"Rationalizing" an equation $$x=\sqrt[3]{p}+\sqrt[3]{q}$$
I'm trying to figure out some way to "rationalize" the previous equation, meaning to rewrite it purely in terms of whole number powers of $p$, $q$, and $x$. It seems quite simple but I've been stuck trying to do it. I'd appreciate anyone's help with this.

Hint: Use the high school identity:
$$(a+b)(a^2-ab+b^2)=a^3+b^3.$$
