Answer
Solutions of the provided equation in polar form are:
$\begin{align}
& 2\left[ \cos 18{}^\circ +i\sin 18{}^\circ \right],2\left[ \cos 90{}^\circ +i\sin 90{}^\circ \right],2\left[ \cos 162{}^\circ +i\sin 162{}^\circ \right], \\
& 2\left[ \cos 234{}^\circ +i\sin 234{}^\circ \right]\text{ and }2\left[ \cos 306{}^\circ +i\sin 306{}^\circ \right] \\
\end{align}$
Solutions of the provided equation in rectangular form are:
$1.9021+0.6180i,0+2i,-1.9021+0.6180i,-1.1756-1.6180i\text{ and }1.1756-1.6180i$
Work Step by Step
We apply Demoivre’s Theorem.
Demoivre’s Theorem for finding complex roots states that every complex number has two distinct complex square roots, three distinct complex cube roots, four distinct complex fourth roots, and so on. Each root has the same modulus.
$\begin{align}
& r=z_{k}^{n} \\
& {{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +360{}^\circ k}{n} \right)+i\sin \left( \frac{\theta +360{}^\circ k}{n} \right) \right] \\
\end{align}$
Here $k$ is the number of distinct ${{n}^{th}}$ solutions and $\theta $ is in degrees.
There are exactly five solutions of the provided equation as the degree of the equation is $5$ .
$\begin{align}
& {{x}^{5}}=32i \\
& \sqrt[5]{{{x}^{5}}}=\sqrt[5]{32i} \\
& x=\sqrt[5]{32i} \\
& =\sqrt[5]{32\left( \cos 90{}^\circ +i\sin 90{}^\circ \right)}
\end{align}$
Apply Demoivre’s Theorem to find all solutions of the provided equation.
Here, $n=5$ as the number of solutions is five.
${{z}_{k}}=\sqrt[5]{32}\left[ \cos \left( \frac{90{}^\circ +360{}^\circ k}{5} \right)+i\sin \left( \frac{90{}^\circ +360{}^\circ k}{5} \right) \right]$
Here, $k=0,1,2,3,4$
Insert above values of $k$ to find five distinct solutions of the provided equation.
For $k=0$
$\begin{align}
& {{z}_{0}}=\sqrt[5]{32}\left[ \cos \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 0 \right)}{5} \right)+i\sin \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 0 \right)}{5} \right) \right] \\
& =2\left[ \cos 18{}^\circ +i\sin 18{}^\circ \right]
\end{align}$
Above solution is in the polar form. Substitute the value of $\cos 18{}^\circ $ and $\sin 18{}^\circ $ to find solution in rectangular form.
$\begin{align}
& {{z}_{0}}=2\left[ 0.9511+0.3090i \right] \\
& \approx 1.9022+0.6180i
\end{align}$
For $k=1$
$\begin{align}
& {{z}_{1}}=\sqrt[5]{32}\left[ \cos \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 1 \right)}{5} \right)+i\sin \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 1 \right)}{5} \right) \right] \\
& =2\left[ \cos 90{}^\circ +i\sin 90{}^\circ \right]
\end{align}$
Above solution is in the polar form. Substitute the value of $\cos 90{}^\circ $ and $\sin 90{}^\circ $ to find solution in rectangular form.
$\begin{align}
& {{z}_{1}}=2\left[ 0+1.00i \right] \\
& =0+2.0i
\end{align}$
For $k=2$
$\begin{align}
& {{z}_{2}}=\sqrt[5]{32}\left[ \cos \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 2 \right)}{5} \right)+i\sin \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 2 \right)}{5} \right) \right] \\
& =2\left[ \cos 162{}^\circ +i\sin 162{}^\circ \right]
\end{align}$
Above solution is in the polar form. Substitute the value of $\cos 162{}^\circ $ and $\sin 162{}^\circ $ to find solution in rectangular form.
$\begin{align}
& {{z}_{2}}=2\left[ -0.9511+0.3090i \right] \\
& \approx -1.9022+0.6180i
\end{align}$
For $k=3$
$\begin{align}
& {{z}_{3}}=\sqrt[5]{32}\left[ \cos \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 3 \right)}{5} \right)+i\sin \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 3 \right)}{5} \right) \right] \\
& =2\left[ \cos 234{}^\circ +i\sin 234{}^\circ \right]
\end{align}$
Above solution is in the polar form. Substitute the value of $\cos 234{}^\circ $ and $\sin 234{}^\circ $ to find solution in rectangular form.
$\begin{align}
& {{z}_{3}}=2\left[ -0.5878+\left( -0.8090 \right)i \right] \\
& \approx -1.1756-1.6180i
\end{align}$
For $k=4$
$\begin{align}
& {{z}_{4}}=\sqrt[5]{32}\left[ \cos \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 4 \right)}{5} \right)+i\sin \left( \frac{90{}^\circ +360{}^\circ \cdot \left( 4 \right)}{5} \right) \right] \\
& =2\left[ \cos 306{}^\circ +i\sin 306{}^\circ \right]
\end{align}$
Above solution is in the polar form. Substitute the value of $\cos 234{}^\circ $ and $\sin 234{}^\circ $ to find solution in rectangular form.
$\begin{align}
& {{z}_{4}}=2\left[ 0.5878+\left( -0.8090 \right)i \right] \\
& \approx 1.1756-1.6180i
\end{align}$
