$\text{a)}\frac{{\left(5^{2n+1}\cdot 5^{n-1}\cdot 5^{2+n}\right)}^n}{{\left(5^{n-1}\right)}^{4n+4}}=\frac{{\left(5^{2n+1+n-1+2+n}\right)}^n}{{\left(5^{n-1}\right)}^{4n+4}}=$

$=\frac{{\left(5^{4n+2}\right)}^n}{5^{\left(n-1\right)\left(4n+4\right)}}=\frac{5^{4n^2+2n}}{5^{4\left(n-1\right)\left(n+1\right)}}=\frac{5^{4n^2+2n}}{5^{4n^2-4}}=5^{4n^2+2n-4n^2+4}=5^{2n+4}$   

$\text{b)}{\left(\frac{4^{2n+1}\cdot 8^{n-1}\cdot 2^n}{{16}^{2n+1}}\right)}^n={\left(\frac{{\left(2^2\right)}^{2n+1}\cdot {\left(2^3\right)}^{n-1}\cdot 2^n}{{\left(2^4\right)}^{2n+1}}\right)}^n={\left(\frac{2^{4n+2}\cdot 2^{3n-3}\cdot 2^n}{2^{8n+4}}\right)}^n=$  

$={\left(\frac{2^{4n+2+3n-3+n}}{2^{8n+4}}\right)}^n={\left(\frac{2^{8n-1}}{2^{8n+4}}\right)}^n={\left(2^{8n-1-8n-4}\right)}^n={\left(2^{-5}\right)}^n=2^{-5n}$