Evaluate (243)^{n/5} × 3^{2n + 1} ÷ (9^{n} × 3^{n − 1}).

Given data

• Expression: (243)^{n/5} × 3^{2n+1} / (9^{n} × 3^{n−1}).

Concept / Approach

• Rewrite all as powers of 3 and then subtract exponents for division.

Step-by-step calculation

243 = 3^5 ⇒ (243)^{n/5} = (3^5)^{n/5} = 3^{n}.Numerator exponent: n + (2n + 1) = 3n + 1 ⇒ 3^{3n + 1}.Denominator: 9^{n} × 3^{n − 1} = (3^2)^{n} × 3^{n − 1} = 3^{2n} × 3^{n − 1} = 3^{3n − 1}.Overall = 3^{(3n + 1) − (3n − 1)} = 3^{2} = 9.

Verification

Try n = 1: (243)^{0.2}×3^{3}/(9×3^{0}) = 3×27/9 = 9 ✔.

Common pitfalls

• Forgetting to subtract exponents when dividing powers with the same base.

Final Answer

9.