In "On definable susbsets of p-adic fields", Macintyre 1976, he states the following fact that, let $x\in Q_p^*$ such that $v(x) = 0$ (where $v$ is the p-adic valuation) and $k\in N$ then there exists $n\in Z$ such that $x/n\in (Q_p)^k$ (ie the $k$-th powers).

This property is evident if $k$ is prime to $p$ by Hensel's lemma (the form I use is that let $f\in Z_p[X]$ and $a\in Z_p$ such that res$(f(a)) = 0$ and res$(f'(a))\neq 0$ then there exists $b\in Z_p$ such that $f(b) = 0$ and res$(a) =$ res$(b)$). But if not, then I don't know how to prove it.

If anyone has any ideas on the subject I'd be very grateful.