Question 1026549
At the very least I think you have to consider separating the proof by cases.

When n is even, it is trickier.  For example, {{{(-root(4,11))^4 = (11^(1/4))^4}}} doesn't mean  {{{-root(4,11) = 11^(1/4)}}}.

For n odd, it is automatic.

With that said, we have to impose the restriction that the root itself be a real number for the argument to hold. If complex roots are allowed then the notion of "roots" itself is not a single, particular number, but a class of (complex) numbers, and as such cannot be equal to just one number.