The expression $\frac{a^{-2}}{a^{5}} \times \frac{4 a}{\left(2^{-1} a\right)^{-3}}$ where $a \neq 0$, is equal to:
(a) $\frac{a^{3}}{2}$
(b) $\frac{2}{a^{3}}$
(c) $\frac{1}{2 a^{3}}$
(d) $\frac{a^{5}}{2}$
(e) $\frac{2}{a^{5}}$
We have:

$$
\begin{aligned}
\frac{a^{-2}}{a^{5}} \times \frac{4 a}{\left(2^{-1} a\right)^{-3}} & =a^{-2-5} \times \frac{2^{2} a}{2^{3} a^{-3}} \\
& =a^{-7} \times \frac{a^{1-(-3)}}{2} \\
& =a^{-7} \times \frac{a^{4}}{2} \\
& =\frac{a^{-7+4}}{2}=\frac{a^{-3}}{2}=\frac{1}{2 a^{3}}
\end{aligned}
$$

The correct option is (c)