What is wrong with $\sqrt{-1} = (-1)^{1/2} = (-1)^{2 \times {1/4}} = (-1^2)^{1/4} = 1^{1/4} = 1$? Why $\sqrt{-1} = (-1)^{1/2} = (-1)^{2 \times {1/4}} = (-1^2)^{1/4} = 1^{1/4} = 1$ is not true?
 A: 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 $$x^2+1=0.$$
A: If we map the real numbers $\mathbb{R}$ to itself using powers we end up with an image which is in most cases just a subset of $\mathbb{R}$. An example would be taking the square of a real number. The square of a negative real number yields a positive number, so the set of numbers we end up with after squaring is smaller than what we started with. In algebra we often make use of the inverse map, known as the square root. When dealing with positive numbers taking the square root is the proper inverse of taking the square. When we consider all real numbers one has to take into account that the original number migt have been negative, commonly denoted as $\sqrt{4}=\pm2$.
This $\pm$ symbol is very elegantly extended into a "phase" or "angle" when we generalize to the complex numbers $\mathbb{C}$. Every complex number's phase can have an arbitrary integer amount of full rotations $n2\pi$ added to it. When we take fractional powers, like the cube root, this becomes a $n\frac{2}{3}\pi$.
This gives, for example,
$\sqrt[3]{27}=
\begin{cases}
3\\
3\cdot e^{i\frac{2}{3}\pi}\\
3\cdot e^{i\frac{4}{3}\pi}\quad,
\end{cases}$
which are now three solutions instead of just two for the square root case.
In short, the manipulations you did, changing the square root into a hypercube root, only work for positive numbers, where all roots are proper inverses of powers. When mathematicians decide to use the positive answer for a square root when there also could be nagative numbers involved, they often speak of a "principal value". It should always be clear when or if this is justified, and it is the origin of many "fake proofs".
