Q: Where's the mistake in this calculation? Obviously something is wrong with this, but where is the error and why is it one?
$$
\begin{align}
\sqrt{-1} &= (-1)^{1/2} \\
&= (-1)^{2/4} \\
&= \sqrt[4]{(-1)^2} \\
&= \sqrt[4]{1} \\
&= 1.
\end{align}
$$

A: Multivaluedness has been ignored at several steps, example $\sqrt{-1}=\pm i$. 
In the language of complex analysis, you have to define
an appropriate branch of the square root function.

A: When the exponent is not an integer there may be more than one valid result of any exponentiation of a complex number.   In particular there are always two square roots, and four quaternary roots.   The square root of a square of a number is not necessarily that number.
${(a^2)}^{1/2} = \pm a$
$(r^2e^{2i\theta})^{1/2} = \lvert r\rvert e^{i(\theta+n\pi)} \;\mathbf 1_{n\in\Bbb Z}$

A: Because there is more than one root of $1^{\frac{1}{4}}$. 
