Let $a,b,n\in\mathbb N^*$, $a<b$,
Does a rational $p/q$ such that $p,q\in\mathbb N$ and $a<p^n/q^n<b$ exist ?
If that's the case, Is there a way to find one ?
I know that $\mathbb {\bar Q} = \mathbb R$ ($\mathbb Q$ is dense in $\mathbb R$).
If we let $\mathbb Q_n = \{p^n/q^n|p\in\mathbb Z\,,q\in\mathbb N^*\}$, I would say that $\forall n\in\mathbb N, \mathbb {\bar Q_{2n+1}} = \mathbb R$ and $\mathbb {\bar Q_{2n+2}} = \mathbb R^+$.
But I am not sure how to prove it. What would you do ?
Thank you !
Edit : When I say "to find one", it is by using a formula which should use $a$ and $b$
The map $x \to x^n$ is a homeomorphism of the positive real line. Since the rationals are dense, so is the set of their $n$th powers, so there will be many such in any interval.
How to "find one" depends on the tools you allow.