what are the rules regarding $x,a,b$ for this expression to be true: $(x^a)^b = (x^b)^a$ (i am considering only for when $a,b$ are real)?

Question

in particular i am asking for the case when one of the powers $a$ or $b$ is a fraction. in such a case, i believe the maths expression then may be ambiguous, as when you do to the power of a fraction, it's not clear if the required answer is the principal root, or if it is all roots.

eg i know $(4^{\frac12})^2 = 4$ but is $(4^2)^{\frac12} = 4$? in this case, i think it's ambiguous, some will say also $-4$ is a solution , so we should add some notation that we want the "principal root" , and then it is true

Answer 1

If $n\in\Bbb N$, and $x\geqslant0$, $x^{1/n}$ is the only number $y\geqslant0$ such that $y^n=x$ . In particular, $(4^2)^{1/2}=16^{1/2}=4$.

And, if $x>0$, and $a,b\in\Bbb R$, $\left(x^a\right)^b=\left(x^b\right)^a$. Actually, both numbers are equal to $x^{ab}$. To see why, I will assume that you define $x^a$ as $e^{a\log x}$. Then\begin{align}\left( x^a\right)^b&=e^{b\log\left(x^a\right)}\\&=e^{ab\log x}\\&=x^{ab}.\end{align}