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 solutions to the multiplication of surds problems: 1) sqrt(12) × sqrt(8) Step 1: Simplify each surd. sqrt(12) = sqrt(4 × 3) = 2sqrt(3) sqrt(8) = sqrt(4 × 2) = 2sqrt(2) Step 2: Multiply the simplified surds. 2sqrt(3) × 2sqrt(2) = (2 × 2) × (sqrt(3) × sqrt(2)) = 4sqrt(3 × 2) = 4sqrt(6) 2) sqrt(45) × sqrt(27) Step 1: Simplify each surd. sqrt(45) = sqrt(9 × 5) = 3sqrt(5) sqrt(27) = sqrt(9 × 3) = 3sqrt(3) Step 2: Multiply the simplified surds. 3sqrt(5) × 3sqrt(3) = (3 × 3) × (sqrt(5) × sqrt(3)) = 9sqrt(5 × 3) = 9sqrt(15) 3) sqrt(28) × sqrt(8) × sqrt(12) Step 1: Simplify each surd. sqrt(28) = sqrt(4 × 7) = 2sqrt(7) sqrt(8) = sqrt(4 × 2) = 2sqrt(2) sqrt(12) = sqrt(4 × 3) = 2sqrt(3) Step 2: Multiply the simplified surds. 2sqrt(7) × 2sqrt(2) × 2sqrt(3) = (2 × 2 × 2) × (sqrt(7) × sqrt(2) × sqrt(3)) = 8sqrt(7 × 2 × 3) = 8sqrt(42) 4) sqrt(8) × sqrt(45) × sqrt(90) Step 1: Simplify each surd. sqrt(8) = sqrt(4 × 2) = 2sqrt(2) sqrt(45) = sqrt(9 × 5) = 3sqrt(5) sqrt(90) = sqrt(9 × 10) = 3sqrt(10) Step 2: Multiply the simplified surds. 2sqrt(2) × 3sqrt(5) × 3sqrt(10) = (2 × 3 × 3) × (sqrt(2) × sqrt(5) × sqrt(10)) = 18 × sqrt(2 × 5 × 10) = 18 × sqrt(100) = 18 × 10 = 180 5) sqrt(12) × 3sqrt(20) × 2sqrt(45) Step 1: Simplify each surd. sqrt(12) = sqrt(4 × 3) = 2sqrt(3) 3sqrt(20) = 3sqrt(4 × 5) = 3 × 2sqrt(5) = 6sqrt(5) 2sqrt(45) = 2sqrt(9 × 5) = 2 × 3sqrt(5) = 6sqrt(5) Step 2: Multiply the simplified terms. 2sqrt(3) × 6sqrt(5) × 6sqrt(5) = (2 × 6 × 6) × (sqrt(3) × sqrt(5) × sqrt(5)) = 72 × (sqrt(3) × 5) = 72 × 5sqrt(3) = 360sqrt(3) 6) (2sqrt(3))^2 Step 1: Apply the exponent to both parts of the term. (2sqrt(3))^2 = 2^2 × (sqrt(3))^2 Step 2: Calculate the squares and multiply. 4 × 3 = 12 7) (2sqrt(3))^2 × (2sqrt(2)) Step 1: Calculate (2sqrt(3))^2. (2sqrt(3))^2 = 2^2 × (sqrt(3))^2 = 4 × 3 = 12 Step 2: Multiply this result by 2sqrt(2). 12 × 2sqrt(2) = (12 × 2)sqrt(2) = 24sqrt(2) That's 2 down. 3 left today — send the next one.