\[\frac{\left( 5^{3} \right)^{5} \cdot 5^{7}}{\left( 5^{5} \right)^{4}} = \frac{5^{15} \cdot 5^{7}}{5^{20}} =\]
\[= \frac{5^{22}}{5^{20}} = 5^{2} = 25.\]