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?
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?
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.