Question

Need help This practice asks to fill in the four boxes in the picture

Answer 1

Solution

Find the fourth root of the complex root

$2\sqrt{3}-2i$

Let

$\begin{gathered} 2\sqrt{3}-2i=r(cos\theta+isin\theta) \\ 2\sqrt{3}=rcos\theta.........(1) \\ -2=rsin\theta............(11) \end{gathered}$

Squaring + adding the equations

$\begin{gathered} r^2(cos^2\theta+sin^2\theta)=-2^2+(2\sqrt{3})^2 \\ r^2=4+12 \\ r^2=16 \\ r=\sqrt{16} \\ r=4 \end{gathered}$$2\sqrt{3}-2i=2(cos\frac{2\pi}{3}+isin\frac{2\pi}{3})$