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