Answer
$\dfrac{m}{196}$
Work Step by Step
Using the laws of exponents, the given expression, $
\left( \dfrac{7m^{-2}}{m^{-3}} \right)^{-2} \cdot\dfrac{m^3}{4}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\left( \dfrac{m^{-3}}{7m^{-2}} \right)^{2} \cdot\dfrac{m^3}{4}
\\\\=
\dfrac{m^{-3(2)}}{7^2m^{-2(2)}} \cdot\dfrac{m^3}{4}
\\\\=
\dfrac{m^{-6}}{49m^{-4}} \cdot\dfrac{m^3}{4}
\\\\=
\dfrac{m^{-6+3}}{196m^{-4}}
\\\\=
\dfrac{m^{-3}}{196m^{-4}}
\\\\=
\dfrac{m^{-3-(-4)}}{196}
\\\\=
\dfrac{m^{-3+4}}{196}
\\\\=
\dfrac{m^{1}}{196}
\\\\=
\dfrac{m}{196}
.\end{array}