Answer
(a) The three cube roots are:
$2~(cos~90^{\circ}+i~sin~90^{\circ})$
$2~(cos~210^{\circ}+i~sin~210^{\circ})$
$2~(cos~330^{\circ}+i~sin~330^{\circ})$
(b) We can see the three vectors in the complex plane:
Work Step by Step
(a)
$z = -8i$
$z = 8~(0-i)$
$z = 8~(cos~270^{\circ}+i~sin~270^{\circ})$
$r = 8$ and $\theta = 270^{\circ}$
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:
$z^{1/3} = 8^{1/3}~[cos(\frac{270^{\circ}}{3}+\frac{(360^{\circ})(0)}{3})+i~sin(\frac{270^{\circ}}{3}+\frac{(360^{\circ})(0)}{3})]$
$z^{1/3} = 2~(cos~90^{\circ}+i~sin~90^{\circ})$
When k = 1:
$z^{1/3} = 8^{1/3}~[cos(\frac{270^{\circ}}{3}+\frac{(360^{\circ})(1)}{3})+i~sin(\frac{270^{\circ}}{3}+\frac{(360^{\circ})(1)}{3})]$
$z^{1/3} = 2~(cos~210^{\circ}+i~sin~210^{\circ})$
When k = 2:
$z^{1/3} = 8^{1/3}~[cos(\frac{270^{\circ}}{3}+\frac{(360^{\circ})(2)}{3})+i~sin(\frac{270^{\circ}}{3}+\frac{(360^{\circ})(2)}{3})]$
$z^{1/3} = 2~(cos~330^{\circ}+i~sin~330^{\circ})$
(b) We can see the three vectors in the complex plane: