Answer
$\dfrac{2k}{3}$
Work Step by Step
Using the laws of exponents, the given expression, $
\left( \dfrac{3k^{-2}}{k^4} \right)^{-1} \cdot\dfrac{2}{k}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{k^4}{3k^{-2}} \cdot\dfrac{2}{k}
\\\\=
\dfrac{2k^4}{3k^{-2}\cdot k}
\\\\=
\dfrac{2k^{4-(-2)-1}}{3}
\\\\=
\dfrac{2k^{4+2-1}}{3}
\\\\=
\dfrac{2k^{1}}{3}
\\\\=
\dfrac{2k}{3}
.\end{array}