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 problems: 1. Simplify: sqrt(108) + sqrt(27) - sqrt(3) Step 1: Simplify each radical by finding perfect square factors. sqrt(108) = sqrt(36 × 3) = sqrt(36) × sqrt(3) = 6sqrt(3) sqrt(27) = sqrt(9 × 3) = sqrt(9) × sqrt(3) = 3sqrt(3) Step 2: Substitute the simplified radicals back into the expression. 6sqrt(3) + 3sqrt(3) - sqrt(3) Step 3: Combine the like terms. (6 + 3 - 1)sqrt(3) = 8sqrt(3) The correct option is A. A. 8sqrt(3) 2. Rationalize the denominator: (6)/(sqrt(5) - 1) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is sqrt(5) + 1. (6)/(sqrt(5) - 1) × sqrt(5) + 1sqrt(5) + 1 Step 2: Multiply the numerators and denominators. Use the difference of squares formula (a-b)(a+b) = a^2 - b^2 for the denominator. 6(sqrt(5) + 1)(sqrt(5))^2 - (1)^2 Step 3: Simplify the denominator. 6(sqrt(5) + 1)5 - 1 = 6(sqrt(5) + 1)4 Step 4: Simplify the fraction by dividing the numerator and denominator by 2. 3(sqrt(5) + 1)2 The correct option is B. B. 3(sqrt(5) + 1)/2 3. Simplify: (5^3)^2 ÷ 5^4 Step 1: Apply the power of a power rule, (a^m)^n = a^m × n. (5^3)^2 = 5^3 × 2 = 5^6 Step 2: Substitute this back into the expression. 5^6 ÷ 5^4 Step 3: Apply the quotient rule of exponents, a^m ÷ a^n = a^m-n. 5^6-4 = 5^2 The correct option is A. A. 5^2 4. Solve: 6^x = 216 Step 1: Express 216 as a power of 6. 216 = 6 × 6 × 6 = 6^3 Step 2: Substitute this into the equation. 6^x = 6^3 Step 3: Since the bases are equal, the exponents must be equal. x = 3 The correct option is B. B. x=3 What's next?