Answer
$25x^{4}y^{1/3}$
Work Step by Step
Using $\left(x^my^n\right)^p=x^{mp}y^{np}$ the given expression, $
\left(125x^6y^{1/2}\right)^{2/3}
,$ is equivalent to
\begin{align*}
&
125^{\frac{2}{3}}x^{6\cdot\frac{2}{3}}y^{\frac{1}{2}\cdot\frac{2}{3}}
\\\\&=
125^{\frac{2}{3}}x^{4}y^{\frac{1}{3}}
\\\\&=
125^{\frac{2}{3}}x^{4}y^{1/3}
.\end{align*}
Using $x^{\frac{m}{n}}=\sqrt[n]{x^m}=\left(\sqrt[n]{x}\right)^m$ the expression above is equivalent to
\begin{align*}
&
\left(\sqrt[3]{125}\right)^{2}x^{4}y^{1/3}
\\\\&=
\left(5\right)^{2}x^{4}y^{1/3}
\\\\&=
25x^{4}y^{1/3}
.\end{align*}
Hence, the expression $\left(125x^6y^{1/2}\right)^{2/3}$ simplifies to $25x^{4}y^{1/3}$.
