Evaluate: $\frac{a^{2 n+1} \times a^{(2 n+1)(2 n-1)}}{a^{n(4 n-1)}\times(a^{2})^{2 n+3}}$.

Given: $\frac{a^{2n+1} \times a^{(2n+1)(2n-1)}}{a^{n(4n-1)}\times(a^{2})^{2n+3}}$.

To do: To solve: $\frac{a^{2n+1}\times a^{(2n+1)(2n-1)}}{a^{n(4n-1)}\times(a^{2})^{2n+3}}$.

Solution: 

$\frac{a^{2n+1}\times a^{(2n+1)(2n-1)}}{a^{n(4n-1)}\times(a^{2})^{2n+3}}$

$=\frac{a^{( 2n+1)} \times a^{( 4n^2-1)}}{a^{(4n^2-n)}\times a^{( 4n+6)}}$

$=\frac{a^{( 2n+1+4n^2-1)}}{a^{( 4n^2-n+4n+6)}}$

$=a^{( 2n+1+4n^2-1-4n^2+n-4n-6)}$

$=a^{( -n-6)}$

$=a^{-( n+6)}$

$=\frac{1}{a^{( n+6)}}$

$=\frac{1}{a^n\times a^{6}}$