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 simplified forms of the square root expressions: a) Simplify sqrt(12) Step 1: Find the largest perfect square factor of 12. The factors of 12 are 1, 2, 3, 4, 6, 12. The largest perfect square factor is 4. Step 2: Rewrite the expression using the perfect square factor. sqrt(12) = sqrt(4 × 3) Step 3: Apply the property sqrt(ab) = sqrt(a) × sqrt(b). sqrt(4 × 3) = sqrt(4) × sqrt(3) Step 4: Simplify the perfect square root. sqrt(4) × sqrt(3) = 2sqrt(3) a) 2sqrt(3) b) Simplify sqrt(72) Step 1: Find the largest perfect square factor of 72. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The largest perfect square factor is 36. Step 2: Rewrite the expression using the perfect square factor. sqrt(72) = sqrt(36 × 2) Step 3: Apply the property sqrt(ab) = sqrt(a) × sqrt(b). sqrt(36 × 2) = sqrt(36) × sqrt(2) Step 4: Simplify the perfect square root. sqrt(36) × sqrt(2) = 6sqrt(2) b) 6sqrt(2) c) Simplify sqrt(80) Step 1: Find the largest perfect square factor of 80. The factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, 80. The largest perfect square factor is 16. Step 2: Rewrite the expression using the perfect square factor. sqrt(80) = sqrt(16 × 5) Step 3: Apply the property sqrt(ab) = sqrt(a) × sqrt(b). sqrt(16 × 5) = sqrt(16) × sqrt(5) Step 4: Simplify the perfect square root. sqrt(16) × sqrt(5) = 4sqrt(5) c) 4sqrt(5) d) Simplify sqrt(125) Step 1: Find the largest perfect square factor of 125. The factors of 125 are 1, 5, 25, 125. The largest perfect square factor is 25. Step 2: Rewrite the expression using the perfect square factor. sqrt(125) = sqrt(25 × 5) Step 3: Apply the property sqrt(ab) = sqrt(a) × sqrt(b). sqrt(25 × 5) = sqrt(25) × sqrt(5) Step 4: Simplify the perfect square root. sqrt(25) × sqrt(5) = 5sqrt(5) d) 5sqrt(5) e) Simplify sqrt(500) Step 1: Find the largest perfect square factor of 500. The factors of 500 are 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500. The largest perfect square factor is 100. Step 2: Rewrite the expression using the perfect square factor. sqrt(500) = sqrt(100 × 5) Step 3: Apply the property sqrt(ab) = sqrt(a) × sqrt(b). sqrt(100 × 5) = sqrt(100) × sqrt(5) Step 4: Simplify the perfect square root. sqrt(100) × sqrt(5) = 10sqrt(5) e) 10sqrt(5)