Question

I don't know how to go about solving this, can you help?

Answer 1

Given the complex number : -64 i

To find the cube roots, we will use the formula:

`\sqrt[3]{r}(\cos (\theta+2\pi k)/(3)+i\sin (\theta+2\pi k)/(3))`

k = 0 , 1 , 2

So,

`r=-64i=64\angle270`

So,

`\sqrt[3]{64}=4`

and the angles will be :

`\begin{gathered} (\theta+2\pi k)/(3)=(\theta+360k)/(3) \\ k=0,(\theta+360\cdot0)/(3)=(270)/(3)=90 \\ \\ k=1,(\theta+360\cdot1)/(3)=(270+360)/(3)=210 \\ \\ k=2,(\theta+360\cdot2)/(3)=(270+720)/(3)=330 \end{gathered}`

so, the cube roots of the -64i are:

`\begin{gathered} 4(\cos 90+i\sin 90) \\ 4(\cos 210+i\sin 210) \\ 4(\cos 330+i\sin 330) \end{gathered}`

So, the answer is option C