Indices clean-up: Find m − n if [(9^n * 3^2 * (3^(−n/2))^(−2) − (27)^n) / (3^(3m) * 2^3)] = 1/27.

Introduction / Context:
Simplify a complex-looking indices expression by rewriting every term with a base of 3 and factoring out common powers. Careful exponent algebra quickly collapses the fraction.

Given Data / Assumptions:

• (9^n * 3^2 * (3^(−n/2))^(−2) − (27)^n) / (3^(3m) * 2^3) = 1/27
• 9 = 3^2 and 27 = 3^3
• (3^(−n/2))^(−2) = 3^n

Concept / Approach:
Convert 9^n and 27^n into base 3. Combine exponents in the numerator, factor out the highest common power, and cancel the 2^3 factor with the numerical factor that appears. Then equate powers of 3.

Step-by-Step Solution:
9^n = (3^2)^n = 3^(2n)(3^(−n/2))^(−2) = 3^nNumerator: 3^(2n) * 3^2 * 3^n − (3^3)^n = 3^(3n+2) − 3^(3n) = 3^(3n)(3^2 − 1) = 3^(3n) * 8Denominator: 3^(3m) * 2^3 = 3^(3m) * 8Fraction simplifies to 3^(3n)/3^(3m) = 3^(3n−3m)Set equal to 1/27 = 3^(−3) ⇒ 3n − 3m = −3 ⇒ n − m = −1 ⇒ m − n = 1

Verification / Alternative check:
Pick m = 2, n = 1 (so m − n = 1) to test; the simplified exponent gives −3 as required.

Why Other Options Are Wrong:
They result from sign errors when moving terms or mishandling the (−n/2)^(−2) step.

Common Pitfalls:
Dropping the factor (3^2 − 1) = 8 or forgetting to cancel 2^3 in the denominator.

Final Answer:
1