How do I break $q=\sqrt{a+ib}$ into its real and imaginary parts?

Question

I am unsure on how to do so with I under the square root. I can do so by just simply squaring both sides but I am trying to do so without using this method (need q for part of an equation).

Answer 1

Hint: Converting to polar coordinates may help:

$$q = \sqrt{a+ib} = \sqrt{re^{i\theta}} = \sqrt{r}e^{i\theta/2}$$

Use the conversions $r=\sqrt{a^2+b^2}$ and $\theta = \tan^{-1}(b/a)$. (For the angle, you'll need to take the quadrant of the point into account.)