TITLE: Which step in this process allows me to erroneously conclude that $i = 1$
QUESTION [12 upvotes]: I was playing around with imaginary numbers and exponents and came up with this:
$$ i = \sqrt{-1} $$
$$ \sqrt{-1} = (-1)^{1/2} $$
  $$ (-1)^{1/2} = (-1)^{2/4} $$
  $$ (-1)^{2/4} = ((-1)^{2})^{1/4} $$
 $$ ((-1)^{2})^{1/4} = (1)^{1/4} $$
 $$ (1)^{1/4} = 1 $$
Since we know that $i \ne 1$, one of these steps must be incorrect, but I can't figure out which one it could be. Where have I gone wrong?

REPLY [3 votes]: The general rule $\left(x^a\right)^b = x^{a\cdot b}$ stands only for $\left(a,b\right) \in \mathbb{Z}^2$.
Also the notation $\sqrt{ \times }$ is abusive and should not be used unless speaking of positive real numbers, because each complex number has two square roots (0 has only one) but for positive reals we can arbitrarily decide that this notation maps to the real positive one.