Answer
The complex fifth roots of the given complex numbers in rectangular form are
${{z}_{0}}=1+i\sqrt{3},\text{ }{{z}_{1}}=-1.3+1.5i,\ {{z}_{2}}=-1.8-0.8i,\text{ }{{z}_{3}}=0.2-2.0i,\text{ and }{{z}_{4}}=2.0-0.4i$.
Work Step by Step
Consider the given complex number
$32\left( \cos \frac{5\pi }{3}+i\sin \frac{5\pi }{3} \right)$
For any complex number of the type
$z=r\cos \theta +i\sin \theta $
If n is a positive integer, then ${{z}^{n}}$ is
${{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +2\pi k}{n} \right)+i\sin \left( \frac{\theta +2\pi k}{n} \right) \right]$ ...... (1)
Now, for the given complex number, use (1) to get
$\begin{align}
& {{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +2\pi k}{n} \right)+i\sin \left( \frac{\theta +2\pi k}{n} \right) \right] \\
& {{z}_{0}}=\sqrt[5]{32}\left[ \cos \frac{\frac{5\pi }{3}+2\pi \times 0}{5}+i\sin \frac{\frac{5\pi }{3}+2\pi \times 0}{5} \right] \\
& {{z}_{0}}=2\left[ \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right] \\
& {{z}_{0}}=1+\sqrt{3}i \\
\end{align}$
Now, for the other roots, substitute the value of $k=1,2,3,4$.
That is,
$\begin{align}
& {{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +2\pi k}{n} \right)+i\sin \left( \frac{\theta +2\pi k}{n} \right) \right] \\
& {{z}_{1}}=\sqrt[5]{32}\left[ \cos \frac{\frac{5\pi }{3}+2\pi \times 1}{5}+i\sin \frac{\frac{5\pi }{3}+2\pi \times 0}{5} \right] \\
& {{z}_{1}}=2\left[ \cos \frac{11\pi }{15}+i\sin \frac{11\pi }{15} \right] \\
& {{z}_{1}}=-1.3+1.5i \\
\end{align}$
Similarly,
$\begin{align}
& {{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +2\pi k}{n} \right)+i\sin \left( \frac{\theta +2\pi k}{n} \right) \right] \\
& {{z}_{2}}=\sqrt[5]{32}\left[ \cos \frac{\frac{5\pi }{3}+2\pi \times 2}{5}+i\sin \frac{\frac{5\pi }{3}+2\pi \times 2}{5} \right] \\
& {{z}_{2}}=2\left[ \cos \frac{17\pi }{15}+i\sin \frac{17\pi }{15} \right] \\
& {{z}_{2}}=-1.8-0.8i \\
\end{align}$
Also,
$\begin{align}
& {{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +2\pi k}{n} \right)+i\sin \left( \frac{\theta +2\pi k}{n} \right) \right] \\
& {{z}_{3}}=\sqrt[5]{32}\left[ \cos \frac{\frac{5\pi }{3}+2\pi \times 3}{5}+i\sin \frac{\frac{5\pi }{3}+2\pi \times 3}{5} \right] \\
& {{z}_{3}}=2\left[ \cos \frac{23\pi }{15}+i\sin \frac{23\pi }{15} \right] \\
& {{z}_{3}}=0.2-2.0i \\
\end{align}$
Also,
$\begin{align}
& {{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +2\pi k}{n} \right)+i\sin \left( \frac{\theta +2\pi k}{n} \right) \right] \\
& {{z}_{4}}=\sqrt[5]{32}\left[ \cos \frac{\frac{5\pi }{3}+2\pi \times 4}{5}+i\sin \frac{\frac{5\pi }{3}+2\pi \times 4}{5} \right] \\
& {{z}_{4}}=2\left[ \cos \frac{29\pi }{15}+i\sin \frac{29\pi }{15} \right] \\
& {{z}_{4}}=2.0-0.4i \\
\end{align}$
The fifth roots of the complex numbers in the polar form are
${{z}_{0}}=1+i\sqrt{3},\text{ }{{z}_{1}}=-1.3+1.5i,\ {{z}_{2}}=-1.8-0.8i,\text{ }{{z}_{3}}=0.2-2.0i,\text{ and }{{z}_{4}}=2.0-0.4i$.
