Question

Simplify:
${\left(\frac{2}{5}\right)}^{-3}\times {\left(\frac{25}{4}\right)}^{-2}$

Answer 1

${\left(\frac{2}{5}\right)}^{-3}\times {\left(\frac{25}{4}\right)}^{-2}$

$={\left(\frac{2}{5}\right)}^{-3}\times {\left(\frac{5^2}{2^2}\right)}^{-2}$

$={\left(\frac{2}{5}\right)}^{-3}\times {\left({\left(\frac{5}{2}\right)}^2\right)}^{-2}$ since $\frac{a^m}{b^m}={\left(\frac{a}{b}\right)}^m$

$={\left(\frac{2}{5}\right)}^{-3}\times {\left(\frac{5}{2}\right)}^{-4}$ since ${\left(a^m\right)}^n=a^{mn}$

$={\left(\frac{2}{5}\right)}^{-3}\times {\left(\frac{2}{5}\right)}^4$ since $a^{-m}=\frac{1}{a^m}$

$={\left(\frac{2}{5}\right)}^{-3+4}$ since $a^m{a}^n=a^{m+n}$

$={\left(\frac{2}{5}\right)}^1$

$=\frac{2}{5}$