Answer
$\dfrac{2\sqrt[3]{x^2}}{x^2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt[3]{\dfrac{8}{x^4}}
,$ make the denominator a perfect power of the index so that the final result will already be in rationalized form. Then find a factor of the radicand that is a perfect power of the index. Finally, extract the root of that factor.
$\bf{\text{Solution Details:}}$
Multiplying the radicand that will make the denominator a perfect power of the index results to
\begin{array}{l}\require{cancel}
\sqrt[3]{\dfrac{8}{x^4}\cdot\dfrac{x^2}{x^2}}
\\\\=
\sqrt[3]{\dfrac{8x^2}{x^6}}
.\end{array}
Factoring the radicand into an expression that is a perfect power of the index and then extracting its root result to
\begin{array}{l}\require{cancel}
\sqrt[3]{\dfrac{8}{x^6}\cdot x^2}
\\\\=
\sqrt[3]{\left(\dfrac{2}{x^2}\right)^3\cdot x^2}
\\\\=
\dfrac{2}{x^2}\sqrt[3]{x^2}
\\\\=
\dfrac{2\sqrt[3]{x^2}}{x^2}
.\end{array}
Note that all variables are assumed to have positive values.
