(8/125) ​-2/3

\(\frac{1}{\sqrt[3]{\left(\frac{8}{125}\right)^2}}\)

There are two ways to solve this problem.

First way:

\({\frac{8}{125}}^{-\frac{2}{3}}\)

\(\sqrt[3]{{(\frac{8}{125})}^{-2}}\)

\(\sqrt[3]{\frac{{8}^{-2}}{{125}^{-2}}}\)

\(\sqrt[3]{\frac{{125}^{2}}{{8}^{2}}}\)

\(\sqrt[3]{\frac{15625}{{8}^{2}}}\)

\(\sqrt[3]{\frac{15625}{64}}\)

\(\frac{\sqrt[3]{15625}}{\sqrt[3]{64}}\)

\(\frac{\sqrt[3]{25\times25\times25}}{\sqrt[3]{64}}\)

\(\frac{25}{\sqrt[3]{64}}\)

\(\frac{25}{\sqrt[3]{4\times4\times4}}\)

\(\frac{25}{4}\)

\(6.25\)

Second way:

\({\frac{8}{125}}^{-\frac{2}{3}}\)

\(\frac{1}{{(\frac{8}{125})}^{\frac{2}{3}}}\)

\(\frac{1}{\frac{{8}^{\frac{2}{3}}}{{125}^{\frac{2}{3}}}}\)

\(\frac{1}{\frac{({2}^{{3})^{\frac{2}{3}}}}{{125}^{\frac{2}{3}}}}\)

\(\frac{1}{\frac{{2}^{\frac{6}{3}}}{{125}^{\frac{2}{3}}}}\)

\(\frac{1}{\frac{{2}^{2}}{{125}^{\frac{2}{3}}}}\)

\(\frac{1}{\frac{4}{{125}^{\frac{2}{3}}}}\)

\(\frac{1}{\frac{4}{({5}^{{3})^{\frac{2}{3}}}}}\)

\(\frac{1}{\frac{4}{{5}^{\frac{6}{3}}}}\)

\(\frac{1}{\frac{4}{{5}^{2}}}\)

\(\frac{1}{\frac{4}{25}}\)

\(1\times\frac{25}{4}\)

\(\frac{25}{4}\)

\(6.25\)

Thanks Gibson.....here's another "trick".....note     (a /b)^-m/n   =  (b /a)^m/n     ....therefore........

(8/125)^-2/3   =

(125 / 8 )^2/3   =

[(125 / 8)^1/3 ] ²  =

[ 5 / 2 ] ²  =

25 / 4