Answer
$\dfrac{\sqrt[3]{2xy^2}}{xy}$
Work Step by Step
Multiplying the numerator and the denominator by an expression equal to $1$ which will make the denominator a perfect power of the index, the given expression, $
\dfrac{\sqrt[3]{14}}{\sqrt[3]{7x^2y}}
,$ is equivalent to
\begin{align*}\require{cancel}
&
\dfrac{\sqrt[3]{14}}{\sqrt[3]{7x^2y}}\cdot\dfrac{\sqrt[3]{7^2xy^2}}{\sqrt[3]{7^2xy^2}}
\\\\&=
\dfrac{\sqrt[3]{686xy^2}}{\sqrt[3]{7^3x^3y^3}}
\\\\&=
\dfrac{\sqrt[3]{343\cdot2xy^2}}{\sqrt[3]{(7xy)^3}}
\\\\&=
\dfrac{\sqrt[3]{(7)^3\cdot2xy^2}}{7xy}
\\\\&=
\dfrac{7\sqrt[3]{2xy^2}}{7xy}
.\end{align*}
Hence, the simplified form of the given expression is $
\dfrac{\sqrt[3]{2xy^2}}{xy}
$.
