Simplify: $\;$ $\dfrac{\left(a^{-1} + b^{-1}\right) \left(a + b\right)^{-1}}{\sqrt[6]{a^{4} \sqrt[5]{a^{-2}}}}$
$\dfrac{\left(a^{-1} + b^{-1}\right) \left(a + b\right)^{-1}}{\sqrt[6]{a^{4} \sqrt[5]{a^{-2}}}}$
$= \dfrac{\left(\dfrac{1}{a} + \dfrac{1}{b}\right) \left(\dfrac{1}{a + b}\right)}{\left[a^4 \left(a^{-2}\right)^\frac{1}{5}\right]^{\frac{1}{6}}}$
$= \dfrac{a + b}{ab \left(a + b\right) \left[a^4 \times a^{\frac{-2}{5}}\right]^{\frac{1}{6}}}$
$= \dfrac{1}{ab \left(a^{\frac{18}{5}}\right)^{\frac{1}{6}}}$
$= \dfrac{1}{a b \times a^{\frac{3}{5}}}$
$= \dfrac{1}{a^{\frac{8}{5}} b}$
$= \dfrac{1}{b \sqrt[5]{a^8}}$