Generalisation of a well known fact: if $a,b,\sqrt[n]a+\sqrt[n]b\in\mathbb Z$, then is it necessary that $\sqrt[n]a,\sqrt[n]b\in\mathbb Z$?

Question

I can prove that if $a$ and $b$ are integers such that $\sqrt{a} + \sqrt{b}$ is also an integer then $a$ and $b$ are perfect squares.

Is this applicable for $n$-th roots also?

I noticed that for cube roots we must use natural numbers instead of integers.