Answer
$\dfrac{xy}{\sqrt[3]{10xyz}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the numerator of the given expression, $
\sqrt[3]{\dfrac{xy^2}{10z}}
,$ multiply by an expression equal to $1$ which will make the numerator a perfect power of the index. Then use the laws of radicals to simplify the resulting expression.
$\bf{\text{Solution Details:}}$
Multiplying the given expression by an expression equal to $1$ which will make the numerator a perfect power of the index results to
\begin{array}{l}\require{cancel}
\sqrt[3]{\dfrac{xy^2}{10z}\cdot\dfrac{xy}{xy}}
\\\\=
\sqrt[3]{\dfrac{x^3y^3}{10xyz}}
.\end{array}
Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[3]{x^3y^3}}{\sqrt[3]{10xyz}}
\\\\=
\dfrac{\sqrt[3]{(xy)^3}}{\sqrt[3]{10xyz}}
\\\\=
\dfrac{xy}{\sqrt[3]{10xyz}}
.\end{array}
