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 — i) Step 1: Simplify 8^1/3. 8^1/3 = [3]8 = 2. Step 2: Rewrite the expression using the simplified term and express all numbers as powers of 2. 8^1/3 × 8^-3 × 32 = 2 × (2^3)^-3 × 2^5 Step 3: Apply the exponent rule (a^m)^n = a^mn. 2 × 2^3 × (-3) × 2^5 = 2^1 × 2^-9 × 2^5 Step 4: Apply the exponent rule a^m × a^n = a^m+n. 2^1 + (-9) + 5 = 2^1 - 9 + 5 = 2^-3 Step 5: Calculate the final value. 2^-3 = (1)/(2^3) = (1)/(8) The simplified expression is (1)/(8). ii) Step 1: Express all numbers in the expression as powers of 3. 27 = 3^3 9 = 3^2 81 = 3^4 Substitute these into the expression: (3^3)^2/3 × (3^2)^5(3^4)^4 Step 2: Apply the exponent rule (a^m)^n = a^mn to each term. 3^3 × (2/3) × 3^2 × 53^4 × 4 = 3^2 × 3^103^16 Step 3: Apply the exponent rule a^m × a^n = a^m+n to the numerator. 3^2+103^16 = 3^123^16 Step 4: Apply the exponent rule (a^m)/(a^n) = a^m-n. 3^12-16 = 3^-4 Step 5: Calculate the final value. 3^-4 = (1)/(3^4) = (1)/(81) The simplified expression is (1)/(81). Got more? Send 'em.