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 simplification problems. i) Simplify 2sqrt(5)sqrt(8) + sqrt(5) Step 1: Simplify the radical in the denominator. sqrt(8) = sqrt(4 × 2) = 2sqrt(2) The expression becomes: 2sqrt(5)2sqrt(2) + sqrt(5) Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2sqrt(2) - sqrt(5). 2sqrt(5)2sqrt(2) + sqrt(5) × 2sqrt(2) - sqrt(5)2sqrt(2) - sqrt(5) Step 3: Expand the numerator and the denominator. Numerator: 2sqrt(5)(2sqrt(2) - sqrt(5)) = (2sqrt(5))(2sqrt(2)) - (2sqrt(5))(sqrt(5)) = 4sqrt(10) - 2(5) = 4sqrt(10) - 10 Denominator (using the difference of squares formula (a+b)(a-b) = a^2 - b^2): (2sqrt(2) + sqrt(5))(2sqrt(2) - sqrt(5)) = (2sqrt(2))^2 - (sqrt(5))^2 = (4 × 2) - 5 = 8 - 5 = 3 Step 4: Combine the simplified numerator and denominator. 4sqrt(10) - 103 The simplified expression is: 4sqrt(10) - 103 ii) Simplify 2(3+3sqrt(2))3sqrt(2) - 2sqrt(3) Step 1: Distribute the 2 in the numerator. 2(3+3sqrt(2)) = 6 + 6sqrt(2) The expression becomes: 6 + 6sqrt(2)3sqrt(2) - 2sqrt(3) Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 3sqrt(2) + 2sqrt(3). 6 + 6sqrt(2)3sqrt(2) - 2sqrt(3) × 3sqrt(2) + 2sqrt(3)3sqrt(2) + 2sqrt(3) Step 3: Expand the numerator. (6 + 6sqrt(2))(3sqrt(2) + 2sqrt(3)) = 6(3sqrt(2)) + 6(2sqrt(3)) + (6sqrt(2))(3sqrt(2)) + (6sqrt(2))(2sqrt(3)) = 18sqrt(2) + 12sqrt(3) + 18(2) + 12sqrt(6) = 18sqrt(2) + 12sqrt(3) + 36 + 12sqrt(6) Rearranging the terms: 36 + 18sqrt(2) + 12sqrt(3) + 12sqrt(6) Step 4: Expand the denominator (using the difference of squares formula (a-b)(a+b) = a^2 - b^2). (3sqrt(2) - 2sqrt(3))(3sqrt(2) + 2sqrt(3)) = (3sqrt(2))^2 - (2sqrt(3))^2 = (9 × 2) - (4 × 3) = 18 - 12 = 6 Step 5: Combine the simplified numerator and denominator, then simplify by dividing each term in the numerator by the denominator. 36 + 18sqrt(2) + 12sqrt(3) + 12sqrt(6)6 = (36)/(6) + 18sqrt(2)6 + 12sqrt(3)6 + 12sqrt(6)6 = 6 + 3sqrt(2) + 2sqrt(3) + 2sqrt(6) The simplified expression is: 6 + 3sqrt(2) + 2sqrt(3) + 2sqrt(6) That's 2 down. 3 left today — send the next one.