Is $\sqrt{-i \sqrt{5}}=-i\sqrt[4]{-5}$?

Question

I have tried doing

$\sqrt{-i \sqrt{5}}$

$=\sqrt{-\sqrt{-1} \sqrt{5}}$

$=\sqrt{-\sqrt{-5}}$

$=i\sqrt{\sqrt{-5}}$

$=i\sqrt[4]{-5}$

But Wolfram says that the answer is $-i\sqrt[4]{-5}$.

Where is my error?

Answer 1

It depends on the definition of square root over the field of complex numbers. Since equation $x^n=c$ has $n$ roots for any $c\neq 0$, we can either define n-th root as a multivalue function (and thus your answer is just the other value of this function, that was found by WA) or we can understand n-th root as principal root (restrict the answer to have argument in half-interval $(-\pi/n,\pi/n]$), in that case $\sqrt{a}\sqrt{b}\neq\sqrt{ab}$ and that's your mistake.