Simplify :

$\Big(x^{\dfrac{1}{3}} - x^{\dfrac{-1}{3}}\Big)(x^{\dfrac{2}{3}} + 1 + x^{\dfrac{-2}{3}}\Big)$

Given,

$\Big(x^{\dfrac{1}{3}} - x^{-\dfrac{1}{3}}\Big)(x^{\dfrac{2}{3}} + 1 + x^{-\dfrac{2}{3}}\Big)$

Simplifying the expression:

$\Rightarrow \Big(x^{\dfrac{1}{3}} - x^{-\dfrac{1}{3}}\Big)(x^{\dfrac{2}{3}} + 1 + x^{-\dfrac{2}{3}}\Big) \\[1em] \Rightarrow \Big(x^{\dfrac{1}{3}} - x^{-\dfrac{1}{3}}\Big) \Big[(x^{\dfrac{1}{3}})^2 + x^{\dfrac{1}{3}} ⋅ x^{-\dfrac{1}{3}} + (x^{-\dfrac{1}{3}})^2\Big]$

Simplifying the expression using the identity,

(a − b)(a² + ab + b²) = a³ − b³

$\Rightarrow \Big([x^{\dfrac{1}{3}}]^3 - [x^{-\dfrac{1}{3}}]^3\Big) \\[1em] \Rightarrow \Big(x^{\dfrac{1}{3} \times 3} - x^{-\dfrac{1}{3} \times 3}\Big) \\[1em] \Rightarrow (x - x^{-1}) \\[1em] \Rightarrow x - \dfrac{1}{x}$

Hence, $\Big(x^{1/3} - x^{-1/3}\Big)(x^{2/3} + 1 + x^{-2/3}\Big) = x - \dfrac{1}{x}$.