Simplify: $\;$ $\dfrac{\left(\sqrt[8]{a} + \sqrt[8]{b}\right)^2 + \left(\sqrt[8]{a} - \sqrt[8]{b}\right)^2}{a - \sqrt{ab}} : \dfrac{\left(\sqrt[4]{a} + \sqrt[8]{ab} + \sqrt[4]{b}\right) \left(\sqrt[4]{a} - \sqrt[8]{ab} + \sqrt[4]{b}\right)}{\sqrt[4]{a^3 b} - b}$
Given expression:
$\dfrac{\left(\sqrt[8]{a} + \sqrt[8]{b}\right)^2 + \left(\sqrt[8]{a} - \sqrt[8]{b}\right)^2}{a - \sqrt{ab}} : \dfrac{\left(\sqrt[4]{a} + \sqrt[8]{ab} + \sqrt[4]{b}\right) \left(\sqrt[4]{a} - \sqrt[8]{ab} + \sqrt[4]{b}\right)}{\sqrt[4]{a^3 b} - b}$ $\;\;\; \cdots \; (1)$
Consider the expression $\;\;\;$ $\dfrac{\left(\sqrt[8]{a} + \sqrt[8]{b}\right)^2 + \left(\sqrt[8]{a} - \sqrt[8]{b}\right)^2}{a - \sqrt{ab}}$
$= \dfrac{a^{\frac{1}{4}} + b^{\frac{1}{4}} + 2 a^{\frac{1}{8}} b^{\frac{1}{8}} + a^{\frac{1}{4}} + b^{\frac{1}{4}} - 2 a^{\frac{1}{8}} b^{\frac{1}{8}}}{\sqrt{a} \left(\sqrt{a} - \sqrt{b}\right)}$
$= \dfrac{2 \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{\sqrt{a} \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)}$
$= \dfrac{2 \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{\sqrt{a} \left[\left(a^{\frac{1}{4}}\right)^2 - \left(b^{\frac{1}{4}}\right)^2\right]}$
$= \dfrac{2 \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{\sqrt{a} \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right) \left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right)}$
$= \dfrac{2}{\sqrt{a} \left(\sqrt[4]{a} - \sqrt[4]{b}\right)}$ $\;\;\; \cdots \; (2)$
Consider the expression $\;\;\;$ $\dfrac{\left(\sqrt[4]{a} + \sqrt[8]{ab} + \sqrt[4]{b}\right) \left(\sqrt[4]{a} - \sqrt[8]{ab} + \sqrt[4]{b}\right)}{\sqrt[4]{a^3 b} - b}$
$= \dfrac{\left(\sqrt[4]{a} + \sqrt[4]{b}\right)^2 - \left(\sqrt[8]{ab}\right)^2}{a^{\frac{3}{4}} b^{\frac{1}{4}} - b}$
$= \dfrac{a^{\frac{1}{2}} + b^{\frac{1}{2}} + 2 a^{\frac{1}{4}} b^{\frac{1}{4}} - a^{\frac{1}{4}} b^{\frac{1}{4}}}{b^{\frac{1}{4}} \left(a^{\frac{3}{4}} - b^{\frac{3}{4}}\right)}$
$= \dfrac{a^{\frac{1}{2}} + b^{\frac{1}{2}} + a^{\frac{1}{4}} b^{\frac{1}{4}}}{b^{\frac{1}{4}} \left[\left(a^{\frac{1}{4}}\right)^3 - \left(b^{\frac{1}{4}}\right)^3\right]}$
$= \dfrac{a^{\frac{1}{2}} + b^{\frac{1}{2}} + a^{\frac{1}{4}} b^{\frac{1}{4}}}{b^{\frac{1}{4}} \left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right) \left(a^{\frac{1}{2}} + a^{\frac{1}{4}} b^{\frac{1}{4}} + b^{\frac{1}{2}}\right)}$
$= \dfrac{1}{\sqrt[4]{b} \left(\sqrt[4]{a} - \sqrt[4]{b}\right)}$ $\;\;\; \cdots \; (3)$
In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes
$\dfrac{2}{\sqrt{a} \left(\sqrt[4]{a} - \sqrt[4]{b}\right)} : \dfrac{1}{\sqrt[4]{b} \left(\sqrt[4]{a} - \sqrt[4]{b}\right)}$
$= \dfrac{2 \sqrt[4]{b}}{\sqrt{a}}$
$= 2 \sqrt[4]{\dfrac{b}{a^2}}$