Question

Simplify : $[{\left(\frac{25}{9}\right)}^{\frac{-3}{2}}÷{\left(\frac{5}{3}\right)}^{-3}]$.

Answer 1

firstly, convert the power positive,

${\left(\frac{25}{9}\right)}^{\frac{-3}{2}}÷{\left(\frac{5}{3}\right)}^{-3}={\left(\frac{9}{25}\right)}^{\frac{3}{2}}÷{\left(\frac{3}{5}\right)}^3$

Simplify the numerator.

$\begin{array}{c}{\left(\frac{9}{25}\right)}^{\frac{3}{2}}÷{\left(\frac{3}{5}\right)}^3={\left({\left(\frac{3}{5}\right)}^2\right)}^{\frac{3}{2}}÷{\left(\frac{3}{5}\right)}^3\\ ={\left(\frac{3}{5}\right)}^3÷{\left(\frac{3}{5}\right)}^3\end{array}$

Simplify further,

$\begin{array}{c}{\left(\frac{3}{5}\right)}^3÷{\left(\frac{3}{5}\right)}^3=\frac{{\left(\frac{3}{5}\right)}^3}{{\left(\frac{3}{5}\right)}^3}\\ =1\end{array}$