[(243)n/5 × 32n + 1] ÷ [9n × 3n - 1] = ?

Concept: The laws of exponents let every term be rewritten with a common base before combining powers: am/n = (am)1/n, am × an = am+n, and am ÷ an = am−n. Converting 243 and 9 to powers of 3 lets the whole expression collapse to a single power of 3.

1. Rewrite 243 as 35, so (243)n/5 = (35)n/5 = 3n.

2. Rewrite 9 as 32, so 9n = (32)n = 32n.

3. Substitute into the expression: numerator = 3n × 32n+1 = 3n+2n+1 = 33n+1; denominator = 32n × 3n−1 = 32n+n−1 = 33n−1.

4. Divide powers of the same base by subtracting exponents: 33n+1 ÷ 33n−1 = 3(3n+1)−(3n−1) = 32.

5. Evaluate: 32 = 9.

Cross-check: Substituting n = 1 into the original expression gives (243)1/5 × 33 = 3 × 27 = 81, and 91 × 30 = 9 × 1 = 9; dividing gives 81/9 = 9, matching the simplified result.

Result: The expression simplifies to 9, so the option with value 9 is correct.