¿Where does equality fail?

Question

I know this: $\sqrt{x}^2 = |x|$, but

$\sqrt{(-1)^2} = \sqrt{(-1)^2}$

$(-1)^\frac{2}{2} = \sqrt{-1 * -1}$

$(-1)^1 = \sqrt{1}$

$-1 = 1^2$, then

$-1 = 1$

What step is wrong?

Answer 1

The square root function has a branch cut discontinuity on the negative real axis. That means you can't do this: $\sqrt{(-1)(-1)}=\sqrt{-1}\,\sqrt{-1}$, because you'd be approaching that discontinuity from two different directions, and expecting them to be equal. For your derivation, $((-1)^2)^{1/2}\not=((-1)^{1/2})^2$, a step implicit in your derivation.