Simplify: $\;$ $\dfrac{2a \sqrt[3]{a b^2} - a \sqrt[6]{a b^5} - ab}{\sqrt[3]{a^2 b} - \sqrt{ab}} - 2^{\left(1 + 2 \log_8 a + \log_8 b\right)}$
$\dfrac{2a \sqrt[3]{a b^2} - a \sqrt[6]{a b^5} - ab}{\sqrt[3]{a^2 b} - \sqrt{ab}} - 2^{\left(1 + 2 \log_8 a + \log_8 b\right)}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\left(\log_8 8 + \log_8 a^2 + \log_8 b\right)}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\log_8 8a^2b}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\frac{\log_2 8a^2b}{\log_2 8}}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\frac{\log_2 8a^2b}{\log_2 2^3}}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\frac{\log_2 8a^2b}{3 \log_2 2}}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\frac{1}{3} \log_2 8a^2b}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\left(\log_2 8a^2b\right)^{\frac{1}{3}}}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - \left(8a^2b\right)^{\frac{1}{3}}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2 a^{\frac{2}{3} b^{\frac{1}{3}}}$
$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab - 2 a^{\frac{4}{3}} b^{\frac{2}{3}} + 2 a^{\frac{7}{6}} b^{\frac{5}{6}}}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}}$
$= \dfrac{a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}}$
$= \dfrac{ab \left(a^{\frac{1}{6}} b^{\frac{-1}{6}} - 1\right)}{a^{\frac{1}{2}} b^{\frac{1}{2}} \left(a^{\frac{1}{6}} b^{\frac{-1}{6}} - 1\right)}$
$= \dfrac{ab}{a^{\frac{1}{2}} b^{\frac{1}{2}}}$
$= \dfrac{ab \times \sqrt{ab}}{\sqrt{ab} \times \sqrt{ab}}$
$= \dfrac{ab \times \sqrt{ab}}{ab}$
$= \sqrt{ab}$