Simplify: $\;$ $\left[\dfrac{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 + \left(2x + a\right)}{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 - \left(x + 2a\right)}\right]^3 + \left[\sqrt{\left|a^3 - 3 a^2 x + 3 a x^2 - x^3\right|^{\frac{2}{3}}} : a\right]$ $\;\;$ for $x > a$
$\left[\dfrac{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 + \left(2x + a\right)}{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 - \left(x + 2a\right)}\right]^3 + \left[\sqrt{\left|a^3 - 3 a^2 x + 3 a x^2 - x^3\right|^{\frac{2}{3}}} : a\right]$ $\;\;\; \cdots \; (1)$
Consider the expression $\;\;$ $\left[\dfrac{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 + \left(2x + a\right)}{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 - \left(x + 2a\right)}\right]^3$
$= \left[\dfrac{x - 3 x^{\frac{2}{3}} a^{\frac{1}{3}} + 3 x^{\frac{1}{3}} a^{\frac{2}{3}} - a + 2x + a}{x - 3 x^{\frac{2}{3}} a^{\frac{1}{3}} + 3 x^{\frac{1}{3}} a^{\frac{2}{3}} - a - x - 2a}\right]^3$
$= \left[\dfrac{3x - 3 x^{\frac{2}{3}} a^{\frac{1}{3}} + 3 x^{\frac{1}{3}} a^{\frac{2}{3}}}{-3a - 3 x^{\frac{2}{3}} a^{\frac{1}{3}} + 3 x^{\frac{1}{3}} a^{\frac{2}{3}}}\right]^3$
$= \left[\dfrac{x^{\frac{1}{3}} \left(x^{\frac{2}{3}} - x^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right)}{-a^{\frac{1}{3}} \left(a^{\frac{2}{3}} + x^{\frac{2}{3}} - x^{\frac{1}{3}} a^{\frac{1}{3}}\right)}\right]^3$
$= \left[\left(\dfrac{-x}{a}\right)^{\frac{1}{3}}\right]^3$
$= \dfrac{-x}{a}$ $\;\;\; \cdots \; (2)$
Consider the expression $\;\;$ $\sqrt{\left|a^3 - 3 a^2 x + 3 a x^2 - x^3\right|^{\frac{2}{3}}}$
$= \sqrt{\left|\left(a - x\right)^3\right|^{\frac{2}{3}}}$
$= \sqrt{\left(x - a\right)^2}$ $\;\;$ for $\;$ $x > a$
$= x - a$ $\;\;\; \cdots \; (3)$
$\therefore \;$ In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes
$\dfrac{-x}{a} + \dfrac{x - a}{a}$
$= \dfrac{-a}{a}$
$= -1$