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 for the remaining expressions: 1. Simplify the following Surds b) sqrt(8) + sqrt(32) Step 1: Simplify each radical by finding the largest perfect square factor. sqrt(8) = sqrt(4 × 2) = 2sqrt(2) sqrt(32) = sqrt(16 × 2) = 4sqrt(2) Step 2: Add the simplified radicals. 2sqrt(2) + 4sqrt(2) = (2+4)sqrt(2) = 6sqrt(2) The final answer is 6sqrt(2). c) sqrt(75) - sqrt(18) Step 1: Simplify each radical. sqrt(75) = sqrt(25 × 3) = 5sqrt(3) sqrt(18) = sqrt(9 × 2) = 3sqrt(2) Step 2: Subtract the simplified radicals. Since the radicals are different (sqrt(3) and sqrt(2)), they cannot be combined further. 5sqrt(3) - 3sqrt(2) The final answer is 5sqrt(3) - 3sqrt(2). d) 3sqrt(125) - 2sqrt(80) Step 1: Simplify each radical. sqrt(125) = sqrt(25 × 5) = 5sqrt(5) sqrt(80) = sqrt(16 × 5) = 4sqrt(5) Step 2: Substitute the simplified radicals back into the expression. 3(5sqrt(5)) - 2(4sqrt(5)) Step 3: Multiply the coefficients. 15sqrt(5) - 8sqrt(5) Step 4: Combine the like terms. (15-8)sqrt(5) = 7sqrt(5) The final answer is 7sqrt(5). e) sqrt(x^3) + sqrt(x) - sqrt(4x) Step 1: Simplify each radical. sqrt(x^3) = sqrt(x^2 × x) = xsqrt(x) sqrt(x) sqrt(4x) = sqrt(4 × x) = 2sqrt(x) Step 2: Substitute the simplified radicals back into the expression. xsqrt(x) + sqrt(x) - 2sqrt(x) Step 3: Combine the like terms (all terms have sqrt(x)). (x+1-2)sqrt(x) = (x-1)sqrt(x) The final answer is (x-1)sqrt(x). f) 7sqrt(24) - 4sqrt(5) + 2sqrt(6) Step 1: Simplify the radical sqrt(24). sqrt(24) = sqrt(4 × 6) = 2sqrt(6) Step 2: Substitute the simplified radical back into the expression. 7(2sqrt(6)) - 4sqrt(5) + 2sqrt(6) Step 3: Multiply the coefficients. 14sqrt(6) - 4sqrt(5) + 2sqrt(6) Step 4: Combine the like terms (terms with sqrt(6)). (14+2)sqrt(6) - 4sqrt(5) = 16sqrt(6) - 4sqrt(5) The final answer is 16sqrt(6) - 4sqrt(5). 2. Rationalize these surds A) (4)/(sqrt(5)) Step 1: Multiply the numerator and denominator by sqrt(5) to rationalize the denominator. (4)/(sqrt(5)) × sqrt(5)sqrt(5) Step 2: Simplify the expression. 4sqrt(5)5 The final answer is 4sqrt(5)5. B) (7)/(sqrt(7)) Step 1: Multiply the numerator and denominator by sqrt(7) to rationalize the denominator. (7)/(sqrt(7)) × sqrt(7)sqrt(7) Step 2: Simplify the expression. 7sqrt(7)7 = sqrt(7) The final answer is sqrt(7). 3 done, 2 left today. You're making progress.