Generally, over any field (such as $\Bbb Z_p$) if a polynomial $f(x)$ has nonzero constant term $f(0)\neq 0$ then $x=0$ is not a root, so every root is nonzero, so invertible. Thus when doing algebra with roots of $f$ we can invert (and cancel) them. OP is the special binomial case $\,f(x) = x^n - a,\ a\!\neq\! 0$.
Generally the above shows that an $\rm\color{#c00}{algebraic}$ root of $f$ divides its simpler $\rm\color{#0a0}{integer}$ constant term, and this simpler multiple allows us to reduce $\rm\color{#c00}{algebraic}$ number inversion to $\rm\color{#0a0}{integer}$ inversion, e.g. using norms or rationalizing denominators.