Why is $\sqrt{-i} = -(-1)^{\frac{3}{4}}$ I was doing some research to look for fun on the properties of $i$, the square root of $-1$, and it got me thinking about what the square root of $-i$ was, or the square root of $-1$.  I found on wolfram alpha that it is $\sqrt{-i} = -(-1)^{\frac{3}{4}}$
, but I can't find an explanation anywhere.  Can someone help me?

If two complex numbers are equal, if a+ bi= c+ di then the real and imaginary parts are equal, a= c and b= d.  That is part of the definition of the "a+ bi" notation.  Her you have $A^2- B^2+ 2ABi= 0+ i$.  You must have the real parts $A^2- B^2= 0$ and imaginary part 2AB= 1.
Personally I would have used the "polar form".  $-i= e^{3\pi/2}$.  The square root of that is $e^{3\pi/4}$.  On the other hand, $-1= e^{i\pi}$ so that $(-1)^{3/4}= e^{3i\pi/4}$.
