Answer
(a) The three cube roots are:
$4^{1/3}~(cos~50^{\circ}+i~sin~50^{\circ})$
$4^{1/3}~(cos~170^{\circ}+i~sin~170^{\circ})$
$4^{1/3}~(cos~290^{\circ}+i~sin~290^{\circ})$
(b) We can see the three vectors in the complex plane:
Work Step by Step
(a)
$z = -2~\sqrt{3}+2~i$
$z = 4~(cos~150^{\circ}+i~sin~150^{\circ})$
$r = 4$ and $\theta = 150^{\circ}$
We can use this equation to find the cube roots:
$z^{1/n} = r^{1/n}~[cos(\frac{\theta}{n}+\frac{360^{\circ}~k}{n})+i~sin(\frac{\theta}{n}+\frac{360^{\circ}~k}{n})]$, where $k \in \{0, 1, 2,...,n-1\}$
When k = 0:
$z^{1/3} = 4^{1/3}~[cos(\frac{150^{\circ}}{3}+\frac{(360^{\circ})(0)}{3})+i~sin(\frac{150^{\circ}}{3}+\frac{(360^{\circ})(0)}{3})]$
$z^{1/3} = 4^{1/3}~(cos~50^{\circ}+i~sin~50^{\circ})$
When k = 1:
$z^{1/3} = 4^{1/3}~[cos(\frac{150^{\circ}}{3}+\frac{(360^{\circ})(1)}{3})+i~sin(\frac{150^{\circ}}{3}+\frac{(360^{\circ})(1)}{3})]$
$z^{1/3} = 4^{1/3}~(cos~170^{\circ}+i~sin~170^{\circ})$
When k = 2:
$z^{1/3} = 4^{1/3}~[cos(\frac{150^{\circ}}{3}+\frac{(360^{\circ})(2)}{3})+i~sin(\frac{150^{\circ}}{3}+\frac{(360^{\circ})(2)}{3})]$
$z^{1/3} = 4^{1/3}~(cos~290^{\circ}+i~sin~290^{\circ})$
(b) We can see the three vectors in the complex plane:
