Simplify: $\;$ $\dfrac{a^{\frac{4}{3}} - 8 a^{\frac{1}{3}} b}{a^{\frac{2}{3}} + 2\sqrt[3]{ab} + 4 b^{\frac{2}{3}}} : \left(1 - 2 \sqrt[3]{\dfrac{b}{a}}\right)$
$\dfrac{a^{\frac{4}{3}} - 8 a^{\frac{1}{3}} b}{a^{\frac{2}{3}} + 2\sqrt[3]{ab} + 4 b^{\frac{2}{3}}} : \left(1 - 2 \sqrt[3]{\dfrac{b}{a}}\right)$
$= \dfrac{a^{\frac{1}{3}} \left(a - 8b\right)}{a^{\frac{2}{3}} + 2 a^{\frac{1}{3}} b^{\frac{1}{3}} + 4 b^{\frac{2}{3}}} \times \dfrac{1}{1 - 2 b^{\frac{1}{3}} a^{\frac{-1}{3}}}$
$= \dfrac{a^{\frac{1}{3}} \left(a - 8b\right)}{a^{\frac{2}{3}} + 2 a^{\frac{1}{3}} b^{\frac{1}{3}} + 4 b^{\frac{2}{3}} - 2 a^{\frac{1}{3}} b^{\frac{1}{3}} - 4b^{\frac{2}{3}} - 8 a^{\frac{-1}{3}} b}$
$= \dfrac{a^{\frac{1}{3}} \left(a - 8b\right)}{a^{\frac{2}{3}} - 8 a^{\frac{-1}{3}} b}$
$= \dfrac{a^{\frac{1}{3}} \left(a - 8b\right)}{a^{\frac{-1}{3}} \left(a - 8b\right)}$
$= \dfrac{a^{\frac{1}{3}}}{a^{\frac{-1}{3}}}$
$= a^{\frac{2}{3}}$