Answer
$\dfrac{\sqrt[6]{x^4y^3}}{|y|}$
Work Step by Step
Using $a^{-m}=\dfrac{1}{a^m}$ and $\dfrac{1}{a^{-m}}=a^m,$ the given expression, $
\sqrt[6]{\dfrac{y^{-3}}{x^{-4}}}
,$ is equivalent to
\begin{align*}\require{cancel}
&\sqrt[6]{\dfrac{x^4}{y^3}}
.\end{align*}
Multiplying the numerator and the denominator by an expression equal to $1$ which will make the denominator a perfect power of the index, the expression above is equivalent to
\begin{align*}\require{cancel}
&
\sqrt[6]{\dfrac{x^4}{y^3}\cdot\dfrac{y^3}{y^3}}
\\\\&=
\sqrt[6]{\dfrac{x^4y^3}{y^6}}
.\end{align*}
Using the properties of radicals, the expression above is equivalent to
\begin{align*}\require{cancel}
&
\dfrac{\sqrt[6]{x^4y^3}}{\sqrt[6]{y^6}}
\\\\&=
\dfrac{\sqrt[6]{x^4y^3}}{\sqrt[6]{(y)^6}}
\\\\&=
\dfrac{\sqrt[6]{x^4y^3}}{|y|}
&\left(\sqrt[n]{a^n}=|a|\text{ if $n$ is even} \right)
.\end{align*}
Hence, the simplified form of the given expression is $
\dfrac{\sqrt[6]{x^4y^3}}{|y|}
.$