Question

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

Answer 1

Find the fourth root of the complex root

`2√(3)-2i`

Let

`\begin{gathered} 2√(3)-2i=r(cos\theta+isin\theta) \\ 2√(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√(3))^2 \\ r^2=4+12 \\ r^2=16 \\ r=√(16) \\ r=4 \end{gathered}`
`2√(3)-2i=2(cos(2\pi)/(3)+isin(2\pi)/(3))`