How can we prove that if $a_1\sqrt{n_1}+a_2\sqrt{n_2}+\dots+a_k\sqrt{n_k}\in\mathbb{Q}$ with $a_1,a_2\dots, a_k\in\mathbb{Q}^{*}_{+}$, then the natural numbers $n_1,n_2,\dots, n_k$ are perfect squares?
I have proved it for $k=1,2,3$ but for $k\geq 4$ I don't find anything. I couldn't show even that $\sqrt{2}+\sqrt{3}+\sqrt{5}+\sqrt{7}\notin\mathbb{Q}$.
