Calculate $\;$ $\left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right) \left(x^{\frac{2}{3}} - x^{\frac{1}{3}} y^{\frac{1}{3}} + y^{\frac{2}{3}}\right)$ $\;\;\;$ for $\;$ $x = 4 \dfrac{5}{7}$, $\;$ $y = 5 \dfrac{2}{7}$
Let $\;$ $x^{\frac{1}{3}} = p$ $\;\;\; \cdots \; (1a)$, $\;$ $y^{\frac{1}{3}} = q$ $\;\;\; \cdots \; (1b)$
Then,
$\begin{aligned}
\left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right) \left(x^{\frac{2}{3}} - x^{\frac{1}{3}} y^{\frac{1}{3}} + y^{\frac{2}{3}}\right) & = \left(p + q\right) \left(p^2 - pq + q^2\right) \\\\
& = p^3 + q^3 \\\\
& = \left(x^{\frac{1}{3}}\right)^3 + \left(y^{\frac{1}{3}}\right)^3 \;\; \left[\text{by equations (1a) and (1b)}\right] \\\\
& = x + y \\\\
& = \dfrac{33}{7} + \dfrac{37}{7} \;\;\; \text{when } x = 4 \dfrac{5}{7} = \dfrac{33}{7}, \; y = 5 \dfrac{2}{7} = \dfrac{37}{7} \\\\
& = \dfrac{70}{7} = 10
\end{aligned}$