Question : Find the value of $\frac{(243)^{\frac{n}{5}}\times 3^{2n+1}}{9^{n}\times 3^{n-1}}$.
Option 1: 3
Option 2: 9
Option 3: 27
Option 4: 4
Correct Answer: 9
Solution :
Consider, $\frac{(243)^{\frac{n}{5}}\times 3^{2n+1}}{9^{n}\times 3^{n-1}}$
$=\frac{((3)^5)^{\frac{n}{5}}\times 3^{2n+1}}{((3)^2)^{n}\times 3^{n-1}}$
$=\frac{3^n\times3^{2n+1}}{3^{2n}\times3^{n-1}}$
$=\frac{3^{n+2n+1}}{3^{2n+n-1}}$
$=\frac{3^{3n}\times 3}{3^{3n}\times3^{-1}}$
$=3 \times 3$
$=9$
Hence, the correct answer is 9.
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