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