Proving that if $m,n,p,q\in\mathbb{Z^+}, \sqrt[p]{m}\in\mathbb{R}\setminus\mathbb{Q}$ then $\sqrt[p]{m}+\sqrt[q]{n}\in\mathbb{R}\setminus\mathbb{Q}$

Question

If $\sqrt{m}+\sqrt[q]{n}=r$ rational, the rationality of $\sqrt{m}$ is derived expanding $(r-\sqrt{m})^q$ using the binomial theorem: after rearrangement, isolating the terms containing odd powers of $\sqrt{m}$ and factoring it out, we're done.

A similar trick does not work for the more general case stated in the title. Is it provable by elementary means?