"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.

Another way (maybe it's the same as Bernard's, but I don't see how):
$${x\over \sqrt[3]{q}}=1+\sqrt[3]{p\over q}\\$$
$${x^3\over q}=1+{p\over q}+3\sqrt[3]{p\over q}\left(1+\sqrt[3]{p\over q}\right)=1+{p\over q}+3x\sqrt[3]{p\over q^2}$$
And you can continue from here.
