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