A solution to equation $x^{\frac{3}{2}}=(-e^{\frac{2\pi i}{3}})^{\frac{1}{2}}$ in complex numbers

Question

Let $x$ be in $\mathbb{C}$. What is the solution of the equation

$$x^{\frac{3}{2}}=(-e^{\frac{2\pi i}{3}})^{\frac{1}{2}}$$

Answer 1

Let $x=e^{i\theta}$. It then follows that for $n,k\in\mathbb Z$,

$$x^{3/2}=e^{3i(\theta+2\pi k)/2}=(-e^{2\pi i/3})^{1/2}=(e^{i(5\pi/3+2\pi n)})^{1/2}=e^{i(5\pi/6+\pi n)}$$

Thus,

$$9(\theta+2\pi k)=5\pi+6\pi n$$

Or,

$$\theta=\pi\left(\frac{5+6n}9-2k\right)$$

which simplifies down to

$$\theta=\pi\left(\frac{5+6n}9\right)$$

and $x=e^{i\theta}$.