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.
ScanSolve AI Answer
You're on a roll — let's tackle these calculations. 1. Simplify using the order of operations: *a) 5^2 - 2^2 Step 1: Calculate the exponents. 5^2 = 25 2^2 = 4 Step 2: Perform the subtraction. 25 - 4 = 21 The simplified expression is 21. *b) 4^2 + 1^3 Step 1: Calculate the exponents. 4^2 = 16 1^3 = 1 Step 2: Perform the addition. 16 + 1 = 17 The simplified expression is 17. *c) 10^3 + 8^2 Step 1: Calculate the exponents. 10^3 = 1000 8^2 = 64 Step 2: Perform the addition. 1000 + 64 = 1064 The simplified expression is 1064. *d) 7^2 + 6^3 Step 1: Calculate the exponents. 7^2 = 49 6^3 = 216 Step 2: Perform the addition. 49 + 216 = 265 The simplified expression is 265. *e) 9^2 - 2^4 Step 1: Calculate the exponents. 9^2 = 81 2^4 = 16 Step 2: Perform the subtraction. 81 - 16 = 65 The simplified expression is 65. *f) 2^3 - 2^2 + 2^1 Step 1: Calculate the exponents. 2^3 = 8 2^2 = 4 2^1 = 2 Step 2: Perform the subtraction and addition from left to right. 8 - 4 + 2 = 4 + 2 = 6 The simplified expression is 6. *g) 10^3 - 3 Step 1: Calculate the exponent. 10^3 = 1000 Step 2: Perform the subtraction. 1000 - 3 = 997 The simplified expression is 997. *h) 10^2 + 4^3 - 1^3 Step 1: Calculate the exponents. 10^2 = 100 4^3 = 64 1^3 = 1 Step 2: Perform the addition and subtraction from left to right. 100 + 64 - 1 = 164 - 1 = 163 The simplified expression is 163. 2. Calculate the square roots: *a) sqrt(1) + sqrt(9) Step 1: Calculate the square roots. sqrt(1) = 1 sqrt(9) = 3 Step 2: Perform the addition. 1 + 3 = 4 The result is 4. *b) sqrt(4) + sqrt(25) Step 1: Calculate the square roots. sqrt(4) = 2 sqrt(25) = 5 Step 2: Perform the addition. 2 + 5 = 7 The result is 7. *c) sqrt(25) + sqrt(16) + sqrt(121) Step 1: Calculate the square roots. sqrt(25) = 5 sqrt(16) = 4 sqrt(121) = 11 Step 2: Perform the addition. 5 + 4 + 11 = 9 + 11 = 20 The result is 20. *d) sqrt(16) + sqrt(144) Step 1: Calculate the square roots. sqrt(16) = 4 sqrt(144) = 12 Step 2: Perform the addition. 4 + 12 = 16 The result is 16. *e) sqrt(49) + sqrt(36) Step 1: Calculate the square roots. sqrt(49) = 7 sqrt(36) = 6 Step 2: Perform the addition. 7 + 6 = 13 The result is 13. *f) sqrt(100) + sqrt(49) Step 1: Calculate the square roots. sqrt(100) = 10 sqrt(49) = 7 Step 2: Perform the addition. 10 + 7 = 17 The result is 17. 3. Calculate using a calculator: *a) sqrt(25 - 9) Step 1: Perform the subtraction inside the square root. 25 - 9 = 16 Step 2: Calculate the square root. sqrt(16) = 4 The result is 4. *b) sqrt(25) - 9 Step 1: Calculate the square root. sqrt(25) = 5 Step 2: Perform the subtraction. 5 - 9 = -4 The result is -4. *c) sqrt(36) + sqrt(9) Step 1: Calculate the square roots. sqrt(36) = 6 sqrt(9) = 3 Step 2: Perform the addition. 6 + 3 = 9 The result is 9. *d) sqrt(100 - 36) Step 1: Perform the subtraction inside the square root. 100 - 36 = 64 Step 2: Calculate the square root. sqrt(64) = 8 The result is 8. *e) sqrt(100) - 36 Step 1: Calculate the square root. sqrt(100) = 10 Step 2: Perform the subtraction. 10 - 36 = -26 The result is -26. *f) sqrt(16 + 9) Step 1: Perform the addition inside the square root. 16 + 9 = 25 Step 2: Calculate the square root. sqrt(25) = 5 The result is 5. 4. Simplify using the order of operations: *a) (2+3)^3 Step 1: Perform the addition inside the parentheses. 2 + 3 = 5 Step 2: Calculate the exponent. 5^3 = 5 × 5 × 5 = 125 The simplified expression is 125. *b) 2^3 + 3^3 Step 1: Calculate the exponents. 2^3 = 8 3^3 = 27 Step 2: Perform the addition. 8 + 27 = 35 The simplified expression is 35. *c) 10^3 - 4^3 Step 1: Calculate the exponents. 10^3 = 1000 4^3 = 64 Step 2: Perform the subtraction. 1000 - 64 = 936 The simplified expression is 936. *d) (10-4)^3 Step 1: Perform the subtraction inside the parentheses. 10 - 4 = 6 Step 2: Calculate the exponent. 6^3 = 6 × 6 × 6 = 216 The simplified expression is 216. *e) [3]216 + [3]125 Step 1: Calculate the cube roots. [3]216 = 6 (since 6 × 6 × 6 = 216) [3]125 = 5 (since 5 × 5 × 5 = 125) Step 2: Perform the addition. 6 + 5 = 11 The simplified expression is 11. *f) [3]8000 Step 1: Calculate the cube root. [3]8000 = 20 (since 20 × 20 × 20 = 8000) The simplified expression is 20. *g) [3]8 + [3]1000 Step 1: Calculate the cube roots. [3]8 = 2 (since 2 × 2 × 2 = 8) [3]1000 = 10 (since 10 × 10 × 10 = 1000) Step 2: Perform the addition. 2 + 10 = 12 The simplified expression is 12. *h) [3]64 - [3]27 Step 1: Calculate the cube roots. [3]64 = 4 (since 4 × 4 × 4 = 64) [3]27 = 3 (since 3 × 3 × 3 = 27) Step 2: Perform the subtraction. 4 - 3 = 1 The simplified expression is 1. *i) [3]27 - sqrt(1) Step 1: Calculate the roots. [3]27 = 3 (since 3 × 3 × 3 = 27) sqrt(1) = 1 (since 1 × 1 = 1) Step 2: Perform the subtraction. 3 - 1 = 2 The simplified expression is 2. Got more? Send 'em!