Simplify: $\;$ $\left[\left(\dfrac{\sqrt[3]{x^2 y^2} + x \sqrt[3]{x}}{x \sqrt[3]{y} + y \sqrt[3]{x}} - 1\right)^{-1} \left(1 + \sqrt[3]{\dfrac{x}{y}} + \sqrt[3]{\dfrac{x^2}{y^2}}\right)^{-1} + 1\right]^{\frac{1}{3}} \sqrt[3]{x - y}$
$\left[\left(\dfrac{\sqrt[3]{x^2 y^2} + x \sqrt[3]{x}}{x \sqrt[3]{y} + y \sqrt[3]{x}} - 1\right)^{-1} \left(1 + \sqrt[3]{\dfrac{x}{y}} + \sqrt[3]{\dfrac{x^2}{y^2}}\right)^{-1} + 1\right]^{\frac{1}{3}} \sqrt[3]{x - y}$ $\;\;\; \cdots \; (1)$
Consider the expression $\;\;$ $\left(\dfrac{\sqrt[3]{x^2 y^2} + x \sqrt[3]{x}}{x \sqrt[3]{y} + y \sqrt[3]{x}} - 1\right)^{-1}$
$= \left(\dfrac{x^{\frac{2}{3}} y^{\frac{2}{3}} + x^{\frac{4}{3}} - x y^{\frac{1}{3}} - y x^{\frac{1}{3}}}{x y^{\frac{1}{3}} + y x^{\frac{1}{3}}} \right)^{-1}$
$= \left[\dfrac{x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + y^{\frac{2}{3}}\right) - x^{\frac{1}{3}} y^{\frac{1}{3}} \left(x^{\frac{2}{3}} + y^{\frac{2}{3}}\right)}{x^{\frac{1}{3}} y^{\frac{1}{3}} \left(x^{\frac{2}{3}} + y^{\frac{2}{3}}\right)} \right]^{-1}$
$= \left[\dfrac{x^{\frac{1}{3}} \left(x^{\frac{1}{3}} - y^{\frac{1}{3}}\right)}{x^{\frac{1}{3}} y^{\frac{1}{3}}}\right]^{-1}$
$\dfrac{y^{\frac{1}{3}}}{x^{\frac{1}{3}} - y^{\frac{1}{3}}}$ $\;\;\; \cdots \; (2)$
Consider the expression $\;\;$ $\left(1 + \sqrt[3]{\dfrac{x}{y}} + \sqrt[3]{\dfrac{x^2}{y^2}}\right)^{-1}$
$= \left(1 + x^{\frac{1}{3}} y^{\frac{-1}{3}} + x^{\frac{2}{3}} y^{\frac{-2}{3}}\right)^{-1}$
$= \left[1 + x^{\frac{1}{3}} y^{\frac{-2}{3}} \left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right)\right]^{-1}$
$= \left[\dfrac{y^{\frac{2}{3}} + x^{\frac{1}{3}} \left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right)}{y^{\frac{2}{3}}}\right]^{-1}$
$= \dfrac{y^{\frac{2}{3}}}{y^{\frac{2}{3}} + x^{\frac{2}{3}} + x^{\frac{1}{3}} y^{\frac{1}{3}}}$ $\;\;\; \cdots \; (3)$
$\therefore \;$ In view of expressions $(2)$ and $(3)$, expression $\;$ $\left[\left(\dfrac{\sqrt[3]{x^2 y^2} + x \sqrt[3]{x}}{x \sqrt[3]{y} + y \sqrt[3]{x}} - 1\right)^{-1} \left(1 + \sqrt[3]{\dfrac{x}{y}} + \sqrt[3]{\dfrac{x^2}{y^2}}\right)^{-1} + 1\right]^{\frac{1}{3}}$ $\;$ becomes
$\left[\left(\dfrac{y^{\frac{1}{3}}}{x^{\frac{1}{3}} - y^{\frac{1}{3}}}\right) \left(\dfrac{y^{\frac{2}{3}}}{x^{\frac{2}{3}} + x^{\frac{1}{3}} y^{\frac{1}{3}} + y^{\frac{2}{3}}}\right) + 1\right]^{\frac{1}{3}}$
$= \left[\dfrac{y}{\left(x^{\frac{1}{3}}\right)^3 - \left(y^{\frac{1}{3}}\right)^3} + 1\right]^{\frac{1}{3}}$
$= \left[\dfrac{y}{x - y} + 1\right]^{\frac{1}{3}}$
$= \left[\dfrac{x}{x - y}\right]^{\frac{1}{3}}$ $\;\;\; \cdots \; (4)$
$\therefore \;$ In view of expression $(4)$, expression $(1)$ becomes
$\dfrac{\sqrt[3]{x}}{\sqrt[3]{x - y}} \times \sqrt[3]{x - y}$
$= \sqrt[3]{x}$