Answer
$|y|\sqrt[12]{x^5y^2}$
Work Step by Step
Converting the radicals to like indices, the given expression, $
\dfrac{\sqrt[3]{x^2y^5}}{\sqrt[4]{xy^2}}
,$ is equivalent to
\begin{align*}
&
\dfrac{\left(\sqrt[3(4)]{x^2y^5}\right)^4}{\left(\sqrt[4(3)]{xy^2}\right)^3}
\\\\&=
\dfrac{\sqrt[12]{\left(x^2y^5\right)^4}}{\sqrt[12]{\left(xy^2\right)^3}}
&\left(\left(\sqrt[n]{x}\right)^m=\sqrt[n]{x^m}\right)
\\\\&=
\dfrac{\sqrt[12]{x^{2(4)}y^{5(4)}}}{\sqrt[12]{x^{1(3)}y^{2(3)}}}
&\left((x^my^n)^p=x^{mp}y^{np}\right)
\\\\&=
\dfrac{\sqrt[12]{x^{8}y^{20}}}{\sqrt[12]{x^{3}y^{6}}}
.\end{align*}
Using $\dfrac{\sqrt[n]{x^m}}{\sqrt[n]{x^n}}=\sqrt[n]{\dfrac{x^m}{x^n}},$ the expression above is equivalent to
\begin{align*}
&
\sqrt[12]{\dfrac{x^{8}y^{20}}{x^{3}y^{6}}}
\\\\&=
\sqrt[12]{x^{8-3}y^{20-6}}
&\left(\dfrac{x^m}{x^n}=x^{m-n}\right)
\\\\&=
\sqrt[12]{x^{5}y^{14}}
\\\\&=
\sqrt[12]{y^{12}\cdot x^5y^2}
\\\\&=
\sqrt[12]{y^{12}}\cdot\sqrt[12]{x^5y^2}
\\\\&=
|y|\sqrt[12]{x^5y^2}
&\left(\sqrt[n]{x^n}=|x|\text{, if $n$ is even}\right)
.\end{align*}
Hence, the expression $\dfrac{\sqrt[3]{x^2y^5}}{\sqrt[4]{xy^2}}$ simplifies to $|y|\sqrt[12]{x^5y^2}$.
