Simplify the expression

$$
\frac{x \sqrt[3]{\frac{1}{y}}-y \sqrt[3]{\frac{1}{x}}}{\sqrt[3]{\frac{x}{y}}-\sqrt[3]{\frac{y}{x}}}
$$
Our expression can also be written as:

$$
\left(\frac{x}{\sqrt[3]{y}}-\frac{y}{\sqrt[3]{x}}\right):\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}-\frac{\sqrt[3]{y}}{\sqrt[3]{x}}\right)
$$

For simplicity, let $\sqrt[3]{x}=u$ and $\sqrt[3]{y}=v$, then $x=u^{3}, y=v^{3}$, and thus

$$
\left(\frac{u^{3}}{v}-\frac{v^{3}}{u}\right):\left(\frac{u}{v}-\frac{v}{u}\right)=\frac{u^{4}-v^{4}}{u v} \div \frac{u^{2}-v^{2}}{u v}=u^{2}+v^{2}=\sqrt[3]{x^{2}}+\sqrt[3]{y^{2}}
$$

Pataki György (Bp. VII., Madách g. IV. o. t.)

Division according to the rule of polynomial division also leads to the goal.