Convert all terms to the same base: 3^7.5 ÷ 27^1.5 × 9^2 = 3^?. Find the value of ?.

Introduction / Context:When expressions involve powers of numbers that are themselves powers of a common base, convert everything to that common base to simplify. Here 27 and 9 are powers of 3, so rewriting yields a single exponent arithmetic problem.

Given Data / Assumptions:

• Expression: 3^7.5 ÷ 27^1.5 × 9^2
• Note: 27 = 3^3 and 9 = 3^2

Concept / Approach:Use power rules: (a^m)^n = a^(m*n). Convert 27^1.5 to (3^3)^1.5 = 3^(4.5). Convert 9^2 to (3^2)^2 = 3^4. Then perform exponent arithmetic: addition when multiplying, subtraction when dividing.

Step-by-Step Solution:

Verification / Alternative check:Check order: division first (subtract 4.5 from 7.5), then multiplication (add 4). The final exponent 7.0 is consistent regardless of grouping due to associativity of multiplication/division in exponent arithmetic with the same base.

Why Other Options Are Wrong:

• 5, 4.5, 6, 6.50: do not match the simplified exponent sum 7.0.

Common Pitfalls:Forgetting that dividing like bases subtracts exponents, or miscomputing (3^3)^1.5 as 3^3*1.5 (which is correct) but then making a decimal slip in the subtraction or addition step.

Final Answer:7