So assume we are given some $a\in\mathbb{Z}_p^\times$ and we want to figure out if $X^p-a$ has a root in $\mathbb{Q}_p$. We know that such a root must be unique, because given two such roots $\alpha,\beta$, the quotient $\alpha/\beta$ would need to be a non-trivial $p^\textrm{th}$ root of unity and $\mathbb{Q}_p$ does not contain any.
Now we can't apply Hensel, which is the canonical thing to do when looking for roots in $\mathbb{Q}_p$. What other approaches are available?