Where's the mistake in this calculation?

Question

Obviously something is wrong with this, but where is the error and why is it one?

$$ \begin{align} \sqrt{-1} &= (-1)^{1/2} \\ &= (-1)^{2/4} \\ &= \sqrt[4]{(-1)^2} \\ &= \sqrt[4]{1} \\ &= 1. \end{align} $$

Answer 1

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.

Answer 2

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.

${(a^2)}^{1/2} = \pm a$

$(r^2e^{2i\theta})^{1/2} = \lvert r\rvert e^{i(\theta+n\pi)} \;\mathbf 1_{n\in\Bbb Z}$

Answer 3

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