Simplify: $\;$ $- \left[\left(\dfrac{\sqrt{a} + \sqrt{b}}{\sqrt[4]{a} - \sqrt[4]{b}}\right)^{-1} - \dfrac{2 \sqrt[4]{ab}}{b^{\frac{3}{4}} - a^{\frac{1}{4}} b^{\frac{1}{2}} + a^{\frac{1}{2}} b^{\frac{1}{4}} - a^{\frac{3}{4}}}\right]^{-1} + \sqrt{2}^{\log_4 a}$
$- \left[\left(\dfrac{\sqrt{a} + \sqrt{b}}{\sqrt[4]{a} - \sqrt[4]{b}}\right)^{-1} - \dfrac{2 \sqrt[4]{ab}}{b^{\frac{3}{4}} - a^{\frac{1}{4}} b^{\frac{1}{2}} + a^{\frac{1}{2}} b^{\frac{1}{4}} - a^{\frac{3}{4}}}\right]^{-1} + \sqrt{2}^{\log_4 a}$
$= - \left[\dfrac{\sqrt[4]{a} - \sqrt[4]{b}}{\sqrt{a} + \sqrt{b}} - \dfrac{2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{b^{\frac{1}{4}} \left(b^{\frac{1}{2}} + a^{\frac{1}{2}}\right) - a^{\frac{1}{4}} \left(b^{\frac{1}{2}} + a^{\frac{1}{2}}\right)}\right]^{-1} + \sqrt{2}^{\frac{\log_{\sqrt{2}} a}{\log_{\sqrt{2}} 4}}$
$= - \left[\dfrac{a^{\frac{1}{4}} - b^{\frac{1}{4}}}{a^{\frac{1}{2}} + b^{\frac{1}{2}}} - \dfrac{2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{\left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right) \left(b^{\frac{1}{2}} + a^{\frac{1}{2}}\right)}\right]^{-1} + \sqrt{2}^{\left(\frac{\log_{\sqrt{2}} a}{\log_{\sqrt{2}} \sqrt{2}^4}\right)}$
$= - \left[\dfrac{- \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)^2 - 2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)}\right]^{-1} + \sqrt{2}^{\left(\frac{\log_{\sqrt{2}}a}{4 \log_{\sqrt{2}} \sqrt{2}}\right)}$
$= - \left[\dfrac{- \left(b^{\frac{1}{2}} + a^{\frac{1}{2}} - 2a^{\frac{1}{4}} b^{\frac{1}{4}}\right) - 2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)}\right]^{-1} + \sqrt{2}^{\left(\frac{1}{4} \log_{\sqrt{2}}a\right)}$
$= - \left[\dfrac{2 a^{\frac{1}{4}} b^{\frac{1}{4}} - a^{\frac{1}{2}} - b^{\frac{1}{2}} - 2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)}\right]^{-1} + \sqrt{2}^{\left(\log_{\sqrt{2}}a^{\frac{1}{4}}\right)}$
$= - \left[\dfrac{- \left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)}\right]^{-1} + a^{\frac{1}{4}}$
$= - \left[\dfrac{-1}{b^{\frac{1}{4}} - a^{\frac{1}{4}}}\right]^{-1} + a^{\frac{1}{4}}$
$= b^{\frac{1}{4}} - a^{\frac{1}{4}} + a^{\frac{1}{4}}$
$= b^{\frac{1}{4}}$
$= \sqrt[4]{b}$