Answer
$\dfrac{x\sqrt{xy}}{y^{2}}$
Work Step by Step
Since $\sqrt[4]{x^2y^8}$ is equivalent to $\sqrt{\sqrt{x^2y^8}},$ the given expression, $
\dfrac{\sqrt{x^4y}}{\sqrt[4]{x^2y^8}}
,$ is equivalent to
\begin{align*}
&
\dfrac{\sqrt{x^4y}}{\sqrt{\sqrt{x^2y^8}}}
\\\\&=
\dfrac{\sqrt{x^4y}}{\sqrt{\sqrt{(xy^4)^2}}}
\\\\&=
\dfrac{\sqrt{x^4y}}{\sqrt{xy^4}}
.\end{align*}
Using the laws of exponents, the expression above is equivalent to
\begin{align*}
&
\sqrt{\dfrac{x^4y}{xy^4}}
&\left( \text{use }\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}} \right)
\\\\&=
\sqrt{x^{4-1}y^{1-4}}
&\left( \text{use }\dfrac{a^x}{a^y}=a^{x-y} \right)
\\\\&=
\sqrt{x^{3}y^{-3}}
\\\\&=
\sqrt{\dfrac{x^{3}}{y^{3}}}
&\left( \text{use }a^{-x}=\dfrac{1}{a^x} \right)
\\\\&=
\sqrt{\dfrac{x^{3}}{y^{3}}\cdot\dfrac{y}{y}}
&\left( \text{rationalize the denominator} \right)
\\\\&=
\sqrt{\dfrac{x^{3}y}{y^{4}}}
\\\\&=
\dfrac{\sqrt{x^{3}y}}{\sqrt{y^{4}}}
\\\\&=
\dfrac{\sqrt{x^{2}\cdot xy}}{\sqrt{(y^{2})^2}}
\\\\&=
\dfrac{x\sqrt{xy}}{y^{2}}
.\end{align*}
Hence, the simplified form of the given expression is $
\dfrac{x\sqrt{xy}}{y^{2}}
.$
