What is wrong with $\sqrt{-1} = (-1)^{1/2} = (-1)^{2 \times {1/4}} = (-1^2)^{1/4} = 1^{1/4} = 1$?

Question

Why $\sqrt{-1} = (-1)^{1/2} = (-1)^{2 \times {1/4}} = (-1^2)^{1/4} = 1^{1/4} = 1$ is not true?

Answer 1

You use the fact that $(x^a)^b = x^{ab}$, a fact that is only true for $x>0$. In fact, for negative values of $x$, the term $$x^{a}$$

is only really properly defined for integer values of $a$, so even the expression $\sqrt{-1}$ is not well defined. Sure, you could say that $i=\sqrt{-1}$, but you could also say that $\sqrt{-1} = -i$, since both $i$ and $-i$ solve the equation $$x^2+1=0.$$