Finding one of the roots of an equation

Question

I am trying to one of the roots of the following equation $$z^5 = -16 + (16\sqrt 3)i$$ which is $$z = 2e^{\frac{(6k+2)\pi}{15}i}$$ However, I have trouble getting that root. Here is what I have done: $$|z^5| = \sqrt {(-16)^2 + (16\sqrt3)^2}$$ $$|z| = \sqrt[5] {32} = 2$$ Since the coordinate lies in the 2nd quadrant, $$arg(z) = \pi - tan^{-1} (\frac{16\sqrt3}{16}) = \frac{2\pi}{3}$$ Therefore, $$\begin{align} z & = re^{i\theta}\\ & = 2e^{(\frac{2k\pi}{5}+\frac{2\pi}{3})i}\\ & = 2e^{\frac{(6k+10)\pi}{15}i}\\ \end{align}$$ Please explain to me where have I gone wrong.

Answer 1

In the two last lines you forgot divide $\dfrac{2\pi}{3}$ by $5$. You wrote \begin{align} z & = re^{i\theta}\\ & = 2e^{(\frac{2k\pi}{5}+\color{red}{\frac{2\pi}{3}})i}\\ & = 2e^{\frac{(6k+10)\pi}{15}i}\\ \end{align}

When you must do \begin{align} z & = re^{i\theta}\\ & = 2e^{(\frac{2k\pi}{5}+\color{red}{\frac{1}{5}\cdot\frac{2\pi}{3}})i}\\ & = \boxed{\color{blue}{2e^{\frac{(6k+2)\pi}{15}i}}}\\ \end{align}