What is fundamentally wrong in writing $(-a)^{1/2}$ as $((-a)^{2})^{1/4}$ when $a$ is positive and thus equating it to $a^{1/2}$?
Edit: I'm basically asking if there is anything wrong with this operation like multiplying $1$ and $2$ with $0$ and equating it to "prove" that $1=2$.
When defining $a^{m/n}$ for $a > 0$, $m, n \in \Bbb N_{\ge 1},$ it is an exercise to verify that the following things hold: $$a^{m/n} = (a^m)^{1/n} = (a^{1/n})^m = a^{m'/n'}$$ for any $m', n'\in \Bbb N_{\ge 1}$ such that $m'n = mn'$.
The point I want to make is that in a manipulation like $$a^{1/2} = a^{2/4} = (a^{2})^{1/4},$$ each equality has to be justified and does not just follow on its own.
It is to be checked that the usual manipulation of rational numbers goes through and that everything works the way you would desire to.
When you have $a < 0$, the things just don't hold anymore even if you involve complex numbers.