This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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7. Calculate. a) 6^2 - 5^2 Step 1: Calculate the squares. 6^2 = 36 5^2 = 25 Step 2: Subtract the values. 36 - 25 = 11 The calculated value is 11. b) 2^3 + 3^3 Step 1: Calculate the cubes. 2^3 = 2 × 2 × 2 = 8 3^3 = 3 × 3 × 3 = 27 Step 2: Add the values. 8 + 27 = 35 The calculated value is 35. 8. Calculate. a) [3]8 Step 1: Find the number that, when multiplied by itself three times, equals 8. 2 × 2 × 2 = 8 The cube root of 8 is 2. b) [3]64 Step 1: Find the number that, when multiplied by itself three times, equals 64. 4 × 4 × 4 = 64 The cube root of 64 is 4. 9. Calculate. a) sqrt(9) + sqrt(16) Step 1: Calculate the square roots. sqrt(9) = 3 sqrt(16) = 4 Step 2: Add the values. 3 + 4 = 7 The calculated value is 7. b) sqrt(100) + sqrt(81) Step 1: Calculate the square roots. sqrt(100) = 10 sqrt(81) = 9 Step 2: Add the values. 10 + 9 = 19 The calculated value is 19. 10. Calculate. a) [3]216 + [3]27 Step 1: Calculate the cube roots. [3]216 = 6 [3]27 = 3 Step 2: Add the values. 6 + 3 = 9 The calculated value is 9. b) [3]27 - [3]8 Step 1: Calculate the cube roots. [3]27 = 3 [3]8 = 2 Step 2: Subtract the values. 3 - 2 = 1 The calculated value is 1. 11. Calculate. a) sqrt(25) + [3]8 Step 1: Calculate the square root and cube root. sqrt(25) = 5 [3]8 = 2 Step 2: Add the values. 5 + 2 = 7 The calculated value is 7. b) sqrt(25) - [3]27 Step 1: Calculate the square root and cube root. sqrt(25) = 5 [3]27 = 3 Step 2: Subtract the values. 5 - 3 = 2 The calculated value is 2. 12. Calculate. a) [3]216 + 4^2 - sqrt(16) Step 1: Calculate each term. [3]216 = 6 4^2 = 16 sqrt(16) = 4 Step 2: Perform the operations from left to right. 6 + 16 - 4 = 22 - 4 = 18 The calculated value is 18. b) 9^2 - [3]27 + sqrt(4) Step 1: Calculate each term. 9^2 = 81 [3]27 = 3 sqrt(4) = 2 Step 2: Perform the operations from left to right. 81 - 3 + 2 = 78 + 2 = 80 The calculated value is 80. 13. Calculate the following as fast as you can: a) 10 × 10 Step 1: Multiply the numbers. 10 × 10 = 100 The product is 100. b) 10 × 10 × 10 × 10 × 10 Step 1: Multiply the numbers. 10 × 10 × 10 × 10 × 10 = 100000 The product is 100000. That's 2 down. 3 left today — send the next one.