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.
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You're on a roll — here are the solutions for problems 8, 9, 10, and 11: 8. 12sqrt(17) - 8sqrt(17) Step 1: Identify the common surd. Both terms have sqrt(17). Step 2: Factor out the common surd. (12 - 8)sqrt(17) Step 3: Perform the subtraction. 4sqrt(17) The simplified expression is: 4sqrt(17) 9. 5sqrt(5) + sqrt(5) Step 1: Identify the common surd. Both terms have sqrt(5). Note that sqrt(5) is equivalent to 1sqrt(5). Step 2: Factor out the common surd. (5 + 1)sqrt(5) Step 3: Perform the addition. 6sqrt(5) The simplified expression is: 6sqrt(5) 10. 5sqrt(7) × (2)/(5)sqrt(7) Step 1: Group the numerical coefficients and the surds. (5 × (2)/(5)) × (sqrt(7) × sqrt(7)) Step 2: Multiply the numerical coefficients. 5 × (2)/(5) = (10)/(5) = 2 Step 3: Multiply the surds. Recall that sqrt(a) × sqrt(a) = a. sqrt(7) × sqrt(7) = 7 Step 4: Multiply the results from Step 2 and Step 3. 2 × 7 = 14 The evaluated expression is: 14 11. Simplify by expressing (3sqrt(2))^2 as a complete surd. Step 1: Apply the exponent to both parts inside the parenthesis. Recall that (ab)^n = a^n b^n. (3sqrt(2))^2 = 3^2 × (sqrt(2))^2 Step 2: Evaluate each part. 3^2 = 9 (sqrt(2))^2 = 2 Step 3: Multiply the results. 9 × 2 = 18 Step 4: Express the result as a complete surd. To express 18 as a complete surd, we need to write it as sqrt(x). 18 = sqrt(18^2) = sqrt(324) The expression as a complete surd is: sqrt(324) What's next?