Given Expression = \(\frac{\left(243\right)^{\frac{n}{5}}\times3^{2n+1}}{9^{n}\times3^{n-1}}\)

= \(\frac{\left(3^{5}\right)^{\left(\frac{n}{5}\right)} \times3^{2n+1}}{\left(3^{2}\right)^{n}\times3^{n-1}}\)

= \(\frac{\left(3^{5\times(\frac{n}{5})}\times3^{2n-1}\right)}{\left(3^{2n}\times3^{n-1}\right)}\)

= \(\frac{3^{n}\times3^{2n-+1}}{3^{2n}\times3^{n-1}}\)

= \(\frac{3^{(n+2n+1)}}{3^{(2n+n-1)}}\)

= \(\frac{3^{3n+1}}{3^{3n-1}}\)

= 3^{(3n + 1 - 3n + 1)}   

= 3²   = 9.