Sum of roots (number theory)

Question

Let $k,m\in\mathbb{N}$. Let $a_1,a_2,\ldots, a_k>0$ and $b_1,b_2,\ldots, b_m>0$

Let $\sqrt[n]{a_1} + \sqrt[n]{a_2} + \cdots + \sqrt[n]{a_k} = \sqrt[n]{b_1} + \sqrt[n]{b_2} + \cdots + \sqrt[n]{b_m}$ for all natural $n \in \mathbb{N}$.

1. Prove that $k = m$.
2. Prove that $a_1,a_2,\ldots, a_k>0$ and $b_1,b_2,\ldots, b_k>0$
3. Prove that if each of the two sets of numbers sort of growth, then these sets will be the same.