The value of $$\frac{{{{\left( {243} \right)}^{\frac{n}{5}}} \times {3^{2n + 1}}}}{{{9^n} \times {3^{n - 1}}}}$$   is?

Solution (By Examveda Team)

$$\eqalign{ & \frac{{{{\left( {243} \right)}^{\frac{n}{5}}} \times {3^{2n + 1}}}}{{{9^n} \times {3^{n - 1}}}} \cr & \Rightarrow \frac{{{3^{5 \times \frac{n}{5}}} \times {3^{2n + 1}}}}{{{3^{2n}} \times {3^{n - 1}}}} \cr & \Rightarrow \frac{{{3^n} \times {3^{2n + 1}}}}{{{3^{2n}} \times {3^{n - 1}}}} \cr & \Rightarrow \frac{{{3^{n + 2n + 1}}}}{{{3^{2n + n - 1}}}} \cr & \Rightarrow \frac{{{3^{3n + 1}}}}{{{3^{3n - 1}}}} \cr & \Rightarrow {3^{\left( {3n + 1} \right) - \left( {3n - 1} \right)}} \cr & \Rightarrow {3^{3n + 1 - 3n + 1}} \cr & \Rightarrow {3^2} \cr & \Rightarrow 9 \cr} $$