Simplify: $\;$ $\dfrac{\sqrt[3]{a^5 b^{\frac{1}{2}} \sqrt[4]{a^{-1}}}}{\left(a^2 \sqrt[5]{ab^3}\right)^2}$
$\dfrac{\sqrt[3]{a^5 b^{\frac{1}{2}} \sqrt[4]{a^{-1}}}}{\left(a^2 \sqrt[5]{ab^3}\right)^2}$
$= \dfrac{\sqrt[3]{a^5 \times b^{\frac{1}{2}} \times a^{\frac{-1}{4}}}}{\left(a^2 \times a^{\frac{1}{5}} \times b^{\frac{3}{5}}\right)^2}$
$= \dfrac{\left(a^{\frac{19}{4}} \times b^{\frac{1}{2}}\right)^{\frac{1}{3}}}{\left(a^{\frac{11}{5}} \times b^{\frac{3}{5}}\right)^2}$
$= \dfrac{a^{\frac{19}{12}} \times b^{\frac{1}{6}}}{a^{\frac{22}{5}} \times b^{\frac{6}{5}}}$
$= a^{\frac{19}{12} - \frac{22}{5}} \times b^{\frac{1}{6} - \frac{6}{5}}$
$= a^{\frac{-169}{60}} \times b^{\frac{-31}{30}}$