Solution to $(-1)^{2x}=1$

Question

So I encounter this problem when studying quantum mechanics:

(-1)^2x=1, find the set of values of x

Obviously, x ∈ {Z}, as 2x is even for all integral x.

Just to be rigorous, I wrote down the steps and come out with something like this:

(-1)^2x=1

[(-1)²]^x=1

1^x=1, so x can be any real numbers.

Where is my mistake?

Answer 1

Your mistake is that non-integer powers of negative numbers are not uniquely defined,

and the "general rule" $(a^m)^n=a^{m\times n}$ does not always work when $m$ and $n$ are not integers.