Question

Let α be a complex number that satisfies both the equations from the previous question. Which of the following are possible arguments of the cube roots of α?

a) π/6

b) 5π/6

c) π/12

d) π/2

e) 5π/12

f) 0

g) π

h) -π/2

Answer 1

$\alpha=r(cos\varphi+isin\varphi)$

$\sqrt[3]{\alpha}=\sqrt[3]{r}(cos(\frac{\varphi+2k\pi}{3})+isin(\frac{\varphi+2k\pi}{3}))$ , $0\leq k\leq 2$

For example:

if $\varphi=\pi/2$ then arguments $\theta$ of the cube roots of $\alpha$ :

$k=0$ : $\theta=\pi/6$

$k=1$ : $\theta=5\pi/6$

$k=2$ : $\theta=3\pi/2$