What is the fundamental error in my reasoning?

Question

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 $1=2$.

How can you write $-a$ as $(-a)^2$? This question is not clear. — Trebor, Jun 25 '20 at 08:29
$\sqrt{-a}$ is not defined for $a>0$, at least not within the real numbers. — Peter, Jun 25 '20 at 08:34
Basically the problem is that many of the usual rules for exponentials and logarithms that are true over positive real numbers are no longer true over the complex numbers — Louis Hainaut, Jun 25 '20 at 08:40

score 4 · Accepted Answer · answered Jun 25 '20 at 09:14

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.

What is the fundamental error in my reasoning?

1 Answers1

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