Question

Evaluate:

${\displaystyle {\left(\frac{27}{125}\right)}^{\frac{2}{3}}\times {\left(\frac{9}{25}\right)}^{-\frac{3}{2}}}$

Answer 1

${\displaystyle {\left(\frac{27}{125}\right)}^{\frac{2}{3}}\times {\left(\frac{9}{25}\right)}^{-\frac{3}{2}}}$

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

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

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

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

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

= ${\displaystyle {\left(\frac{3}{5}\right)}^{-1}}$

= ${\displaystyle \frac{1}{\frac{3}{5}}}$

${\displaystyle =\frac{5}{3}}$