How to prove this set is a Borel set?

Question

I need to prove that the set $$ \left\{x \in \mathbb{R} \mid \text{for all} \ n \in \mathbb{Z} \setminus \left\{0\right\} \ \text{is} \ x^n \in \mathbb{R} \setminus \mathbb{Q} \right\} $$ is a Borel set. I rewrote this set as $$ \bigcap_{n \in \mathbb{Z} \setminus \left\{0\right\}} \left\{x \in \mathbb{R} \mid x^n \in \mathbb{R} \setminus \mathbb{Q} \right\}. $$ I let $n \in \mathbb{Z} \setminus \left\{0\right\}$. I want to write the set $ \left\{x \in \mathbb{R} \mid x^n \in \mathbb{R} \setminus \mathbb{Q} \right\} $ in terms of intervals if possible, but I don't see how. Help is appreciated.

Answer 1

Just write $$ \{x : x^n \in \Bbb {R} \setminus \Bbb {Q} \} = \Bbb {R} \setminus \{x : x^n \in \Bbb {Q} \} = \Bbb {R } \setminus \bigcup_m Q_{n,m} $$ where $(y_m)_m $ is an enumeration of $\Bbb {Q} $ and where each set $Q_{n,m} := \{x : x^n = y_m\} $ is finite (why?) and thus Borel measurable (why?).