4996²) = (5004 + 4996)(5004 - 4996). The Right Hand Side of this equation can be simplified to 10000 x 8 = 80000. ---903. Find the value of (30008 x 29992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 30008 and 29992 are close in value to 30000. We can rewrite the expression (30008 x 29992) as (30000 + 8)(30000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (30000 + 8)(30000 - 8) = 30000² - 8². This is relatively easy to compute; we have 30000² - 8² = 900000000 - 64 = 899999936. ---904. What is the value of the positive square root of [(640 + 192)(640 - 192) + 36864]? -Solution: First, we note that the number 36864 looks approximately like the square of 192. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(640 + 192)(640 - 192) + 36864] = [(640 + 192)(640 - 192) + 192²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(640 + 192)(640 - 192) + 192²] = [640² - 192² + 192²]. The second and third terms cancel, so the expression is just equal to 640².
Clearly, the square root of the expression is equal to 640. ---905. Evaluate the expression (6690² - 3310²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6690 and b = 3310 in the difference-of-squares identity. We then get (6690² - 3310²) = (6690 + 3310)(6690 - 3310). The Right Hand Side of this equation can be simplified to 10000 x 3380 = 33800000. ---906. Evaluate the expression (6007² - 5993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6007 and b = 5993 in the difference-of-squares identity. We then get (6007² - 5993²) = (6007 + 5993)(6007 - 5993). The Right Hand Side of this equation can be simplified to 12000 x 14 = 168000. ---907. Find the value of (30004 x 29996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 30004 and 29996 are close in value to 30000. We can rewrite the expression (30004 x 29996) as (30000 + 4)(30000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (30000 + 4)(30000 - 4) = 30000² - 4². This is relatively easy to compute; we have 30000² - 4² = 900000000 - 16 = 899999984. ---908. What is the value of the positive square root of [(360 + 226)(360 - 226) + 51076]? --
Solution: First, we note that the number 51076 looks approximately like the square of 226. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(360 + 226)(360 - 226) + 51076] = [(360 + 226)(360 - 226) + 226²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(360 + 226)(360 - 226) + 226²] = [360² - 226² + 226²]. The second and third terms cancel, so the expression is just equal to 360². Clearly, the square root of the expression is equal to 360. ---909. Evaluate the expression (9962² - 38²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9962 and b = 38 in the difference-of-squares identity. We then get (9962² - 38²) = (9962 + 38)(9962 - 38). The Right Hand Side of this equation can be simplified to 10000 x 9924 = 99240000. ---910. Evaluate the expression (70004² - 69996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 70004 and b = 69996 in the difference-of-squares identity. We then get (70004² - 69996²) = (70004 + 69996)(70004 - 69996). The Right Hand Side of this equation can be simplified to 140000 x 8 = 1120000. ---911. Find the value of (80003 x 79997), by using algebraic identities to simplify the calculation. -Solution: We observe that both 80003 and 79997 are close in value to 80000. We can rewrite the expression (80003 x 79997) as (80000 + 3)(80000 - 3). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (80000 + 3)(80000 - 3) = 80000² - 3².
This is relatively easy to compute; we have 80000² - 3² = 6400000000 - 9 = 6399999991. ---912. What is the value of the positive square root of [(300 + 220)(300 - 220) + 48400]? -Solution: First, we note that the number 48400 looks approximately like the square of 220. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(300 + 220)(300 - 220) + 48400] = [(300 + 220)(300 - 220) + 220²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(300 + 220)(300 - 220) + 220²] = [300² - 220² + 220²]. The second and third terms cancel, so the expression is just equal to 300². Clearly, the square root of the expression is equal to 300. ---913. Evaluate the expression (9365² - 635²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9365 and b = 635 in the difference-of-squares identity. We then get (9365² - 635²) = (9365 + 635)(9365 - 635). The Right Hand Side of this equation can be simplified to 10000 x 8730 = 87300000. ---914. Evaluate the expression (80004² - 79996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 80004 and b = 79996 in the difference-of-squares identity. We then get (80004² - 79996²) = (80004 + 79996)(80004 - 79996). The Right Hand Side of this equation can be simplified to 160000 x 8 = 1280000. ----
915. Find the value of (8004 x 7996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 8004 and 7996 are close in value to 8000. We can rewrite the expression (8004 x 7996) as (8000 + 4)(8000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (8000 + 4)(8000 - 4) = 8000² - 4². This is relatively easy to compute; we have 8000² - 4² = 64000000 - 16 = 63999984. ---916. What is the value of the positive square root of [(690 + 239)(690 - 239) + 57121]? -Solution: First, we note that the number 57121 looks approximately like the square of 239. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(690 + 239)(690 - 239) + 57121] = [(690 + 239)(690 - 239) + 239²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(690 + 239)(690 - 239) + 239²] = [690² - 239² + 239²]. The second and third terms cancel, so the expression is just equal to 690². Clearly, the square root of the expression is equal to 690. ---917. Evaluate the expression (7681² - 2319²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7681 and b = 2319 in the difference-of-squares identity. We then get (7681² - 2319²) = (7681 + 2319)(7681 - 2319). The Right Hand Side of this equation can be simplified to 10000 x 5362 = 53620000. ---918. Evaluate the expression (9004² - 8996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) =
(a + b)(a - b). To simplify this particular expression, we set a = 9004 and b = 8996 in the difference-of-squares identity. We then get (9004² - 8996²) = (9004 + 8996)(9004 - 8996). The Right Hand Side of this equation can be simplified to 18000 x 8 = 144000. ---919. Find the value of (8003 x 7997), by using algebraic identities to simplify the calculation. -Solution: We observe that both 8003 and 7997 are close in value to 8000. We can rewrite the expression (8003 x 7997) as (8000 + 3)(8000 - 3). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (8000 + 3)(8000 - 3) = 8000² - 3². This is relatively easy to compute; we have 8000² - 3² = 64000000 - 9 = 63999991. ---920. What is the value of the positive square root of [(850 + 216)(850 - 216) + 46656]? -Solution: First, we note that the number 46656 looks approximately like the square of 216. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(850 + 216)(850 - 216) + 46656] = [(850 + 216)(850 - 216) + 216²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(850 + 216)(850 - 216) + 216²] = [850² - 216² + 216²]. The second and third terms cancel, so the expression is just equal to 850². Clearly, the square root of the expression is equal to 850. ---921. Evaluate the expression (6130² - 3870²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6130 and b = 3870 in the difference-of-squares identity. We then get (6130² - 3870²) = (6130 + 3870)(6130 - 3870).
The Right Hand Side of this equation can be simplified to 10000 x 2260 = 22600000. ---922. Evaluate the expression (4007² - 3993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 4007 and b = 3993 in the difference-of-squares identity. We then get (4007² - 3993²) = (4007 + 3993)(4007 - 3993). The Right Hand Side of this equation can be simplified to 8000 x 14 = 112000. ---923. Find the value of (4003 x 3997), by using algebraic identities to simplify the calculation. -Solution: We observe that both 4003 and 3997 are close in value to 4000. We can rewrite the expression (4003 x 3997) as (4000 + 3)(4000 - 3). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (4000 + 3)(4000 - 3) = 4000² - 3². This is relatively easy to compute; we have 4000² - 3² = 16000000 - 9 = 15999991. ---924. What is the value of the positive square root of [(450 + 160)(450 - 160) + 25600]? -Solution: First, we note that the number 25600 looks approximately like the square of 160. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(450 + 160)(450 - 160) + 25600] = [(450 + 160)(450 - 160) + 160²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(450 + 160)(450 - 160) + 160²] = [450² - 160² + 160²]. The second and third terms cancel, so the expression is just equal to 450². Clearly, the square root of the expression is equal to 450. ----
925. Evaluate the expression (5592² - 4408²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5592 and b = 4408 in the difference-of-squares identity. We then get (5592² - 4408²) = (5592 + 4408)(5592 - 4408). The Right Hand Side of this equation can be simplified to 10000 x 1184 = 11840000. ---926. Evaluate the expression (8002² - 7998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8002 and b = 7998 in the difference-of-squares identity. We then get (8002² - 7998²) = (8002 + 7998)(8002 - 7998). The Right Hand Side of this equation can be simplified to 16000 x 4 = 64000. ---927. Find the value of (4004 x 3996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 4004 and 3996 are close in value to 4000. We can rewrite the expression (4004 x 3996) as (4000 + 4)(4000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (4000 + 4)(4000 - 4) = 4000² - 4². This is relatively easy to compute; we have 4000² - 4² = 16000000 - 16 = 15999984. ---928. What is the value of the positive square root of [(810 + 269)(810 - 269) + 72361]? -Solution: First, we note that the number 72361 looks approximately like the square of 269. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(810 + 269)(810 - 269) + 72361] =
[(810 + 269)(810 - 269) + 269²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(810 + 269)(810 - 269) + 269²] = [810² - 269² + 269²]. The second and third terms cancel, so the expression is just equal to 810². Clearly, the square root of the expression is equal to 810. ---929. Evaluate the expression (7672² - 2328²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7672 and b = 2328 in the difference-of-squares identity. We then get (7672² - 2328²) = (7672 + 2328)(7672 - 2328). The Right Hand Side of this equation can be simplified to 10000 x 5344 = 53440000. ---930. Evaluate the expression (20006² - 19994²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 20006 and b = 19994 in the difference-of-squares identity. We then get (20006² - 19994²) = (20006 + 19994)(20006 - 19994). The Right Hand Side of this equation can be simplified to 40000 x 12 = 480000. ---931. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 70005 and 69995 are close in value to 70000. We can rewrite the expression (70005 x 69995) as (70000 + 5)(70000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (70000 + 5)(70000 - 5) = 70000² - 5². This is relatively easy to compute; we have 70000² - 5² = 4900000000 - 25 = 4899999975. ----
932. What is the value of the positive square root of [(540 + 150)(540 - 150) + 22500]? -Solution: First, we note that the number 22500 looks approximately like the square of 150. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(540 + 150)(540 - 150) + 22500] = [(540 + 150)(540 - 150) + 150²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(540 + 150)(540 - 150) + 150²] = [540² - 150² + 150²]. The second and third terms cancel, so the expression is just equal to 540². Clearly, the square root of the expression is equal to 540. ---933. Evaluate the expression (8163² - 1837²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8163 and b = 1837 in the difference-of-squares identity. We then get (8163² - 1837²) = (8163 + 1837)(8163 - 1837). The Right Hand Side of this equation can be simplified to 10000 x 6326 = 63260000. ---934. Evaluate the expression (7007² - 6993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7007 and b = 6993 in the difference-of-squares identity. We then get (7007² - 6993²) = (7007 + 6993)(7007 - 6993). The Right Hand Side of this equation can be simplified to 14000 x 14 = 196000. ---935. Find the value of (6007 x 5993), by using algebraic identities to simplify the calculation. --
Solution: We observe that both 6007 and 5993 are close in value to 6000. We can rewrite the expression (6007 x 5993) as (6000 + 7)(6000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (6000 + 7)(6000 - 7) = 6000² - 7². This is relatively easy to compute; we have 6000² - 7² = 36000000 - 49 = 35999951. ---936. What is the value of the positive square root of [(570 + 179)(570 - 179) + 32041]? -Solution: First, we note that the number 32041 looks approximately like the square of 179. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(570 + 179)(570 - 179) + 32041] = [(570 + 179)(570 - 179) + 179²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(570 + 179)(570 - 179) + 179²] = [570² - 179² + 179²]. The second and third terms cancel, so the expression is just equal to 570². Clearly, the square root of the expression is equal to 570. ---937. Evaluate the expression (6696² - 3304²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6696 and b = 3304 in the difference-of-squares identity. We then get (6696² - 3304²) = (6696 + 3304)(6696 - 3304). The Right Hand Side of this equation can be simplified to 10000 x 3392 = 33920000. ---938. Evaluate the expression (90007² - 89993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 90007 and b = 89993 in the difference-of-squares identity. We then get (90007² - 89993²) = (90007 + 89993)(90007 - 89993).
The Right Hand Side of this equation can be simplified to 180000 x 14 = 2520000. ---939. Find the value of (20009 x 19991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 20009 and 19991 are close in value to 20000. We can rewrite the expression (20009 x 19991) as (20000 + 9)(20000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (20000 + 9)(20000 - 9) = 20000² - 9². This is relatively easy to compute; we have 20000² - 9² = 400000000 - 81 = 399999919. ---940. What is the value of the positive square root of [(430 + 200)(430 - 200) + 40000]? -Solution: First, we note that the number 40000 looks approximately like the square of 200. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(430 + 200)(430 - 200) + 40000] = [(430 + 200)(430 - 200) + 200²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(430 + 200)(430 - 200) + 200²] = [430² - 200² + 200²]. The second and third terms cancel, so the expression is just equal to 430². Clearly, the square root of the expression is equal to 430. ---941. Evaluate the expression (6457² - 3543²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6457 and b = 3543 in the difference-of-squares identity. We then get (6457² - 3543²) = (6457 + 3543)(6457 - 3543). The Right Hand Side of this equation can be simplified to 10000 x 2914 = 29140000. ----
942. Evaluate the expression (8008² - 7992²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8008 and b = 7992 in the difference-of-squares identity. We then get (8008² - 7992²) = (8008 + 7992)(8008 - 7992). The Right Hand Side of this equation can be simplified to 16000 x 16 = 256000. ---943. Find the value of (2002 x 1998), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2002 and 1998 are close in value to 2000. We can rewrite the expression (2002 x 1998) as (2000 + 2)(2000 - 2). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 + 2)(2000 - 2) = 2000² - 2². This is relatively easy to compute; we have 2000² - 2² = 4000000 - 4 = 3999996. ---944. What is the value of the positive square root of [(690 + 185)(690 - 185) + 34225]? -Solution: First, we note that the number 34225 looks approximately like the square of 185. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(690 + 185)(690 - 185) + 34225] = [(690 + 185)(690 - 185) + 185²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(690 + 185)(690 - 185) + 185²] = [690² - 185² + 185²]. The second and third terms cancel, so the expression is just equal to 690². Clearly, the square root of the expression is equal to 690. ---945. Evaluate the expression (9461² - 539²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) =
(a + b)(a - b). To simplify this particular expression, we set a = 9461 and b = 539 in the difference-of-squares identity. We then get (9461² - 539²) = (9461 + 539)(9461 - 539). The Right Hand Side of this equation can be simplified to 10000 x 8922 = 89220000. ---946. Evaluate the expression (20003² - 19997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 20003 and b = 19997 in the difference-of-squares identity. We then get (20003² - 19997²) = (20003 + 19997)(20003 - 19997). The Right Hand Side of this equation can be simplified to 40000 x 6 = 240000. ---947. Find the value of (2005 x 1995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2005 and 1995 are close in value to 2000. We can rewrite the expression (2005 x 1995) as (2000 + 5)(2000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 + 5)(2000 - 5) = 2000² - 5². This is relatively easy to compute; we have 2000² - 5² = 4000000 - 25 = 3999975. ---948. What is the value of the positive square root of [(510 + 214)(510 - 214) + 45796]? -Solution: First, we note that the number 45796 looks approximately like the square of 214. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(510 + 214)(510 - 214) + 45796] = [(510 + 214)(510 - 214) + 214²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(510 + 214)(510 - 214) + 214²] = [510² - 214² + 214²]. The second and third
terms cancel, so the expression is just equal to 510². Clearly, the square root of the expression is equal to 510. ---949. Evaluate the expression (8015² - 1985²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8015 and b = 1985 in the difference-of-squares identity. We then get (8015² - 1985²) = (8015 + 1985)(8015 - 1985). The Right Hand Side of this equation can be simplified to 10000 x 6030 = 60300000. ---950. Evaluate the expression (7009² - 6991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7009 and b = 6991 in the difference-of-squares identity. We then get (7009² - 6991²) = (7009 + 6991)(7009 - 6991). The Right Hand Side of this equation can be simplified to 14000 x 18 = 252000. ---951. Find the value of (6008 x 5992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 6008 and 5992 are close in value to 6000. We can rewrite the expression (6008 x 5992) as (6000 + 8)(6000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (6000 + 8)(6000 - 8) = 6000² - 8². This is relatively easy to compute; we have 6000² - 8² = 36000000 - 64 = 35999936. ---952. What is the value of the positive square root of [(820 + 117)(820 - 117) + 13689]?
-Solution: First, we note that the number 13689 looks approximately like the square of 117. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(820 + 117)(820 - 117) + 13689] = [(820 + 117)(820 - 117) + 117²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(820 + 117)(820 - 117) + 117²] = [820² - 117² + 117²]. The second and third terms cancel, so the expression is just equal to 820². Clearly, the square root of the expression is equal to 820. ---953. Evaluate the expression (9062² - 938²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9062 and b = 938 in the difference-of-squares identity. We then get (9062² - 938²) = (9062 + 938)(9062 - 938). The Right Hand Side of this equation can be simplified to 10000 x 8124 = 81240000. ---954. Evaluate the expression (40002² - 39998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 40002 and b = 39998 in the difference-of-squares identity. We then get (40002² - 39998²) = (40002 + 39998)(40002 - 39998). The Right Hand Side of this equation can be simplified to 80000 x 4 = 320000. ---955. Find the value of (30006 x 29994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 30006 and 29994 are close in value to 30000. We can rewrite the expression (30006 x 29994) as (30000 + 6)(30000 - 6).
Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (30000 + 6)(30000 - 6) = 30000² - 6². This is relatively easy to compute; we have 30000² - 6² = 900000000 - 36 = 899999964. ---956. What is the value of the positive square root of [(510 + 184)(510 - 184) + 33856]? -Solution: First, we note that the number 33856 looks approximately like the square of 184. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(510 + 184)(510 - 184) + 33856] = [(510 + 184)(510 - 184) + 184²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(510 + 184)(510 - 184) + 184²] = [510² - 184² + 184²]. The second and third terms cancel, so the expression is just equal to 510². Clearly, the square root of the expression is equal to 510. ---957. Evaluate the expression (7785² - 2215²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7785 and b = 2215 in the difference-of-squares identity. We then get (7785² - 2215²) = (7785 + 2215)(7785 - 2215). The Right Hand Side of this equation can be simplified to 10000 x 5570 = 55700000. ---958. Evaluate the expression (70005² - 69995²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 70005 and b = 69995 in the difference-of-squares identity. We then get (70005² - 69995²) = (70005 + 69995)(70005 - 69995). The Right Hand Side of this equation can be simplified to 140000 x 10 =
1400000. ---959. Find the value of (30002 x 29998), by using algebraic identities to simplify the calculation. -Solution: We observe that both 30002 and 29998 are close in value to 30000. We can rewrite the expression (30002 x 29998) as (30000 + 2)(30000 - 2). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (30000 + 2)(30000 - 2) = 30000² - 2². This is relatively easy to compute; we have 30000² - 2² = 900000000 - 4 = 899999996. ---960. What is the value of the positive square root of [(690 + 281)(690 - 281) + 78961]? -Solution: First, we note that the number 78961 looks approximately like the square of 281. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(690 + 281)(690 - 281) + 78961] = [(690 + 281)(690 - 281) + 281²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(690 + 281)(690 - 281) + 281²] = [690² - 281² + 281²]. The second and third terms cancel, so the expression is just equal to 690². Clearly, the square root of the expression is equal to 690. ---961. Evaluate the expression (5618² - 4382²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5618 and b = 4382 in the difference-of-squares identity. We then get (5618² - 4382²) = (5618 + 4382)(5618 - 4382). The Right Hand Side of this equation can be simplified to 10000 x 1236 = 12360000. ----
962. Evaluate the expression (3009² - 2991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 3009 and b = 2991 in the difference-of-squares identity. We then get (3009² - 2991²) = (3009 + 2991)(3009 - 2991). The Right Hand Side of this equation can be simplified to 6000 x 18 = 108000. ---963. Find the value of (9009 x 8991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 9009 and 8991 are close in value to 9000. We can rewrite the expression (9009 x 8991) as (9000 + 9)(9000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (9000 + 9)(9000 - 9) = 9000² - 9². This is relatively easy to compute; we have 9000² - 9² = 81000000 - 81 = 80999919. ---964. What is the value of the positive square root of [(880 + 135)(880 - 135) + 18225]? -Solution: First, we note that the number 18225 looks approximately like the square of 135. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(880 + 135)(880 - 135) + 18225] = [(880 + 135)(880 - 135) + 135²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(880 + 135)(880 - 135) + 135²] = [880² - 135² + 135²]. The second and third terms cancel, so the expression is just equal to 880². Clearly, the square root of the expression is equal to 880. ---965. Evaluate the expression (7067² - 2933²), by using algebraic identities to simplify computation. --
Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7067 and b = 2933 in the difference-of-squares identity. We then get (7067² - 2933²) = (7067 + 2933)(7067 - 2933). The Right Hand Side of this equation can be simplified to 10000 x 4134 = 41340000. ---966. Evaluate the expression (3008² - 2992²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 3008 and b = 2992 in the difference-of-squares identity. We then get (3008² - 2992²) = (3008 + 2992)(3008 - 2992). The Right Hand Side of this equation can be simplified to 6000 x 16 = 96000. ---967. Find the value of (80009 x 79991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 80009 and 79991 are close in value to 80000. We can rewrite the expression (80009 x 79991) as (80000 + 9)(80000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (80000 + 9)(80000 - 9) = 80000² - 9². This is relatively easy to compute; we have 80000² - 9² = 6400000000 - 81 = 6399999919. ---968. What is the value of the positive square root of [(840 + 230)(840 - 230) + 52900]? -Solution: First, we note that the number 52900 looks approximately like the square of 230. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(840 + 230)(840 - 230) + 52900] = [(840 + 230)(840 - 230) + 230²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as
[(840 + 230)(840 - 230) + 230²] = [840² - 230² + 230²]. The second and third terms cancel, so the expression is just equal to 840². Clearly, the square root of the expression is equal to 840. ---969. Evaluate the expression (7001² - 2999²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7001 and b = 2999 in the difference-of-squares identity. We then get (7001² - 2999²) = (7001 + 2999)(7001 - 2999). The Right Hand Side of this equation can be simplified to 10000 x 4002 = 40020000. ---970. Evaluate the expression (20002² - 19998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 20002 and b = 19998 in the difference-of-squares identity. We then get (20002² - 19998²) = (20002 + 19998)(20002 - 19998). The Right Hand Side of this equation can be simplified to 40000 x 4 = 160000. ---971. Find the value of (2007 x 1993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2007 and 1993 are close in value to 2000. We can rewrite the expression (2007 x 1993) as (2000 + 7)(2000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 + 7)(2000 - 7) = 2000² - 7². This is relatively easy to compute; we have 2000² - 7² = 4000000 - 49 = 3999951. ---972. What is the value of the positive square root of [(880 + 263)(880 - 263)
+ 69169]? -Solution: First, we note that the number 69169 looks approximately like the square of 263. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(880 + 263)(880 - 263) + 69169] = [(880 + 263)(880 - 263) + 263²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(880 + 263)(880 - 263) + 263²] = [880² - 263² + 263²]. The second and third terms cancel, so the expression is just equal to 880². Clearly, the square root of the expression is equal to 880. ---973. Evaluate the expression (5721² - 4279²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5721 and b = 4279 in the difference-of-squares identity. We then get (5721² - 4279²) = (5721 + 4279)(5721 - 4279). The Right Hand Side of this equation can be simplified to 10000 x 1442 = 14420000. ---974. Evaluate the expression (7009² - 6991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7009 and b = 6991 in the difference-of-squares identity. We then get (7009² - 6991²) = (7009 + 6991)(7009 - 6991). The Right Hand Side of this equation can be simplified to 14000 x 18 = 252000. ---975. Find the value of (60005 x 59995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 60005 and 59995 are close in value to 60000. We can rewrite the expression (60005 x 59995) as (60000 + 5)(60000 - 5).
Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (60000 + 5)(60000 - 5) = 60000² - 5². This is relatively easy to compute; we have 60000² - 5² = 3600000000 - 25 = 3599999975. ---976. What is the value of the positive square root of [(730 + 154)(730 - 154) + 23716]? -Solution: First, we note that the number 23716 looks approximately like the square of 154. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(730 + 154)(730 - 154) + 23716] = [(730 + 154)(730 - 154) + 154²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(730 + 154)(730 - 154) + 154²] = [730² - 154² + 154²]. The second and third terms cancel, so the expression is just equal to 730². Clearly, the square root of the expression is equal to 730. ---977. Evaluate the expression (7696² - 2304²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7696 and b = 2304 in the difference-of-squares identity. We then get (7696² - 2304²) = (7696 + 2304)(7696 - 2304). The Right Hand Side of this equation can be simplified to 10000 x 5392 = 53920000. ---978. Evaluate the expression (5002² - 4998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5002 and b = 4998 in the difference-of-squares identity. We then get (5002² - 4998²) = (5002 + 4998)(5002 - 4998). The Right Hand Side of this equation can be simplified to 10000 x 4 =
40000. ---979. Find the value of (7002 x 6998), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7002 and 6998 are close in value to 7000. We can rewrite the expression (7002 x 6998) as (7000 + 2)(7000 - 2). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 2)(7000 - 2) = 7000² - 2². This is relatively easy to compute; we have 7000² - 2² = 49000000 - 4 = 48999996. ---980. What is the value of the positive square root of [(390 + 278)(390 - 278) + 77284]? -Solution: First, we note that the number 77284 looks approximately like the square of 278. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(390 + 278)(390 - 278) + 77284] = [(390 + 278)(390 - 278) + 278²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(390 + 278)(390 - 278) + 278²] = [390² - 278² + 278²]. The second and third terms cancel, so the expression is just equal to 390². Clearly, the square root of the expression is equal to 390. ---981. Evaluate the expression (6300² - 3700²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6300 and b = 3700 in the difference-of-squares identity. We then get (6300² - 3700²) = (6300 + 3700)(6300 - 3700). The Right Hand Side of this equation can be simplified to 10000 x 2600 = 26000000. ---982. Evaluate the expression (90005² - 89995²), by using algebraic identities
to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 90005 and b = 89995 in the difference-of-squares identity. We then get (90005² - 89995²) = (90005 + 89995)(90005 - 89995). The Right Hand Side of this equation can be simplified to 180000 x 10 = 1800000. ---983. Find the value of (3007 x 2993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 3007 and 2993 are close in value to 3000. We can rewrite the expression (3007 x 2993) as (3000 + 7)(3000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (3000 + 7)(3000 - 7) = 3000² - 7². This is relatively easy to compute; we have 3000² - 7² = 9000000 - 49 = 8999951. ---984. What is the value of the positive square root of [(690 + 152)(690 - 152) + 23104]? -Solution: First, we note that the number 23104 looks approximately like the square of 152. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(690 + 152)(690 - 152) + 23104] = [(690 + 152)(690 - 152) + 152²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(690 + 152)(690 - 152) + 152²] = [690² - 152² + 152²]. The second and third terms cancel, so the expression is just equal to 690². Clearly, the square root of the expression is equal to 690. ---985. Evaluate the expression (8713² - 1287²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) =
(a + b)(a - b). To simplify this particular expression, we set a = 8713 and b = 1287 in the difference-of-squares identity. We then get (8713² - 1287²) = (8713 + 1287)(8713 - 1287). The Right Hand Side of this equation can be simplified to 10000 x 7426 = 74260000. ---986. Evaluate the expression (8003² - 7997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8003 and b = 7997 in the difference-of-squares identity. We then get (8003² - 7997²) = (8003 + 7997)(8003 - 7997). The Right Hand Side of this equation can be simplified to 16000 x 6 = 96000. ---987. Find the value of (90004 x 89996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 90004 and 89996 are close in value to 90000. We can rewrite the expression (90004 x 89996) as (90000 + 4)(90000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (90000 + 4)(90000 - 4) = 90000² - 4². This is relatively easy to compute; we have 90000² - 4² = 8100000000 - 16 = 8099999984. ---988. What is the value of the positive square root of [(310 + 184)(310 - 184) + 33856]? -Solution: First, we note that the number 33856 looks approximately like the square of 184. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(310 + 184)(310 - 184) + 33856] = [(310 + 184)(310 - 184) + 184²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(310 + 184)(310 - 184) + 184²] = [310² - 184² + 184²]. The second and third
terms cancel, so the expression is just equal to 310². Clearly, the square root of the expression is equal to 310. ---989. Evaluate the expression (5842² - 4158²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5842 and b = 4158 in the difference-of-squares identity. We then get (5842² - 4158²) = (5842 + 4158)(5842 - 4158). The Right Hand Side of this equation can be simplified to 10000 x 1684 = 16840000. ---990. Evaluate the expression (2004² - 1996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 2004 and b = 1996 in the difference-of-squares identity. We then get (2004² - 1996²) = (2004 + 1996)(2004 - 1996). The Right Hand Side of this equation can be simplified to 4000 x 8 = 32000. ---991. Find the value of (60007 x 59993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 60007 and 59993 are close in value to 60000. We can rewrite the expression (60007 x 59993) as (60000 + 7)(60000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (60000 + 7)(60000 - 7) = 60000² - 7². This is relatively easy to compute; we have 60000² - 7² = 3600000000 - 49 = 3599999951. ---992. What is the value of the positive square root of [(540 + 285)(540 - 285) + 81225]? --
Solution: First, we note that the number 81225 looks approximately like the square of 285. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(540 + 285)(540 - 285) + 81225] = [(540 + 285)(540 - 285) + 285²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(540 + 285)(540 - 285) + 285²] = [540² - 285² + 285²]. The second and third terms cancel, so the expression is just equal to 540². Clearly, the square root of the expression is equal to 540. ---993. Evaluate the expression (9310² - 690²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9310 and b = 690 in the difference-of-squares identity. We then get (9310² - 690²) = (9310 + 690)(9310 - 690). The Right Hand Side of this equation can be simplified to 10000 x 8620 = 86200000. ---994. Evaluate the expression (9007² - 8993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9007 and b = 8993 in the difference-of-squares identity. We then get (9007² - 8993²) = (9007 + 8993)(9007 - 8993). The Right Hand Side of this equation can be simplified to 18000 x 14 = 252000. ---995. Find the value of (20007 x 19993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 20007 and 19993 are close in value to 20000. We can rewrite the expression (20007 x 19993) as (20000 + 7)(20000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (20000
+ 7)(20000 - 7) = 20000² - 7². This is relatively easy to compute; we have 20000² - 7² = 400000000 - 49 = 399999951. ---996. What is the value of the positive square root of [(580 + 135)(580 - 135) + 18225]? -Solution: First, we note that the number 18225 looks approximately like the square of 135. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(580 + 135)(580 - 135) + 18225] = [(580 + 135)(580 - 135) + 135²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(580 + 135)(580 - 135) + 135²] = [580² - 135² + 135²]. The second and third terms cancel, so the expression is just equal to 580². Clearly, the square root of the expression is equal to 580. ---997. Evaluate the expression (8169² - 1831²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8169 and b = 1831 in the difference-of-squares identity. We then get (8169² - 1831²) = (8169 + 1831)(8169 - 1831). The Right Hand Side of this equation can be simplified to 10000 x 6338 = 63380000. ---998. Evaluate the expression (9004² - 8996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9004 and b = 8996 in the difference-of-squares identity. We then get (9004² - 8996²) = (9004 + 8996)(9004 - 8996). The Right Hand Side of this equation can be simplified to 18000 x 8 = 144000. ----
999. Find the value of (2008 x 1992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2008 and 1992 are close in value to 2000. We can rewrite the expression (2008 x 1992) as (2000 + 8)(2000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 + 8)(2000 - 8) = 2000² - 8². This is relatively easy to compute; we have 2000² - 8² = 4000000 - 64 = 3999936. ---1000. What is the value of the positive square root of [(530 + 110)(530 110) + 12100]? -Solution: First, we note that the number 12100 looks approximately like the square of 110. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(530 + 110)(530 - 110) + 12100] = [(530 + 110)(530 - 110) + 110²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(530 + 110)(530 - 110) + 110²] = [530² - 110² + 110²]. The second and third terms cancel, so the expression is just equal to 530². Clearly, the square root of the expression is equal to 530. ---1001. Evaluate the expression (8972² - 1028²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8972 and b = 1028 in the difference-of-squares identity. We then get (8972² - 1028²) = (8972 + 1028)(8972 - 1028). The Right Hand Side of this equation can be simplified to 10000 x 7944 = 79440000. ---1002. Evaluate the expression (80008² - 79992²), by using algebraic identities to simplify computation. --
Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 80008 and b = 79992 in the difference-of-squares identity. We then get (80008² - 79992²) = (80008 + 79992)(80008 - 79992). The Right Hand Side of this equation can be simplified to 160000 x 16 = 2560000. ---1003. Find the value of (9008 x 8992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 9008 and 8992 are close in value to 9000. We can rewrite the expression (9008 x 8992) as (9000 + 8)(9000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (9000 + 8)(9000 - 8) = 9000² - 8². This is relatively easy to compute; we have 9000² - 8² = 81000000 - 64 = 80999936. ---1004. What is the value of the positive square root of [(770 + 181)(770 181) + 32761]? -Solution: First, we note that the number 32761 looks approximately like the square of 181. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(770 + 181)(770 - 181) + 32761] = [(770 + 181)(770 - 181) + 181²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(770 + 181)(770 - 181) + 181²] = [770² - 181² + 181²]. The second and third terms cancel, so the expression is just equal to 770². Clearly, the square root of the expression is equal to 770. ---1005. Evaluate the expression (7925² - 2075²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7925 and b = 2075 in the difference-of-squares identity. We then get (7925² - 2075²) = (7925 + 2075)(7925 - 2075).
The Right Hand Side of this equation can be simplified to 10000 x 5850 = 58500000. ---1006. Evaluate the expression (80002² - 79998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 80002 and b = 79998 in the difference-of-squares identity. We then get (80002² - 79998²) = (80002 + 79998)(80002 - 79998). The Right Hand Side of this equation can be simplified to 160000 x 4 = 640000. ---1007. Find the value of (2006 x 1994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2006 and 1994 are close in value to 2000. We can rewrite the expression (2006 x 1994) as (2000 + 6)(2000 - 6). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 + 6)(2000 - 6) = 2000² - 6². This is relatively easy to compute; we have 2000² - 6² = 4000000 - 36 = 3999964. ---1008. What is the value of the positive square root of [(670 + 162)(670 162) + 26244]? -Solution: First, we note that the number 26244 looks approximately like the square of 162. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(670 + 162)(670 - 162) + 26244] = [(670 + 162)(670 - 162) + 162²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(670 + 162)(670 - 162) + 162²] = [670² - 162² + 162²]. The second and third terms cancel, so the expression is just equal to 670². Clearly, the square root of the expression is equal to 670. ----
1009. Evaluate the expression (5607² - 4393²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5607 and b = 4393 in the difference-of-squares identity. We then get (5607² - 4393²) = (5607 + 4393)(5607 - 4393). The Right Hand Side of this equation can be simplified to 10000 x 1214 = 12140000. ---1010. Evaluate the expression (8003² - 7997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8003 and b = 7997 in the difference-of-squares identity. We then get (8003² - 7997²) = (8003 + 7997)(8003 - 7997). The Right Hand Side of this equation can be simplified to 16000 x 6 = 96000. ---1011. Find the value of (20007 x 19993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 20007 and 19993 are close in value to 20000. We can rewrite the expression (20007 x 19993) as (20000 + 7)(20000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (20000 + 7)(20000 - 7) = 20000² - 7². This is relatively easy to compute; we have 20000² - 7² = 400000000 - 49 = 399999951. ---1012. What is the value of the positive square root of [(310 + 229)(310 229) + 52441]? -Solution: First, we note that the number 52441 looks approximately like the square of 229. We can verify this through a quick computation. Therefore,
the given expression can be rewritten as [(310 + 229)(310 - 229) + 52441] = [(310 + 229)(310 - 229) + 229²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(310 + 229)(310 - 229) + 229²] = [310² - 229² + 229²]. The second and third terms cancel, so the expression is just equal to 310². Clearly, the square root of the expression is equal to 310. ---1013. Evaluate the expression (8793² - 1207²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8793 and b = 1207 in the difference-of-squares identity. We then get (8793² - 1207²) = (8793 + 1207)(8793 - 1207). The Right Hand Side of this equation can be simplified to 10000 x 7586 = 75860000. ---1014. Evaluate the expression (7003² - 6997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7003 and b = 6997 in the difference-of-squares identity. We then get (7003² - 6997²) = (7003 + 6997)(7003 - 6997). The Right Hand Side of this equation can be simplified to 14000 x 6 = 84000. ---1015. Find the value of (90005 x 89995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 90005 and 89995 are close in value to 90000. We can rewrite the expression (90005 x 89995) as (90000 + 5)(90000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (90000 + 5)(90000 - 5) = 90000² - 5². This is relatively easy to compute; we have 90000² - 5² = 8100000000 - 25 = 8099999975.
---1016. What is the value of the positive square root of [(770 + 211)(770 211) + 44521]? -Solution: First, we note that the number 44521 looks approximately like the square of 211. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(770 + 211)(770 - 211) + 44521] = [(770 + 211)(770 - 211) + 211²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(770 + 211)(770 - 211) + 211²] = [770² - 211² + 211²]. The second and third terms cancel, so the expression is just equal to 770². Clearly, the square root of the expression is equal to 770. ---1017. Evaluate the expression (7078² - 2922²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7078 and b = 2922 in the difference-of-squares identity. We then get (7078² - 2922²) = (7078 + 2922)(7078 - 2922). The Right Hand Side of this equation can be simplified to 10000 x 4156 = 41560000. ---1018. Evaluate the expression (40003² - 39997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 40003 and b = 39997 in the difference-of-squares identity. We then get (40003² - 39997²) = (40003 + 39997)(40003 - 39997). The Right Hand Side of this equation can be simplified to 80000 x 6 = 480000. ---1019. Find the value of (80003 x 79997), by using algebraic identities to simplify the calculation.
-Solution: We observe that both 80003 and 79997 are close in value to 80000. We can rewrite the expression (80003 x 79997) as (80000 + 3)(80000 - 3). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (80000 + 3)(80000 - 3) = 80000² - 3². This is relatively easy to compute; we have 80000² - 3² = 6400000000 - 9 = 6399999991. ---1020. What is the value of the positive square root of [(810 + 128)(810 128) + 16384]? -Solution: First, we note that the number 16384 looks approximately like the square of 128. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(810 + 128)(810 - 128) + 16384] = [(810 + 128)(810 - 128) + 128²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(810 + 128)(810 - 128) + 128²] = [810² - 128² + 128²]. The second and third terms cancel, so the expression is just equal to 810². Clearly, the square root of the expression is equal to 810. ---1021. Evaluate the expression (6117² - 3883²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6117 and b = 3883 in the difference-of-squares identity. We then get (6117² - 3883²) = (6117 + 3883)(6117 - 3883). The Right Hand Side of this equation can be simplified to 10000 x 2234 = 22340000. ---1022. Evaluate the expression (2006² - 1994²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 2006 and b =
1994 in the difference-of-squares identity. We then get (2006² - 1994²) = (2006 + 1994)(2006 - 1994). The Right Hand Side of this equation can be simplified to 4000 x 12 = 48000. ---1023. Find the value of (6007 x 5993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 6007 and 5993 are close in value to 6000. We can rewrite the expression (6007 x 5993) as (6000 + 7)(6000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (6000 + 7)(6000 - 7) = 6000² - 7². This is relatively easy to compute; we have 6000² - 7² = 36000000 - 49 = 35999951. ---1024. What is the value of the positive square root of [(420 + 265)(420 265) + 70225]? -Solution: First, we note that the number 70225 looks approximately like the square of 265. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(420 + 265)(420 - 265) + 70225] = [(420 + 265)(420 - 265) + 265²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(420 + 265)(420 - 265) + 265²] = [420² - 265² + 265²]. The second and third terms cancel, so the expression is just equal to 420². Clearly, the square root of the expression is equal to 420. ---1025. Evaluate the expression (6801² - 3199²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6801 and b = 3199 in the difference-of-squares identity. We then get (6801² - 3199²) = (6801 + 3199)(6801 - 3199). The Right Hand Side of this equation can be simplified to 10000 x 3602 =
36020000. ---1026. Evaluate the expression (50002² - 49998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 50002 and b = 49998 in the difference-of-squares identity. We then get (50002² - 49998²) = (50002 + 49998)(50002 - 49998). The Right Hand Side of this equation can be simplified to 100000 x 4 = 400000. ---1027. Find the value of (5005 x 4995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 5005 and 4995 are close in value to 5000. We can rewrite the expression (5005 x 4995) as (5000 + 5)(5000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (5000 + 5)(5000 - 5) = 5000² - 5². This is relatively easy to compute; we have 5000² - 5² = 25000000 - 25 = 24999975. ---1028. What is the value of the positive square root of [(470 + 205)(470 205) + 42025]? -Solution: First, we note that the number 42025 looks approximately like the square of 205. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(470 + 205)(470 - 205) + 42025] = [(470 + 205)(470 - 205) + 205²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(470 + 205)(470 - 205) + 205²] = [470² - 205² + 205²]. The second and third terms cancel, so the expression is just equal to 470². Clearly, the square root of the expression is equal to 470. ---1029. Evaluate the expression (5379² - 4621²), by using algebraic identities
to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5379 and b = 4621 in the difference-of-squares identity. We then get (5379² - 4621²) = (5379 + 4621)(5379 - 4621). The Right Hand Side of this equation can be simplified to 10000 x 758 = 7580000. ---1030. Evaluate the expression (8005² - 7995²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8005 and b = 7995 in the difference-of-squares identity. We then get (8005² - 7995²) = (8005 + 7995)(8005 - 7995). The Right Hand Side of this equation can be simplified to 16000 x 10 = 160000. ---1031. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 70005 and 69995 are close in value to 70000. We can rewrite the expression (70005 x 69995) as (70000 + 5)(70000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (70000 + 5)(70000 - 5) = 70000² - 5². This is relatively easy to compute; we have 70000² - 5² = 4900000000 - 25 = 4899999975. ---1032. What is the value of the positive square root of [(300 + 103)(300 103) + 10609]? -Solution: First, we note that the number 10609 looks approximately like the square of 103. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(300 + 103)(300 - 103) + 10609] =
[(300 + 103)(300 - 103) + 103²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(300 + 103)(300 - 103) + 103²] = [300² - 103² + 103²]. The second and third terms cancel, so the expression is just equal to 300². Clearly, the square root of the expression is equal to 300. ---1033. Evaluate the expression (5490² - 4510²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5490 and b = 4510 in the difference-of-squares identity. We then get (5490² - 4510²) = (5490 + 4510)(5490 - 4510). The Right Hand Side of this equation can be simplified to 10000 x 980 = 9800000. ---1034. Evaluate the expression (9009² - 8991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9009 and b = 8991 in the difference-of-squares identity. We then get (9009² - 8991²) = (9009 + 8991)(9009 - 8991). The Right Hand Side of this equation can be simplified to 18000 x 18 = 324000. ---1035. Find the value of (5006 x 4994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 5006 and 4994 are close in value to 5000. We can rewrite the expression (5006 x 4994) as (5000 + 6)(5000 - 6). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (5000 + 6)(5000 - 6) = 5000² - 6². This is relatively easy to compute; we have 5000² - 6² = 25000000 - 36 = 24999964.
---1036. What is the value of the positive square root of [(610 + 159)(610 159) + 25281]? -Solution: First, we note that the number 25281 looks approximately like the square of 159. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(610 + 159)(610 - 159) + 25281] = [(610 + 159)(610 - 159) + 159²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(610 + 159)(610 - 159) + 159²] = [610² - 159² + 159²]. The second and third terms cancel, so the expression is just equal to 610². Clearly, the square root of the expression is equal to 610. ---1037. Evaluate the expression (9042² - 958²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9042 and b = 958 in the difference-of-squares identity. We then get (9042² - 958²) = (9042 + 958)(9042 - 958). The Right Hand Side of this equation can be simplified to 10000 x 8084 = 80840000. ---1038. Evaluate the expression (3006² - 2994²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 3006 and b = 2994 in the difference-of-squares identity. We then get (3006² - 2994²) = (3006 + 2994)(3006 - 2994). The Right Hand Side of this equation can be simplified to 6000 x 12 = 72000. ---1039. Find the value of (4006 x 3994), by using algebraic identities to
simplify the calculation. -Solution: We observe that both 4006 and 3994 are close in value to 4000. We can rewrite the expression (4006 x 3994) as (4000 + 6)(4000 - 6). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (4000 + 6)(4000 - 6) = 4000² - 6². This is relatively easy to compute; we have 4000² - 6² = 16000000 - 36 = 15999964. ---1040. What is the value of the positive square root of [(670 + 197)(670 197) + 38809]? -Solution: First, we note that the number 38809 looks approximately like the square of 197. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(670 + 197)(670 - 197) + 38809] = [(670 + 197)(670 - 197) + 197²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(670 + 197)(670 - 197) + 197²] = [670² - 197² + 197²]. The second and third terms cancel, so the expression is just equal to 670². Clearly, the square root of the expression is equal to 670. ---1041. Evaluate the expression (7943² - 2057²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7943 and b = 2057 in the difference-of-squares identity. We then get (7943² - 2057²) = (7943 + 2057)(7943 - 2057). The Right Hand Side of this equation can be simplified to 10000 x 5886 = 58860000. ---1042. Evaluate the expression (3007² - 2993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) =
(a + b)(a - b). To simplify this particular expression, we set a = 3007 and b = 2993 in the difference-of-squares identity. We then get (3007² - 2993²) = (3007 + 2993)(3007 - 2993). The Right Hand Side of this equation can be simplified to 6000 x 14 = 84000. ---1043. Find the value of (90003 x 89997), by using algebraic identities to simplify the calculation. -Solution: We observe that both 90003 and 89997 are close in value to 90000. We can rewrite the expression (90003 x 89997) as (90000 + 3)(90000 - 3). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (90000 + 3)(90000 - 3) = 90000² - 3². This is relatively easy to compute; we have 90000² - 3² = 8100000000 - 9 = 8099999991. ---1044. What is the value of the positive square root of [(540 + 188)(540 188) + 35344]? -Solution: First, we note that the number 35344 looks approximately like the square of 188. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(540 + 188)(540 - 188) + 35344] = [(540 + 188)(540 - 188) + 188²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(540 + 188)(540 - 188) + 188²] = [540² - 188² + 188²]. The second and third terms cancel, so the expression is just equal to 540². Clearly, the square root of the expression is equal to 540. ---1045. Evaluate the expression (7030² - 2970²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7030 and b = 2970 in the difference-of-squares identity. We then get (7030² - 2970²) = (7030 + 2970)(7030 - 2970). The Right Hand Side of this equation can be simplified to 10000 x 4060 =
40600000. ---1046. Evaluate the expression (3009² - 2991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 3009 and b = 2991 in the difference-of-squares identity. We then get (3009² - 2991²) = (3009 + 2991)(3009 - 2991). The Right Hand Side of this equation can be simplified to 6000 x 18 = 108000. ---1047. Find the value of (4009 x 3991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 4009 and 3991 are close in value to 4000. We can rewrite the expression (4009 x 3991) as (4000 + 9)(4000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (4000 + 9)(4000 - 9) = 4000² - 9². This is relatively easy to compute; we have 4000² - 9² = 16000000 - 81 = 15999919. ---1048. What is the value of the positive square root of [(470 + 160)(470 160) + 25600]? -Solution: First, we note that the number 25600 looks approximately like the square of 160. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(470 + 160)(470 - 160) + 25600] = [(470 + 160)(470 - 160) + 160²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(470 + 160)(470 - 160) + 160²] = [470² - 160² + 160²]. The second and third terms cancel, so the expression is just equal to 470². Clearly, the square root of the expression is equal to 470. ----
1049. Evaluate the expression (7643² - 2357²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7643 and b = 2357 in the difference-of-squares identity. We then get (7643² - 2357²) = (7643 + 2357)(7643 - 2357). The Right Hand Side of this equation can be simplified to 10000 x 5286 = 52860000. ---1050. Evaluate the expression (20004² - 19996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 20004 and b = 19996 in the difference-of-squares identity. We then get (20004² - 19996²) = (20004 + 19996)(20004 - 19996). The Right Hand Side of this equation can be simplified to 40000 x 8 = 320000. ---1051. Find the value of (90002 x 89998), by using algebraic identities to simplify the calculation. -Solution: We observe that both 90002 and 89998 are close in value to 90000. We can rewrite the expression (90002 x 89998) as (90000 + 2)(90000 - 2). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (90000 + 2)(90000 - 2) = 90000² - 2². This is relatively easy to compute; we have 90000² - 2² = 8100000000 - 4 = 8099999996. ---1052. What is the value of the positive square root of [(300 + 133)(300 133) + 17689]? -Solution: First, we note that the number 17689 looks approximately like the square of 133. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(300 + 133)(300 - 133) + 17689] =
[(300 + 133)(300 - 133) + 133²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(300 + 133)(300 - 133) + 133²] = [300² - 133² + 133²]. The second and third terms cancel, so the expression is just equal to 300². Clearly, the square root of the expression is equal to 300. ---1053. Evaluate the expression (9793² - 207²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9793 and b = 207 in the difference-of-squares identity. We then get (9793² - 207²) = (9793 + 207)(9793 - 207). The Right Hand Side of this equation can be simplified to 10000 x 9586 = 95860000. ---1054. Evaluate the expression (7002² - 6998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7002 and b = 6998 in the difference-of-squares identity. We then get (7002² - 6998²) = (7002 + 6998)(7002 - 6998). The Right Hand Side of this equation can be simplified to 14000 x 4 = 56000. ---1055. Find the value of (60009 x 59991), by using algebraic identities to simplify the calculation. -Solution: We observe that both 60009 and 59991 are close in value to 60000. We can rewrite the expression (60009 x 59991) as (60000 + 9)(60000 - 9). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (60000 + 9)(60000 - 9) = 60000² - 9². This is relatively easy to compute; we have 60000² - 9² = 3600000000 - 81 = 3599999919.
---1056. What is the value of the positive square root of [(890 + 277)(890 277) + 76729]? -Solution: First, we note that the number 76729 looks approximately like the square of 277. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(890 + 277)(890 - 277) + 76729] = [(890 + 277)(890 - 277) + 277²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(890 + 277)(890 - 277) + 277²] = [890² - 277² + 277²]. The second and third terms cancel, so the expression is just equal to 890². Clearly, the square root of the expression is equal to 890. ---1057. Evaluate the expression (9578² - 422²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9578 and b = 422 in the difference-of-squares identity. We then get (9578² - 422²) = (9578 + 422)(9578 - 422). The Right Hand Side of this equation can be simplified to 10000 x 9156 = 91560000. ---1058. Evaluate the expression (80005² - 79995²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 80005 and b = 79995 in the difference-of-squares identity. We then get (80005² - 79995²) = (80005 + 79995)(80005 - 79995). The Right Hand Side of this equation can be simplified to 160000 x 10 = 1600000. ---1059. Find the value of (5004 x 4996), by using algebraic identities to simplify the calculation.
-Solution: We observe that both 5004 and 4996 are close in value to 5000. We can rewrite the expression (5004 x 4996) as (5000 + 4)(5000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (5000 + 4)(5000 - 4) = 5000² - 4². This is relatively easy to compute; we have 5000² - 4² = 25000000 - 16 = 24999984. ---1060. What is the value of the positive square root of [(840 + 255)(840 255) + 65025]? -Solution: First, we note that the number 65025 looks approximately like the square of 255. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(840 + 255)(840 - 255) + 65025] = [(840 + 255)(840 - 255) + 255²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(840 + 255)(840 - 255) + 255²] = [840² - 255² + 255²]. The second and third terms cancel, so the expression is just equal to 840². Clearly, the square root of the expression is equal to 840. ---1061. Evaluate the expression (9536² - 464²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9536 and b = 464 in the difference-of-squares identity. We then get (9536² - 464²) = (9536 + 464)(9536 - 464). The Right Hand Side of this equation can be simplified to 10000 x 9072 = 90720000. ---1062. Evaluate the expression (4004² - 3996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 4004 and b =
3996 in the difference-of-squares identity. We then get (4004² - 3996²) = (4004 + 3996)(4004 - 3996). The Right Hand Side of this equation can be simplified to 8000 x 8 = 64000. ---1063. Find the value of (6005 x 5995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 6005 and 5995 are close in value to 6000. We can rewrite the expression (6005 x 5995) as (6000 + 5)(6000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (6000 + 5)(6000 - 5) = 6000² - 5². This is relatively easy to compute; we have 6000² - 5² = 36000000 - 25 = 35999975. ---1064. What is the value of the positive square root of [(810 + 197)(810 197) + 38809]? -Solution: First, we note that the number 38809 looks approximately like the square of 197. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(810 + 197)(810 - 197) + 38809] = [(810 + 197)(810 - 197) + 197²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(810 + 197)(810 - 197) + 197²] = [810² - 197² + 197²]. The second and third terms cancel, so the expression is just equal to 810². Clearly, the square root of the expression is equal to 810. ---1065. Evaluate the expression (6169² - 3831²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6169 and b = 3831 in the difference-of-squares identity. We then get (6169² - 3831²) = (6169 + 3831)(6169 - 3831). The Right Hand Side of this equation can be simplified to 10000 x 2338 = 23380000.
---1066. Evaluate the expression (70009² - 69991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 70009 and b = 69991 in the difference-of-squares identity. We then get (70009² - 69991²) = (70009 + 69991)(70009 - 69991). The Right Hand Side of this equation can be simplified to 140000 x 18 = 2520000. ---1067. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 70005 and 69995 are close in value to 70000. We can rewrite the expression (70005 x 69995) as (70000 + 5)(70000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (70000 + 5)(70000 - 5) = 70000² - 5². This is relatively easy to compute; we have 70000² - 5² = 4900000000 - 25 = 4899999975. ---1068. What is the value of the positive square root of [(600 + 260)(600 260) + 67600]? -Solution: First, we note that the number 67600 looks approximately like the square of 260. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(600 + 260)(600 - 260) + 67600] = [(600 + 260)(600 - 260) + 260²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(600 + 260)(600 - 260) + 260²] = [600² - 260² + 260²]. The second and third terms cancel, so the expression is just equal to 600². Clearly, the square root of the expression is equal to 600. ---1069. Evaluate the expression (6410² - 3590²), by using algebraic identities to simplify computation.
-Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6410 and b = 3590 in the difference-of-squares identity. We then get (6410² - 3590²) = (6410 + 3590)(6410 - 3590). The Right Hand Side of this equation can be simplified to 10000 x 2820 = 28200000. ---1070. Evaluate the expression (90004² - 89996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 90004 and b = 89996 in the difference-of-squares identity. We then get (90004² - 89996²) = (90004 + 89996)(90004 - 89996). The Right Hand Side of this equation can be simplified to 180000 x 8 = 1440000. ---1071. Find the value of (70004 x 69996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 70004 and 69996 are close in value to 70000. We can rewrite the expression (70004 x 69996) as (70000 + 4)(70000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (70000 + 4)(70000 - 4) = 70000² - 4². This is relatively easy to compute; we have 70000² - 4² = 4900000000 - 16 = 4899999984. ---1072. What is the value of the positive square root of [(450 + 205)(450 205) + 42025]? -Solution: First, we note that the number 42025 looks approximately like the square of 205. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(450 + 205)(450 - 205) + 42025] = [(450 + 205)(450 - 205) + 205²].
Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(450 + 205)(450 - 205) + 205²] = [450² - 205² + 205²]. The second and third terms cancel, so the expression is just equal to 450². Clearly, the square root of the expression is equal to 450. ---1073. Evaluate the expression (7939² - 2061²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7939 and b = 2061 in the difference-of-squares identity. We then get (7939² - 2061²) = (7939 + 2061)(7939 - 2061). The Right Hand Side of this equation can be simplified to 10000 x 5878 = 58780000. ---1074. Evaluate the expression (80007² - 79993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 80007 and b = 79993 in the difference-of-squares identity. We then get (80007² - 79993²) = (80007 + 79993)(80007 - 79993). The Right Hand Side of this equation can be simplified to 160000 x 14 = 2240000. ---1075. Find the value of (90005 x 89995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 90005 and 89995 are close in value to 90000. We can rewrite the expression (90005 x 89995) as (90000 + 5)(90000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (90000 + 5)(90000 - 5) = 90000² - 5². This is relatively easy to compute; we have 90000² - 5² = 8100000000 - 25 = 8099999975. ----
1076. What is the value of the positive square root of [(680 + 178)(680 178) + 31684]? -Solution: First, we note that the number 31684 looks approximately like the square of 178. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(680 + 178)(680 - 178) + 31684] = [(680 + 178)(680 - 178) + 178²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(680 + 178)(680 - 178) + 178²] = [680² - 178² + 178²]. The second and third terms cancel, so the expression is just equal to 680². Clearly, the square root of the expression is equal to 680. ---1077. Evaluate the expression (9786² - 214²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9786 and b = 214 in the difference-of-squares identity. We then get (9786² - 214²) = (9786 + 214)(9786 - 214). The Right Hand Side of this equation can be simplified to 10000 x 9572 = 95720000. ---1078. Evaluate the expression (30006² - 29994²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 30006 and b = 29994 in the difference-of-squares identity. We then get (30006² - 29994²) = (30006 + 29994)(30006 - 29994). The Right Hand Side of this equation can be simplified to 60000 x 12 = 720000. ---1079. Find the value of (7008 x 6992), by using algebraic identities to simplify the calculation. --
Solution: We observe that both 7008 and 6992 are close in value to 7000. We can rewrite the expression (7008 x 6992) as (7000 + 8)(7000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 8)(7000 - 8) = 7000² - 8². This is relatively easy to compute; we have 7000² - 8² = 49000000 - 64 = 48999936. ---1080. What is the value of the positive square root of [(580 + 198)(580 198) + 39204]? -Solution: First, we note that the number 39204 looks approximately like the square of 198. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(580 + 198)(580 - 198) + 39204] = [(580 + 198)(580 - 198) + 198²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(580 + 198)(580 - 198) + 198²] = [580² - 198² + 198²]. The second and third terms cancel, so the expression is just equal to 580². Clearly, the square root of the expression is equal to 580. ---1081. Evaluate the expression (9171² - 829²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9171 and b = 829 in the difference-of-squares identity. We then get (9171² - 829²) = (9171 + 829)(9171 - 829). The Right Hand Side of this equation can be simplified to 10000 x 8342 = 83420000. ---1082. Evaluate the expression (5006² - 4994²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5006 and b = 4994 in the difference-of-squares identity. We then get (5006² - 4994²) = (5006 + 4994)(5006 - 4994).
The Right Hand Side of this equation can be simplified to 10000 x 12 = 120000. ---1083. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 70005 and 69995 are close in value to 70000. We can rewrite the expression (70005 x 69995) as (70000 + 5)(70000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (70000 + 5)(70000 - 5) = 70000² - 5². This is relatively easy to compute; we have 70000² - 5² = 4900000000 - 25 = 4899999975. ---1084. What is the value of the positive square root of [(580 + 252)(580 252) + 63504]? -Solution: First, we note that the number 63504 looks approximately like the square of 252. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(580 + 252)(580 - 252) + 63504] = [(580 + 252)(580 - 252) + 252²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(580 + 252)(580 - 252) + 252²] = [580² - 252² + 252²]. The second and third terms cancel, so the expression is just equal to 580². Clearly, the square root of the expression is equal to 580. ---1085. Evaluate the expression (9239² - 761²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9239 and b = 761 in the difference-of-squares identity. We then get (9239² - 761²) = (9239 + 761)(9239 - 761). The Right Hand Side of this equation can be simplified to 10000 x 8478 = 84780000. ----
1086. Evaluate the expression (3003² - 2997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 3003 and b = 2997 in the difference-of-squares identity. We then get (3003² - 2997²) = (3003 + 2997)(3003 - 2997). The Right Hand Side of this equation can be simplified to 6000 x 6 = 36000. ---1087. Find the value of (7008 x 6992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7008 and 6992 are close in value to 7000. We can rewrite the expression (7008 x 6992) as (7000 + 8)(7000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 8)(7000 - 8) = 7000² - 8². This is relatively easy to compute; we have 7000² - 8² = 49000000 - 64 = 48999936. ---1088. What is the value of the positive square root of [(470 + 112)(470 112) + 12544]? -Solution: First, we note that the number 12544 looks approximately like the square of 112. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(470 + 112)(470 - 112) + 12544] = [(470 + 112)(470 - 112) + 112²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(470 + 112)(470 - 112) + 112²] = [470² - 112² + 112²]. The second and third terms cancel, so the expression is just equal to 470². Clearly, the square root of the expression is equal to 470. ---1089. Evaluate the expression (8671² - 1329²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) =
(a + b)(a - b). To simplify this particular expression, we set a = 8671 and b = 1329 in the difference-of-squares identity. We then get (8671² - 1329²) = (8671 + 1329)(8671 - 1329). The Right Hand Side of this equation can be simplified to 10000 x 7342 = 73420000. ---1090. Evaluate the expression (70008² - 69992²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 70008 and b = 69992 in the difference-of-squares identity. We then get (70008² - 69992²) = (70008 + 69992)(70008 - 69992). The Right Hand Side of this equation can be simplified to 140000 x 16 = 2240000. ---1091. Find the value of (2006 x 1994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2006 and 1994 are close in value to 2000. We can rewrite the expression (2006 x 1994) as (2000 + 6)(2000 - 6). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 + 6)(2000 - 6) = 2000² - 6². This is relatively easy to compute; we have 2000² - 6² = 4000000 - 36 = 3999964. ---1092. What is the value of the positive square root of [(400 + 208)(400 208) + 43264]? -Solution: First, we note that the number 43264 looks approximately like the square of 208. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(400 + 208)(400 - 208) + 43264] = [(400 + 208)(400 - 208) + 208²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(400 + 208)(400 - 208) + 208²] = [400² - 208² + 208²]. The second and third terms cancel, so the expression is just equal to 400².
Clearly, the square root of the expression is equal to 400. ---1093. Evaluate the expression (8216² - 1784²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8216 and b = 1784 in the difference-of-squares identity. We then get (8216² - 1784²) = (8216 + 1784)(8216 - 1784). The Right Hand Side of this equation can be simplified to 10000 x 6432 = 64320000. ---1094. Evaluate the expression (80007² - 79993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 80007 and b = 79993 in the difference-of-squares identity. We then get (80007² - 79993²) = (80007 + 79993)(80007 - 79993). The Right Hand Side of this equation can be simplified to 160000 x 14 = 2240000. ---1095. Find the value of (50002 x 49998), by using algebraic identities to simplify the calculation. -Solution: We observe that both 50002 and 49998 are close in value to 50000. We can rewrite the expression (50002 x 49998) as (50000 + 2)(50000 - 2). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (50000 + 2)(50000 - 2) = 50000² - 2². This is relatively easy to compute; we have 50000² - 2² = 2500000000 - 4 = 2499999996. ---1096. What is the value of the positive square root of [(430 + 294)(430 294) + 86436]? --
Solution: First, we note that the number 86436 looks approximately like the square of 294. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(430 + 294)(430 - 294) + 86436] = [(430 + 294)(430 - 294) + 294²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(430 + 294)(430 - 294) + 294²] = [430² - 294² + 294²]. The second and third terms cancel, so the expression is just equal to 430². Clearly, the square root of the expression is equal to 430. ---1097. Evaluate the expression (7724² - 2276²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7724 and b = 2276 in the difference-of-squares identity. We then get (7724² - 2276²) = (7724 + 2276)(7724 - 2276). The Right Hand Side of this equation can be simplified to 10000 x 5448 = 54480000. ---1098. Evaluate the expression (90007² - 89993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 90007 and b = 89993 in the difference-of-squares identity. We then get (90007² - 89993²) = (90007 + 89993)(90007 - 89993). The Right Hand Side of this equation can be simplified to 180000 x 14 = 2520000. ---1099. Find the value of (2005 x 1995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2005 and 1995 are close in value to 2000. We can rewrite the expression (2005 x 1995) as (2000 + 5)(2000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 +
5)(2000 - 5) = 2000² - 5². This is relatively easy to compute; we have 2000² - 5² = 4000000 - 25 = 3999975. ---1100. What is the value of the positive square root of [(470 + 257)(470 257) + 66049]? -Solution: First, we note that the number 66049 looks approximately like the square of 257. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(470 + 257)(470 - 257) + 66049] = [(470 + 257)(470 - 257) + 257²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(470 + 257)(470 - 257) + 257²] = [470² - 257² + 257²]. The second and third terms cancel, so the expression is just equal to 470². Clearly, the square root of the expression is equal to 470. ---1101. Evaluate the expression (7505² - 2495²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7505 and b = 2495 in the difference-of-squares identity. We then get (7505² - 2495²) = (7505 + 2495)(7505 - 2495). The Right Hand Side of this equation can be simplified to 10000 x 5010 = 50100000. ---1102. Evaluate the expression (60002² - 59998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 60002 and b = 59998 in the difference-of-squares identity. We then get (60002² - 59998²) = (60002 + 59998)(60002 - 59998). The Right Hand Side of this equation can be simplified to 120000 x 4 = 480000. ----
1103. Find the value of (20004 x 19996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 20004 and 19996 are close in value to 20000. We can rewrite the expression (20004 x 19996) as (20000 + 4)(20000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (20000 + 4)(20000 - 4) = 20000² - 4². This is relatively easy to compute; we have 20000² - 4² = 400000000 - 16 = 399999984. ---1104. What is the value of the positive square root of [(660 + 146)(660 146) + 21316]? -Solution: First, we note that the number 21316 looks approximately like the square of 146. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(660 + 146)(660 - 146) + 21316] = [(660 + 146)(660 - 146) + 146²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(660 + 146)(660 - 146) + 146²] = [660² - 146² + 146²]. The second and third terms cancel, so the expression is just equal to 660². Clearly, the square root of the expression is equal to 660. ---1105. Evaluate the expression (9831² - 169²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9831 and b = 169 in the difference-of-squares identity. We then get (9831² - 169²) = (9831 + 169)(9831 - 169). The Right Hand Side of this equation can be simplified to 10000 x 9662 = 96620000. ---1106. Evaluate the expression (50009² - 49991²), by using algebraic identities to simplify computation.
-Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 50009 and b = 49991 in the difference-of-squares identity. We then get (50009² - 49991²) = (50009 + 49991)(50009 - 49991). The Right Hand Side of this equation can be simplified to 100000 x 18 = 1800000. ---1107. Find the value of (9007 x 8993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 9007 and 8993 are close in value to 9000. We can rewrite the expression (9007 x 8993) as (9000 + 7)(9000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (9000 + 7)(9000 - 7) = 9000² - 7². This is relatively easy to compute; we have 9000² - 7² = 81000000 - 49 = 80999951. ---1108. What is the value of the positive square root of [(900 + 155)(900 155) + 24025]? -Solution: First, we note that the number 24025 looks approximately like the square of 155. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(900 + 155)(900 - 155) + 24025] = [(900 + 155)(900 - 155) + 155²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(900 + 155)(900 - 155) + 155²] = [900² - 155² + 155²]. The second and third terms cancel, so the expression is just equal to 900². Clearly, the square root of the expression is equal to 900. ---1109. Evaluate the expression (8082² - 1918²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8082 and b = 1918 in the difference-of-squares identity. We then get (8082² - 1918²) =
(8082 + 1918)(8082 - 1918). The Right Hand Side of this equation can be simplified to 10000 x 6164 = 61640000. ---1110. Evaluate the expression (4007² - 3993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 4007 and b = 3993 in the difference-of-squares identity. We then get (4007² - 3993²) = (4007 + 3993)(4007 - 3993). The Right Hand Side of this equation can be simplified to 8000 x 14 = 112000. ---1111. Find the value of (4006 x 3994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 4006 and 3994 are close in value to 4000. We can rewrite the expression (4006 x 3994) as (4000 + 6)(4000 - 6). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (4000 + 6)(4000 - 6) = 4000² - 6². This is relatively easy to compute; we have 4000² - 6² = 16000000 - 36 = 15999964. ---1112. What is the value of the positive square root of [(620 + 174)(620 174) + 30276]? -Solution: First, we note that the number 30276 looks approximately like the square of 174. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(620 + 174)(620 - 174) + 30276] = [(620 + 174)(620 - 174) + 174²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(620 + 174)(620 - 174) + 174²] = [620² - 174² + 174²]. The second and third terms cancel, so the expression is just equal to 620². Clearly, the square root of the expression is equal to 620.
---1113. Evaluate the expression (7056² - 2944²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7056 and b = 2944 in the difference-of-squares identity. We then get (7056² - 2944²) = (7056 + 2944)(7056 - 2944). The Right Hand Side of this equation can be simplified to 10000 x 4112 = 41120000. ---1114. Evaluate the expression (30009² - 29991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 30009 and b = 29991 in the difference-of-squares identity. We then get (30009² - 29991²) = (30009 + 29991)(30009 - 29991). The Right Hand Side of this equation can be simplified to 60000 x 18 = 1080000. ---1115. Find the value of (8004 x 7996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 8004 and 7996 are close in value to 8000. We can rewrite the expression (8004 x 7996) as (8000 + 4)(8000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (8000 + 4)(8000 - 4) = 8000² - 4². This is relatively easy to compute; we have 8000² - 4² = 64000000 - 16 = 63999984. ---1116. What is the value of the positive square root of [(340 + 113)(340 113) + 12769]? -Solution: First, we note that the number 12769 looks approximately like the
square of 113. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(340 + 113)(340 - 113) + 12769] = [(340 + 113)(340 - 113) + 113²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(340 + 113)(340 - 113) + 113²] = [340² - 113² + 113²]. The second and third terms cancel, so the expression is just equal to 340². Clearly, the square root of the expression is equal to 340. ---1117. Evaluate the expression (8508² - 1492²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8508 and b = 1492 in the difference-of-squares identity. We then get (8508² - 1492²) = (8508 + 1492)(8508 - 1492). The Right Hand Side of this equation can be simplified to 10000 x 7016 = 70160000. ---1118. Evaluate the expression (6009² - 5991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6009 and b = 5991 in the difference-of-squares identity. We then get (6009² - 5991²) = (6009 + 5991)(6009 - 5991). The Right Hand Side of this equation can be simplified to 12000 x 18 = 216000. ---1119. Find the value of (70005 x 69995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 70005 and 69995 are close in value to 70000. We can rewrite the expression (70005 x 69995) as (70000 + 5)(70000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (70000 + 5)(70000 - 5) = 70000² - 5². This is relatively easy to compute; we have 70000² - 5² = 4900000000 - 25 =
4899999975. ---1120. What is the value of the positive square root of [(600 + 147)(600 147) + 21609]? -Solution: First, we note that the number 21609 looks approximately like the square of 147. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(600 + 147)(600 - 147) + 21609] = [(600 + 147)(600 - 147) + 147²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(600 + 147)(600 - 147) + 147²] = [600² - 147² + 147²]. The second and third terms cancel, so the expression is just equal to 600². Clearly, the square root of the expression is equal to 600. ---1121. Evaluate the expression (8087² - 1913²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8087 and b = 1913 in the difference-of-squares identity. We then get (8087² - 1913²) = (8087 + 1913)(8087 - 1913). The Right Hand Side of this equation can be simplified to 10000 x 6174 = 61740000. ---1122. Evaluate the expression (70003² - 69997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 70003 and b = 69997 in the difference-of-squares identity. We then get (70003² - 69997²) = (70003 + 69997)(70003 - 69997). The Right Hand Side of this equation can be simplified to 140000 x 6 = 840000. ---1123. Find the value of (60003 x 59997), by using algebraic identities to
simplify the calculation. -Solution: We observe that both 60003 and 59997 are close in value to 60000. We can rewrite the expression (60003 x 59997) as (60000 + 3)(60000 - 3). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (60000 + 3)(60000 - 3) = 60000² - 3². This is relatively easy to compute; we have 60000² - 3² = 3600000000 - 9 = 3599999991. ---1124. What is the value of the positive square root of [(470 + 291)(470 291) + 84681]? -Solution: First, we note that the number 84681 looks approximately like the square of 291. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(470 + 291)(470 - 291) + 84681] = [(470 + 291)(470 - 291) + 291²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(470 + 291)(470 - 291) + 291²] = [470² - 291² + 291²]. The second and third terms cancel, so the expression is just equal to 470². Clearly, the square root of the expression is equal to 470. ---1125. Evaluate the expression (8416² - 1584²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8416 and b = 1584 in the difference-of-squares identity. We then get (8416² - 1584²) = (8416 + 1584)(8416 - 1584). The Right Hand Side of this equation can be simplified to 10000 x 6832 = 68320000. ---1126. Evaluate the expression (30003² - 29997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 30003 and b
= 29997 in the difference-of-squares identity. We then get (30003² - 29997²) = (30003 + 29997)(30003 - 29997). The Right Hand Side of this equation can be simplified to 60000 x 6 = 360000. ---1127. Find the value of (8007 x 7993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 8007 and 7993 are close in value to 8000. We can rewrite the expression (8007 x 7993) as (8000 + 7)(8000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (8000 + 7)(8000 - 7) = 8000² - 7². This is relatively easy to compute; we have 8000² - 7² = 64000000 - 49 = 63999951. ---1128. What is the value of the positive square root of [(750 + 249)(750 249) + 62001]? -Solution: First, we note that the number 62001 looks approximately like the square of 249. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(750 + 249)(750 - 249) + 62001] = [(750 + 249)(750 - 249) + 249²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(750 + 249)(750 - 249) + 249²] = [750² - 249² + 249²]. The second and third terms cancel, so the expression is just equal to 750². Clearly, the square root of the expression is equal to 750. ---1129. Evaluate the expression (7133² - 2867²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7133 and b = 2867 in the difference-of-squares identity. We then get (7133² - 2867²) = (7133 + 2867)(7133 - 2867). The Right Hand Side of this equation can be simplified to 10000 x 4266 =
42660000. ---1130. Evaluate the expression (80002² - 79998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 80002 and b = 79998 in the difference-of-squares identity. We then get (80002² - 79998²) = (80002 + 79998)(80002 - 79998). The Right Hand Side of this equation can be simplified to 160000 x 4 = 640000. ---1131. Find the value of (7002 x 6998), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7002 and 6998 are close in value to 7000. We can rewrite the expression (7002 x 6998) as (7000 + 2)(7000 - 2). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 2)(7000 - 2) = 7000² - 2². This is relatively easy to compute; we have 7000² - 2² = 49000000 - 4 = 48999996. ---1132. What is the value of the positive square root of [(690 + 191)(690 191) + 36481]? -Solution: First, we note that the number 36481 looks approximately like the square of 191. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(690 + 191)(690 - 191) + 36481] = [(690 + 191)(690 - 191) + 191²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(690 + 191)(690 - 191) + 191²] = [690² - 191² + 191²]. The second and third terms cancel, so the expression is just equal to 690². Clearly, the square root of the expression is equal to 690. ---1133. Evaluate the expression (9147² - 853²), by using algebraic identities to
simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9147 and b = 853 in the difference-of-squares identity. We then get (9147² - 853²) = (9147 + 853)(9147 - 853). The Right Hand Side of this equation can be simplified to 10000 x 8294 = 82940000. ---1134. Evaluate the expression (3002² - 2998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 3002 and b = 2998 in the difference-of-squares identity. We then get (3002² - 2998²) = (3002 + 2998)(3002 - 2998). The Right Hand Side of this equation can be simplified to 6000 x 4 = 24000. ---1135. Find the value of (20002 x 19998), by using algebraic identities to simplify the calculation. -Solution: We observe that both 20002 and 19998 are close in value to 20000. We can rewrite the expression (20002 x 19998) as (20000 + 2)(20000 - 2). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (20000 + 2)(20000 - 2) = 20000² - 2². This is relatively easy to compute; we have 20000² - 2² = 400000000 - 4 = 399999996. ---1136. What is the value of the positive square root of [(300 + 108)(300 108) + 11664]? -Solution: First, we note that the number 11664 looks approximately like the square of 108. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(300 + 108)(300 - 108) + 11664] = [(300 + 108)(300 - 108) + 108²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as
[(300 + 108)(300 - 108) + 108²] = [300² - 108² + 108²]. The second and third terms cancel, so the expression is just equal to 300². Clearly, the square root of the expression is equal to 300. ---1137. Evaluate the expression (6619² - 3381²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6619 and b = 3381 in the difference-of-squares identity. We then get (6619² - 3381²) = (6619 + 3381)(6619 - 3381). The Right Hand Side of this equation can be simplified to 10000 x 3238 = 32380000. ---1138. Evaluate the expression (8008² - 7992²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8008 and b = 7992 in the difference-of-squares identity. We then get (8008² - 7992²) = (8008 + 7992)(8008 - 7992). The Right Hand Side of this equation can be simplified to 16000 x 16 = 256000. ---1139. Find the value of (2007 x 1993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 2007 and 1993 are close in value to 2000. We can rewrite the expression (2007 x 1993) as (2000 + 7)(2000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (2000 + 7)(2000 - 7) = 2000² - 7². This is relatively easy to compute; we have 2000² - 7² = 4000000 - 49 = 3999951. ---1140. What is the value of the positive square root of [(410 + 256)(410 -
256) + 65536]? -Solution: First, we note that the number 65536 looks approximately like the square of 256. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(410 + 256)(410 - 256) + 65536] = [(410 + 256)(410 - 256) + 256²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(410 + 256)(410 - 256) + 256²] = [410² - 256² + 256²]. The second and third terms cancel, so the expression is just equal to 410². Clearly, the square root of the expression is equal to 410. ---1141. Evaluate the expression (8211² - 1789²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8211 and b = 1789 in the difference-of-squares identity. We then get (8211² - 1789²) = (8211 + 1789)(8211 - 1789). The Right Hand Side of this equation can be simplified to 10000 x 6422 = 64220000. ---1142. Evaluate the expression (3002² - 2998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 3002 and b = 2998 in the difference-of-squares identity. We then get (3002² - 2998²) = (3002 + 2998)(3002 - 2998). The Right Hand Side of this equation can be simplified to 6000 x 4 = 24000. ---1143. Find the value of (7005 x 6995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7005 and 6995 are close in value to 7000. We can rewrite the expression (7005 x 6995) as (7000 + 5)(7000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 +
5)(7000 - 5) = 7000² - 5². This is relatively easy to compute; we have 7000² - 5² = 49000000 - 25 = 48999975. ---1144. What is the value of the positive square root of [(510 + 295)(510 295) + 87025]? -Solution: First, we note that the number 87025 looks approximately like the square of 295. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(510 + 295)(510 - 295) + 87025] = [(510 + 295)(510 - 295) + 295²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(510 + 295)(510 - 295) + 295²] = [510² - 295² + 295²]. The second and third terms cancel, so the expression is just equal to 510². Clearly, the square root of the expression is equal to 510. ---1145. Evaluate the expression (5537² - 4463²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5537 and b = 4463 in the difference-of-squares identity. We then get (5537² - 4463²) = (5537 + 4463)(5537 - 4463). The Right Hand Side of this equation can be simplified to 10000 x 1074 = 10740000. ---1146. Evaluate the expression (9002² - 8998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9002 and b = 8998 in the difference-of-squares identity. We then get (9002² - 8998²) = (9002 + 8998)(9002 - 8998). The Right Hand Side of this equation can be simplified to 18000 x 4 = 72000. ----
1147. Find the value of (6004 x 5996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 6004 and 5996 are close in value to 6000. We can rewrite the expression (6004 x 5996) as (6000 + 4)(6000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (6000 + 4)(6000 - 4) = 6000² - 4². This is relatively easy to compute; we have 6000² - 4² = 36000000 - 16 = 35999984. ---1148. What is the value of the positive square root of [(850 + 126)(850 126) + 15876]? -Solution: First, we note that the number 15876 looks approximately like the square of 126. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(850 + 126)(850 - 126) + 15876] = [(850 + 126)(850 - 126) + 126²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(850 + 126)(850 - 126) + 126²] = [850² - 126² + 126²]. The second and third terms cancel, so the expression is just equal to 850². Clearly, the square root of the expression is equal to 850. ---1149. Evaluate the expression (5849² - 4151²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5849 and b = 4151 in the difference-of-squares identity. We then get (5849² - 4151²) = (5849 + 4151)(5849 - 4151). The Right Hand Side of this equation can be simplified to 10000 x 1698 = 16980000. ---1150. Evaluate the expression (50007² - 49993²), by using algebraic identities to simplify computation. --
Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 50007 and b = 49993 in the difference-of-squares identity. We then get (50007² - 49993²) = (50007 + 49993)(50007 - 49993). The Right Hand Side of this equation can be simplified to 100000 x 14 = 1400000. ---1151. Find the value of (7008 x 6992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7008 and 6992 are close in value to 7000. We can rewrite the expression (7008 x 6992) as (7000 + 8)(7000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 8)(7000 - 8) = 7000² - 8². This is relatively easy to compute; we have 7000² - 8² = 49000000 - 64 = 48999936. ---1152. What is the value of the positive square root of [(690 + 246)(690 246) + 60516]? -Solution: First, we note that the number 60516 looks approximately like the square of 246. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(690 + 246)(690 - 246) + 60516] = [(690 + 246)(690 - 246) + 246²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(690 + 246)(690 - 246) + 246²] = [690² - 246² + 246²]. The second and third terms cancel, so the expression is just equal to 690². Clearly, the square root of the expression is equal to 690. ---1153. Evaluate the expression (9193² - 807²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9193 and b = 807 in the difference-of-squares identity. We then get (9193² - 807²) = (9193
+ 807)(9193 - 807). The Right Hand Side of this equation can be simplified to 10000 x 8386 = 83860000. ---1154. Evaluate the expression (8009² - 7991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 8009 and b = 7991 in the difference-of-squares identity. We then get (8009² - 7991²) = (8009 + 7991)(8009 - 7991). The Right Hand Side of this equation can be simplified to 16000 x 18 = 288000. ---1155. Find the value of (7006 x 6994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7006 and 6994 are close in value to 7000. We can rewrite the expression (7006 x 6994) as (7000 + 6)(7000 - 6). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 6)(7000 - 6) = 7000² - 6². This is relatively easy to compute; we have 7000² - 6² = 49000000 - 36 = 48999964. ---1156. What is the value of the positive square root of [(320 + 261)(320 261) + 68121]? -Solution: First, we note that the number 68121 looks approximately like the square of 261. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(320 + 261)(320 - 261) + 68121] = [(320 + 261)(320 - 261) + 261²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(320 + 261)(320 - 261) + 261²] = [320² - 261² + 261²]. The second and third terms cancel, so the expression is just equal to 320². Clearly, the square root of the expression is equal to 320. ----
1157. Evaluate the expression (7761² - 2239²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7761 and b = 2239 in the difference-of-squares identity. We then get (7761² - 2239²) = (7761 + 2239)(7761 - 2239). The Right Hand Side of this equation can be simplified to 10000 x 5522 = 55220000. ---1158. Evaluate the expression (9005² - 8995²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9005 and b = 8995 in the difference-of-squares identity. We then get (9005² - 8995²) = (9005 + 8995)(9005 - 8995). The Right Hand Side of this equation can be simplified to 18000 x 10 = 180000. ---1159. Find the value of (6005 x 5995), by using algebraic identities to simplify the calculation. -Solution: We observe that both 6005 and 5995 are close in value to 6000. We can rewrite the expression (6005 x 5995) as (6000 + 5)(6000 - 5). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (6000 + 5)(6000 - 5) = 6000² - 5². This is relatively easy to compute; we have 6000² - 5² = 36000000 - 25 = 35999975. ---1160. What is the value of the positive square root of [(820 + 163)(820 163) + 26569]? -Solution: First, we note that the number 26569 looks approximately like the square of 163. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(820 + 163)(820 - 163) + 26569] =
[(820 + 163)(820 - 163) + 163²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(820 + 163)(820 - 163) + 163²] = [820² - 163² + 163²]. The second and third terms cancel, so the expression is just equal to 820². Clearly, the square root of the expression is equal to 820. ---1161. Evaluate the expression (6544² - 3456²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6544 and b = 3456 in the difference-of-squares identity. We then get (6544² - 3456²) = (6544 + 3456)(6544 - 3456). The Right Hand Side of this equation can be simplified to 10000 x 3088 = 30880000. ---1162. Evaluate the expression (70003² - 69997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 70003 and b = 69997 in the difference-of-squares identity. We then get (70003² - 69997²) = (70003 + 69997)(70003 - 69997). The Right Hand Side of this equation can be simplified to 140000 x 6 = 840000. ---1163. Find the value of (60004 x 59996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 60004 and 59996 are close in value to 60000. We can rewrite the expression (60004 x 59996) as (60000 + 4)(60000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (60000 + 4)(60000 - 4) = 60000² - 4². This is relatively easy to compute; we have 60000² - 4² = 3600000000 - 16 = 3599999984.
---1164. What is the value of the positive square root of [(870 + 160)(870 160) + 25600]? -Solution: First, we note that the number 25600 looks approximately like the square of 160. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(870 + 160)(870 - 160) + 25600] = [(870 + 160)(870 - 160) + 160²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(870 + 160)(870 - 160) + 160²] = [870² - 160² + 160²]. The second and third terms cancel, so the expression is just equal to 870². Clearly, the square root of the expression is equal to 870. ---1165. Evaluate the expression (5573² - 4427²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5573 and b = 4427 in the difference-of-squares identity. We then get (5573² - 4427²) = (5573 + 4427)(5573 - 4427). The Right Hand Side of this equation can be simplified to 10000 x 1146 = 11460000. ---1166. Evaluate the expression (50007² - 49993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 50007 and b = 49993 in the difference-of-squares identity. We then get (50007² - 49993²) = (50007 + 49993)(50007 - 49993). The Right Hand Side of this equation can be simplified to 100000 x 14 = 1400000. ---1167. Find the value of (8004 x 7996), by using algebraic identities to simplify the calculation.
-Solution: We observe that both 8004 and 7996 are close in value to 8000. We can rewrite the expression (8004 x 7996) as (8000 + 4)(8000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (8000 + 4)(8000 - 4) = 8000² - 4². This is relatively easy to compute; we have 8000² - 4² = 64000000 - 16 = 63999984. ---1168. What is the value of the positive square root of [(490 + 154)(490 154) + 23716]? -Solution: First, we note that the number 23716 looks approximately like the square of 154. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(490 + 154)(490 - 154) + 23716] = [(490 + 154)(490 - 154) + 154²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(490 + 154)(490 - 154) + 154²] = [490² - 154² + 154²]. The second and third terms cancel, so the expression is just equal to 490². Clearly, the square root of the expression is equal to 490. ---1169. Evaluate the expression (7291² - 2709²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7291 and b = 2709 in the difference-of-squares identity. We then get (7291² - 2709²) = (7291 + 2709)(7291 - 2709). The Right Hand Side of this equation can be simplified to 10000 x 4582 = 45820000. ---1170. Evaluate the expression (90002² - 89998²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 90002 and b = 89998 in the difference-of-squares identity. We then get (90002² - 89998²)
= (90002 + 89998)(90002 - 89998). The Right Hand Side of this equation can be simplified to 180000 x 4 = 720000. ---1171. Find the value of (20004 x 19996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 20004 and 19996 are close in value to 20000. We can rewrite the expression (20004 x 19996) as (20000 + 4)(20000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (20000 + 4)(20000 - 4) = 20000² - 4². This is relatively easy to compute; we have 20000² - 4² = 400000000 - 16 = 399999984. ---1172. What is the value of the positive square root of [(370 + 128)(370 128) + 16384]? -Solution: First, we note that the number 16384 looks approximately like the square of 128. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(370 + 128)(370 - 128) + 16384] = [(370 + 128)(370 - 128) + 128²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(370 + 128)(370 - 128) + 128²] = [370² - 128² + 128²]. The second and third terms cancel, so the expression is just equal to 370². Clearly, the square root of the expression is equal to 370. ---1173. Evaluate the expression (9095² - 905²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 9095 and b = 905 in the difference-of-squares identity. We then get (9095² - 905²) = (9095 + 905)(9095 - 905). The Right Hand Side of this equation can be simplified to 10000 x 8190 = 81900000. ----
1174. Evaluate the expression (6007² - 5993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6007 and b = 5993 in the difference-of-squares identity. We then get (6007² - 5993²) = (6007 + 5993)(6007 - 5993). The Right Hand Side of this equation can be simplified to 12000 x 14 = 168000. ---1175. Find the value of (3004 x 2996), by using algebraic identities to simplify the calculation. -Solution: We observe that both 3004 and 2996 are close in value to 3000. We can rewrite the expression (3004 x 2996) as (3000 + 4)(3000 - 4). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (3000 + 4)(3000 - 4) = 3000² - 4². This is relatively easy to compute; we have 3000² - 4² = 9000000 - 16 = 8999984. ---1176. What is the value of the positive square root of [(730 + 286)(730 286) + 81796]? -Solution: First, we note that the number 81796 looks approximately like the square of 286. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(730 + 286)(730 - 286) + 81796] = [(730 + 286)(730 - 286) + 286²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(730 + 286)(730 - 286) + 286²] = [730² - 286² + 286²]. The second and third terms cancel, so the expression is just equal to 730². Clearly, the square root of the expression is equal to 730. ---1177. Evaluate the expression (5338² - 4662²), by using algebraic identities to simplify computation. --
Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5338 and b = 4662 in the difference-of-squares identity. We then get (5338² - 4662²) = (5338 + 4662)(5338 - 4662). The Right Hand Side of this equation can be simplified to 10000 x 676 = 6760000. ---1178. Evaluate the expression (30007² - 29993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 30007 and b = 29993 in the difference-of-squares identity. We then get (30007² - 29993²) = (30007 + 29993)(30007 - 29993). The Right Hand Side of this equation can be simplified to 60000 x 14 = 840000. ---1179. Find the value of (20003 x 19997), by using algebraic identities to simplify the calculation. -Solution: We observe that both 20003 and 19997 are close in value to 20000. We can rewrite the expression (20003 x 19997) as (20000 + 3)(20000 - 3). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (20000 + 3)(20000 - 3) = 20000² - 3². This is relatively easy to compute; we have 20000² - 3² = 400000000 - 9 = 399999991. ---1180. What is the value of the positive square root of [(790 + 194)(790 194) + 37636]? -Solution: First, we note that the number 37636 looks approximately like the square of 194. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(790 + 194)(790 - 194) + 37636] = [(790 + 194)(790 - 194) + 194²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(790 + 194)(790 - 194) + 194²] = [790² - 194² + 194²]. The second and third
terms cancel, so the expression is just equal to 790². Clearly, the square root of the expression is equal to 790. ---1181. Evaluate the expression (7274² - 2726²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7274 and b = 2726 in the difference-of-squares identity. We then get (7274² - 2726²) = (7274 + 2726)(7274 - 2726). The Right Hand Side of this equation can be simplified to 10000 x 4548 = 45480000. ---1182. Evaluate the expression (70007² - 69993²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 70007 and b = 69993 in the difference-of-squares identity. We then get (70007² - 69993²) = (70007 + 69993)(70007 - 69993). The Right Hand Side of this equation can be simplified to 140000 x 14 = 1960000. ---1183. Find the value of (7006 x 6994), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7006 and 6994 are close in value to 7000. We can rewrite the expression (7006 x 6994) as (7000 + 6)(7000 - 6). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 6)(7000 - 6) = 7000² - 6². This is relatively easy to compute; we have 7000² - 6² = 49000000 - 36 = 48999964. ---1184. What is the value of the positive square root of [(530 + 165)(530 165) + 27225]?
-Solution: First, we note that the number 27225 looks approximately like the square of 165. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(530 + 165)(530 - 165) + 27225] = [(530 + 165)(530 - 165) + 165²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(530 + 165)(530 - 165) + 165²] = [530² - 165² + 165²]. The second and third terms cancel, so the expression is just equal to 530². Clearly, the square root of the expression is equal to 530. ---1185. Evaluate the expression (5923² - 4077²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5923 and b = 4077 in the difference-of-squares identity. We then get (5923² - 4077²) = (5923 + 4077)(5923 - 4077). The Right Hand Side of this equation can be simplified to 10000 x 1846 = 18460000. ---1186. Evaluate the expression (6004² - 5996²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6004 and b = 5996 in the difference-of-squares identity. We then get (6004² - 5996²) = (6004 + 5996)(6004 - 5996). The Right Hand Side of this equation can be simplified to 12000 x 8 = 96000. ---1187. Find the value of (7007 x 6993), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7007 and 6993 are close in value to 7000. We can rewrite the expression (7007 x 6993) as (7000 + 7)(7000 - 7). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 +
7)(7000 - 7) = 7000² - 7². This is relatively easy to compute; we have 7000² - 7² = 49000000 - 49 = 48999951. ---1188. What is the value of the positive square root of [(480 + 252)(480 252) + 63504]? -Solution: First, we note that the number 63504 looks approximately like the square of 252. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(480 + 252)(480 - 252) + 63504] = [(480 + 252)(480 - 252) + 252²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(480 + 252)(480 - 252) + 252²] = [480² - 252² + 252²]. The second and third terms cancel, so the expression is just equal to 480². Clearly, the square root of the expression is equal to 480. ---1189. Evaluate the expression (6126² - 3874²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 6126 and b = 3874 in the difference-of-squares identity. We then get (6126² - 3874²) = (6126 + 3874)(6126 - 3874). The Right Hand Side of this equation can be simplified to 10000 x 2252 = 22520000. ---1190. Evaluate the expression (3009² - 2991²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 3009 and b = 2991 in the difference-of-squares identity. We then get (3009² - 2991²) = (3009 + 2991)(3009 - 2991). The Right Hand Side of this equation can be simplified to 6000 x 18 = 108000.
---1191. Find the value of (30002 x 29998), by using algebraic identities to simplify the calculation. -Solution: We observe that both 30002 and 29998 are close in value to 30000. We can rewrite the expression (30002 x 29998) as (30000 + 2)(30000 - 2). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (30000 + 2)(30000 - 2) = 30000² - 2². This is relatively easy to compute; we have 30000² - 2² = 900000000 - 4 = 899999996. ---1192. What is the value of the positive square root of [(350 + 241)(350 241) + 58081]? -Solution: First, we note that the number 58081 looks approximately like the square of 241. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(350 + 241)(350 - 241) + 58081] = [(350 + 241)(350 - 241) + 241²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(350 + 241)(350 - 241) + 241²] = [350² - 241² + 241²]. The second and third terms cancel, so the expression is just equal to 350². Clearly, the square root of the expression is equal to 350. ---1193. Evaluate the expression (5096² - 4904²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5096 and b = 4904 in the difference-of-squares identity. We then get (5096² - 4904²) = (5096 + 4904)(5096 - 4904). The Right Hand Side of this equation can be simplified to 10000 x 192 = 1920000. ---1194. Evaluate the expression (7007² - 6993²), by using algebraic identities to simplify computation.
-Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 7007 and b = 6993 in the difference-of-squares identity. We then get (7007² - 6993²) = (7007 + 6993)(7007 - 6993). The Right Hand Side of this equation can be simplified to 14000 x 14 = 196000. ---1195. Find the value of (7008 x 6992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 7008 and 6992 are close in value to 7000. We can rewrite the expression (7008 x 6992) as (7000 + 8)(7000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (7000 + 8)(7000 - 8) = 7000² - 8². This is relatively easy to compute; we have 7000² - 8² = 49000000 - 64 = 48999936. ---1196. What is the value of the positive square root of [(340 + 110)(340 110) + 12100]? -Solution: First, we note that the number 12100 looks approximately like the square of 110. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(340 + 110)(340 - 110) + 12100] = [(340 + 110)(340 - 110) + 110²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(340 + 110)(340 - 110) + 110²] = [340² - 110² + 110²]. The second and third terms cancel, so the expression is just equal to 340². Clearly, the square root of the expression is equal to 340. ---1197. Evaluate the expression (5092² - 4908²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 5092 and b =
4908 in the difference-of-squares identity. We then get (5092² - 4908²) = (5092 + 4908)(5092 - 4908). The Right Hand Side of this equation can be simplified to 10000 x 184 = 1840000. ---1198. Evaluate the expression (20003² - 19997²), by using algebraic identities to simplify computation. -Solution: The expression's form suggests that we use the identity (a² - b²) = (a + b)(a - b). To simplify this particular expression, we set a = 20003 and b = 19997 in the difference-of-squares identity. We then get (20003² - 19997²) = (20003 + 19997)(20003 - 19997). The Right Hand Side of this equation can be simplified to 40000 x 6 = 240000. ---1199. Find the value of (70008 x 69992), by using algebraic identities to simplify the calculation. -Solution: We observe that both 70008 and 69992 are close in value to 70000. We can rewrite the expression (70008 x 69992) as (70000 + 8)(70000 - 8). Next, we use the identity (a² - b²) = (a + b)(a - b). Using this, we get (70000 + 8)(70000 - 8) = 70000² - 8². This is relatively easy to compute; we have 70000² - 8² = 4900000000 - 64 = 4899999936. ---1200. What is the value of the positive square root of [(670 + 182)(670 182) + 33124]? -Solution: First, we note that the number 33124 looks approximately like the square of 182. We can verify this through a quick computation. Therefore, the given expression can be rewritten as [(670 + 182)(670 - 182) + 33124] = [(670 + 182)(670 - 182) + 182²]. Now, we use the identity (a² - b²) = (a + b)(a - b) to rewrite the expression as [(670 + 182)(670 - 182) + 182²] = [670² - 182² + 182²]. The second and third terms cancel, so the expression is just equal to 670². Clearly, the square root of the expression is equal to 670.
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