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
You're on a roll — Step 1: Evaluate 3^2 × 2^3 × 4^1 × 3^1 × 4^2. Group terms with the same base: (3^2 × 3^1) × 2^3 × (4^1 × 4^2) Apply the rule a^m × a^n = a^m+n: 3^2+1 × 2^3 × 4^1+2 3^3 × 2^3 × 4^3 Calculate the values: 3^3 = 3 × 3 × 3 = 27 2^3 = 2 × 2 × 2 = 8 4^3 = 4 × 4 × 4 = 64 Multiply these values: 27 × 8 × 64 216 × 64 13824 13824 Step 2: Evaluate 3^3 × 4^4 × 5^-23^1 × 5^-1 × 4^1. Group terms with the same base and apply the rule (a^m)/(a^n) = a^m-n: ((3^3)/(3^1)) × ((4^4)/(4^1)) × (5^-25^-1) 3^3-1 × 4^4-1 × 5^-2 - (-1) 3^2 × 4^3 × 5^-2+1 3^2 × 4^3 × 5^-1 Calculate the values: 3^2 = 9 4^3 = 64 5^-1 = (1)/(5) Multiply these values: 9 × 64 × (1)/(5) 576 × (1)/(5) (576)/(5) (576)/(5) Step 3: Find the value of (3^1)^2. Apply the rule (a^m)^n = a^mn: (3^1)^2 = 3^1 × 2 = 3^2 Calculate the value: 3^2 = 3 × 3 = 9 9 Step 4: Expand (2^3 x^-1 (1)/(3^2))^-1 and simplify. Write your answer as a positive power. First, rewrite (1)/(3^2) as 3^-2: (2^3 x^-1 3^-2)^-1 Apply the outer exponent of -1 to each term inside the parentheses using the rule (abc)^n = a^n b^n c^n and (a^m)^n = a^mn: (2^3)^-1 (x^-1)^-1 (3^-2)^-1 2^3 × (-1) x^-1 × (-1) 3^-2 × (-1) 2^-3 x^1 3^2 Rewrite terms with negative exponents as positive exponents using the rule a^-n = (1)/(a^n): 2^-3 = (1)/(2^3) So the expression becomes: (1)/(2^3) × x × 3^2 (x × 3^2)/(2^3) (9x)/(8) What's next?