Simplify $\left(\dfrac{-2p^{3}}{q^{-1}}\right)^{2}\div\left(\dfrac{8q^{4}}{p^{3}}\right)^{\tfrac{1}{3}}$, giving your answer in positive index form.

1. Simplify the first bracket and square it.

$\left(\dfrac{-2p^3}{q^{-1}}\right)^2=\left(-2p^3\times q\right)^2=(-2p^3q)^2=4p^6q^2$

2. Simplify the second bracket using the fractional index.

$\left(\dfrac{8q^4}{p^3}\right)^{\frac13}=\dfrac{8^{\frac13}q^{\frac43}}{p}=\dfrac{2q^{\frac43}}{p}$

3. Divide the two simplified expressions and write the final answer with positive indices.

$4p^6q^2\div \dfrac{2q^{\frac43}}{p}=4p^6q^2\times \dfrac{p}{2q^{\frac43}}$
$=2p^7q^{2-\frac43}=2p^7q^{\frac23}$