Simplify: $( \frac{4}{5})^{2}\times5^{4}\times( \frac{2}{5})^{-2}\p( \frac{5}{2})^{-3}$.

 Given: $( \frac{4}{5})^{2}\times5^{4}\times( \frac{2}{5})^{-2}\div( \frac{5}{2})^{-3}$.

To do: To simplify: $( \frac{4}{5})^{2}\times5^{4}\times( \frac{2}{5})^{-2}\div( \frac{5}{2})^{-3}$.

Solution:

$( \frac{4}{5})^{2}\times5^{4}\times( \frac{2}{5})^{-2}\div( \frac{5}{2})^{-3}$

$=( \frac{4}{5})^{2}\times5^{4}\times( \frac{5}{2})^{2}\times( \frac{5}{2})^{3}$       [$\because ( \frac{a}{b})^{-n}=\frac{b}{a})^n$]

$=\frac{4^2\times5^4\times5^2\times5^3}{5^2\times2^2\times2^3}$

$=\frac{(2^2)^2\times5^{( 4+2+3)}}{5^2\times2^{( 2+3)}}$        [$\because a^m\times a^n=a^{(m+n)}$]

$=\frac{2^4\times5^9}{5^2\times2^5}$

$=2^{4-5}\times5^{9-2}$

$=2^{-1}\times5^7$

$=\frac{5^7}{2}$

Thus, $( \frac{4}{5})^{2}\times5^{4}\times( \frac{2}{5})^{-2}\div( \frac{5}{2})^{-3}=\frac{5^7}{2}$.