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
Here are the simplifications for each expression: i) Simplify sqrt(27) Step 1: Find the largest perfect square factor of 27. The factors of 27 are 1, 3, 9, 27. The largest perfect square factor is 9. Step 2: Rewrite the expression using the perfect square factor. sqrt(27) = sqrt(9 × 3) Step 3: Apply the property sqrt(ab) = sqrt(a) × sqrt(b). sqrt(9 × 3) = sqrt(9) × sqrt(3) Step 4: Simplify the perfect square root. sqrt(9) × sqrt(3) = 3sqrt(3) The simplified expression is 3sqrt(3). ii) Simplify sqrt(32) Step 1: Find the largest perfect square factor of 32. The factors of 32 are 1, 2, 4, 8, 16, 32. The largest perfect square factor is 16. Step 2: Rewrite the expression using the perfect square factor. sqrt(32) = sqrt(16 × 2) Step 3: Apply the property sqrt(ab) = sqrt(a) × sqrt(b). sqrt(16 × 2) = sqrt(16) × sqrt(2) Step 4: Simplify the perfect square root. sqrt(16) × sqrt(2) = 4sqrt(2) The simplified expression is 4sqrt(2). iii) Simplify sqrt(8) Step 1: Find the largest perfect square factor of 8. The factors of 8 are 1, 2, 4, 8. The largest perfect square factor is 4. Step 2: Rewrite the expression using the perfect square factor. sqrt(8) = sqrt(4 × 2) Step 3: Apply the property sqrt(ab) = sqrt(a) × sqrt(b). sqrt(4 × 2) = sqrt(4) × sqrt(2) Step 4: Simplify the perfect square root. sqrt(4) × sqrt(2) = 2sqrt(2) The simplified expression is 2sqrt(2). iv) Simplify sqrt(128) Step 1: Find the largest perfect square factor of 128. The factors of 128 include 1, 2, 4, 8, 16, 32, 64, 128. The largest perfect square factor is 64. Step 2: Rewrite the expression using the perfect square factor. sqrt(128) = sqrt(64 × 2) Step 3: Apply the property sqrt(ab) = sqrt(a) × sqrt(b). sqrt(64 × 2) = sqrt(64) × sqrt(2) Step 4: Simplify the perfect square root. sqrt(64) × sqrt(2) = 8sqrt(2) The simplified expression is 8sqrt(2). That's 2 down. 3 left today — send the next one.