Simplify: $\;$ $\left(\dfrac{\sqrt[4]{a^3} - b}{\sqrt[4]{a} - \sqrt[3]{b}} - 3 \sqrt[12]{a^3 b^4}\right)^{\frac{-1}{2}} \left(\dfrac{\sqrt[4]{a^3} + b}{\sqrt[4]{a} + \sqrt[3]{b}} - \sqrt[3]{b^2}\right)$, $\;$ $b > 0$, $\;$ $\sqrt[4]{a} > \sqrt[3]{b}$
$\left(\dfrac{\sqrt[4]{a^3} - b}{\sqrt[4]{a} - \sqrt[3]{b}} - 3 \sqrt[12]{a^3 b^4}\right)^{\frac{-1}{2}} \left(\dfrac{\sqrt[4]{a^3} + b}{\sqrt[4]{a} + \sqrt[3]{b}} - \sqrt[3]{b^2}\right)$
$= \left(\dfrac{a^{\frac{3}{4}} - b}{a^{\frac{1}{4}} - b^{\frac{1}{3}}} - 3 a^{\frac{1}{4}} b^{\frac{1}{3}}\right)^{\frac{-1}{2}} \left(\dfrac{a^{\frac{3}{4}} + b}{a^{\frac{1}{4}} + b^{\frac{1}{3}}} - b^{\frac{2}{3}}\right)$
$= \left(\dfrac{a^{\frac{3}{4}} - b - 3 a^{\frac{1}{2}} b^{\frac{1}{3}} + 3 a^{\frac{1}{4}} b^{\frac{2}{3}}}{a^{\frac{1}{4}} - b^{\frac{1}{3}}}\right)^{\frac{-1}{2}} \left(\dfrac{a^{\frac{3}{4}} + b - a^{\frac{1}{4}} b^{\frac{2}{3}} - b}{a^{\frac{1}{4}} + b^{\frac{1}{3}}}\right)$
$= \left[\dfrac{a^{\frac{1}{4}} - b^{\frac{1}{3}}}{\left(a^{\frac{1}{4}} - b^{\frac{1}{3}}\right)^3}\right]^{\frac{1}{2}} \left(\dfrac{a^{\frac{3}{4}} - a^{\frac{1}{4}} b^{\frac{2}{3}}}{a^{\frac{1}{4}} + b^{\frac{1}{3}}}\right)$
$= \left[\dfrac{1}{\left(a^{\frac{1}{4}} - b^{\frac{1}{3}}\right)^2}\right]^{\frac{1}{2}} \left[\dfrac{a^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{2}{3}}\right)}{a^{\frac{1}{4}} + b^{\frac{1}{3}}}\right]$
$= \dfrac{a^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{2}{3}}\right)}{\left(a^{\frac{1}{4}} - b^{\frac{1}{3}}\right) \left(a^{\frac{1}{4}} + b^{\frac{1}{3}}\right)}$
$= \dfrac{a^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{2}{3}}\right)}{\left(a^{\frac{1}{4}}\right)^2 - \left(b^{\frac{1}{3}}\right)^2}$
$= \dfrac{a^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{2}{3}}\right)}{a^{\frac{1}{2}} - b^{\frac{2}{3}}}$
$= a^{\frac{1}{4}}$ $\;\;\;$ $\left[\because \; \sqrt[4]{a} > \sqrt[3]{b} \implies \left(\sqrt[4]{a}\right)^2 > \left(\sqrt[3]{b}\right)^2 \implies a^{\frac{1}{2}} - b^{\frac{2}{3}} > 0 \right]$