Prove that: if $a_i$ are postive integers, then $\Sigma \sqrt[n]{a_i}$ is rational iff $\forall _i \sqrt[n]{a_i} \in \mathbb{Q}$.

Question

If for $1\leq i \leq k$: $a_i$ are postive integers and $k$ is a arbitrary integer , then $\Sigma \sqrt[n]{a_i}$ is rational iff $\forall _i \sqrt[n]{a_i} \in \mathbb{Q}$.

I tried a couple of times, but I still cannot solve it. Thanks in advance.