I was reading some question on this site and stream of thought led me to the creation of another question that could be trivial for someone but I am unable even to start solving it. I wanna share this question with you in hope that someone will be able to answer it.
Let $S_2$ be the set of square roots of all positive integers that are not of the form $a^2$ , $S_3$ the set of cube roots of all positive integers that are not of the form $a^3$, $...$ , $S_n$ the set of $n$-th roots of all positive integers that are not of the form $a^n$...
Now let us define set $S$ as $S=\bigcup_{n=2}^\infty S_n$.
The question is:
Does there exist a natural number $k$ and rational numbers $r_1,r_2,...,r_k$ (all different from zero) such that for some $k$ different elements of the set $S$, denote them as $s_1,s_2,...,s_k$, which are not in the same set of roots, for instance if $s_i$ is in the $S_4$ then $s_j$ is in the $S=(\bigcup_{n=2}^\infty S_n) \setminus S_4$, we have $\sum_{i=1}^{k}r_is_i=0$?
(All roots in this question are unique real roots.)
We can devise examples, for arbitrarily large $k$, where the roots $s_i$ are drawn from different constituent classes, so that $\sum_{i=1}^k r_i s_i = 0$ with all coefficients $r_i$ nonzero integers.
A simple example with just two terms is $2\sqrt[4]{9} - \sqrt{12} = 0$.