$$( \frac { 2 ^ { 2 } } { 5 } ) ^ { 5 } \times ( \frac { 2 ^ { 2 n + 2 } } { 5 } ) = ( \frac { 2 ^ { 3 } } { 5 } ) ^ { 6 }$$
$n=3$
$$\left(\frac{2^{2}}{5}\right)^{5}\times 2^{2n+2}=5\times \left(\frac{2^{3}}{5}\right)^{6}$$
$$\left(\frac{4}{5}\right)^{5}\times 2^{2n+2}=5\times \left(\frac{2^{3}}{5}\right)^{6}$$
$$\frac{1024}{3125}\times 2^{2n+2}=5\times \left(\frac{2^{3}}{5}\right)^{6}$$
$$\frac{1024}{3125}\times 2^{2n+2}=5\times \left(\frac{8}{5}\right)^{6}$$
$$\frac{1024}{3125}\times 2^{2n+2}=5\times \frac{262144}{15625}$$
$$\frac{1024}{3125}\times 2^{2n+2}=\frac{262144}{3125}$$
$$2^{2n+2}=256$$
$$\log(2^{2n+2})=\log(256)$$
$$\left(2n+2\right)\log(2)=\log(256)$$
$$2n+2=\frac{\log(256)}{\log(2)}$$
$$2n+2=\log_{2}\left(256\right)$$
$$2n=8-2$$
$$n=\frac{6}{2}$$
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