Question

How do I find the complex fifth roots of 5-5√3i?

Answer 1

\[z^5=5-5\sqrt3 i\]
First, convert to polar:
\[5-5\sqrt3i=10\left(\frac{1}{2}-\frac{\sqrt3}{2}i\right)\]
This gives the trig ratios,
\[\begin{cases}
\cos\theta=\dfrac{1}{2}\\\\\\
\sin\theta=-\dfrac{\sqrt3}{2}
\end{cases}~~\Rightarrow~~\theta=\frac{5\pi}{3}\]
So, you have
\[\large z^5=10\left(\cos\frac{5\pi}{3}+i\sin\frac{5\pi}{3}\right)=10e^{i5\pi/3}\]
In general, for the equation \(\large z^n=re^{i\theta}\), you have roots:
\[\large r^{1/n}\left[\cos\frac{\theta+2k\pi}{n}+i\sin\frac{\theta+2k\pi}{n}\right]\]
for \(k=0,1,2,\cdots,n-1\).

For your question, \(n=5\) and \(\theta=\dfrac{5\pi}{3}\), so the roots are
\[\large 10^{1/5}\left[\cos\frac{\frac{5\pi}{3}}{5}+i\sin\frac{\frac{5\pi}{3}}{5}\right]\]
\[\large 10^{1/5}\left[\cos\frac{\frac{5\pi}{3}+2\pi}{5}+i\sin\frac{\frac{5\pi}{3}+2\pi}{5}\right]\]
\[\large 10^{1/5}\left[\cos\frac{\frac{5\pi}{3}+4\pi}{5}+i\sin\frac{\frac{5\pi}{3}+4\pi}{5}\right]\]
\[\large 10^{1/5}\left[\cos\frac{\frac{5\pi}{3}+6\pi}{5}+i\sin\frac{\frac{5\pi}{3}+6\pi}{5}\right]\]
\[\large 10^{1/5}\left[\cos\frac{\frac{5\pi}{3}+8\pi}{5}+i\sin\frac{\frac{5\pi}{3}+8\pi}{5}\right]\]