In math textbooks, ${(x^a)}^b = x^{ab}$.
However take x=-1, a=1/2, b=2. Then the left side is undefined (assuming we're limiting to real numbers).
We know that the order of operations is power precedes multiplication. So for $x^{ab}$, why are we used to first perform the multiplication? Any written reference would be helpful.
Step back from the textbook.
We try to define powers, first for integer exponents >= 0, then for negative exponents, then for rational exponents, then for irrational exponents.
We find that some cannot be easily defined, like $0^0$, $0^{-1}$, $(-1)^{\pi}$ etc., so we carefully write down what powers exactly are defined and what powers are not.
And then we find rules how to perform some operations. And again, we have to examine exactly what rules can be applied under which circumstances.
The rule you found cannot be applied under all possible circumstances. You’d need to check it carefully, it may be that if the left and right side are both defined then they are equal. That would be worth checking.