\((X)^{2}=(Y)^{3}\ \&\ Y>1(given)\)

\(\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={\left({X}^{{{2}}}\right)}^{{\frac{{{1}}}{{{3}}}}}-------------{r}\underline{{e}}\)

\((X)^{\frac{m}{n}}=(X)^{m\frac{1}{n}}\)

compensate by \(( X )^{2} = ( Y )^{3}\)

\(\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={\left({Y}^{{{3}}}\right)}^{{\frac{{{1}}}{{{3}}}}}={\left({Y}\right)}^{{3}}\times{\frac{{{1}}}{{{3}}}}={Y}\)

thus

\(\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={Y}\)

another answer

\(\displaystyle{\left({X}\right)}^{{{2}}}={\left({Y}\right)}^{{{3}}}\) ------------ take the cube root of both sides

\(\displaystyle{\left({X}^{{{2}}}\right)}^{{\frac{{{1}}}{{{3}}}}}={\left({\left({Y}\right)}^{{{3}}}\right)}^{{\frac{{{1}}}{{{3}}}}}\)

\(\displaystyle{\left({X}^{{{2}\times{\frac{{{1}}}{{{3}}}}}}={Y}^{{{3}}}\times{\frac{{{1}}}{{{3}}}}\right.}\)

\(\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={Y}\)

answer ------------- \(\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={Y}\)