91Ó°ÊÓ

The given expression is $\sqrt[5]{i}$.

Complex numbers are represented in terms of real numbers and imaginary numbers; a complex can be written in the form of:

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}\mathit{i}\mathit{b}$

Here a and b are real numbers, and i is the imaginary number which is known as iota, whose value is $\sqrt{\mathbf{-}\mathbf{1}}$.

Let z = i

The exponential form of z is given by $z=r×e^{\left({\mathrm{θ}}_i\right)}$.

The root is $z_k=r^{\frac{1}{n}}\exp \left({\mathrm{θ}}_{k}i\right)$.

Where k = 0,1,2,3,4

And n = 5

Angle ${\mathrm{θ}}_k$is written as ${\mathrm{θ}}_k=\frac{{\displaystyle \frac{\mathrm{Ï€}}{2}}+2\mathrm{Ï€°ì}}{n}$.

Find the roots of the complex number z for different values of $\mathrm{θ}$.

Solve z and $\mathrm{θ}$for $k=0,1.$

$$\begin{gathered} {\mathrm{θ}}_0=\frac{\mathrm{Ï€}}{10} \\ {z}_0=e^{\left(\mathrm{Ï€¾\pm }/10\right)} \\ {\mathrm{θ}}_1=\frac{\mathrm{Ï€}}{2} \\ {z}_1=e^{(\mathrm{Ï€¾\pm }/2)} \end{gathered}$$

Solve z and $\mathrm{θ}$for k = 2,3 .

$$\begin{gathered} {\mathrm{θ}}_2=\frac{9\mathrm{Ï€}}{10} \\ {z}_2=e^{(9\mathrm{Ï€¾\pm }/10)} \\ {\mathrm{θ}}_3=\frac{13\mathrm{Ï€}}{10} \\ {\mathrm{z}}_3={\mathrm{e}}^{13\mathrm{Ï€¾\pm }/10} \end{gathered}$$

Solve z and $\mathrm{θ}$for k = 4.

$$\begin{gathered} {\mathrm{Ï‘}}_4=\frac{17\mathrm{Ï€}}{10} \\ {z}_4=e^{17\mathrm{Ï€¾\pm }/10} \end{gathered}$$

Solve for$z_0$

$$\begin{gathered} z_0=\left[\cos \left(\frac{\mathrm{Ï€}}{10}\right)+i\sin \left(\frac{\mathrm{Ï€}}{10}\right)\right] \\ =0.95+0.31i \end{gathered}$$

Solve for$z_1$.

localid="1658739180865" $$\begin{gathered} z_1=\left[\cos \left(\frac{\mathrm{Ï€}}{10}\right)+i\sin \left(\frac{\mathrm{Ï€}}{10}\right)\right] \\ =i \end{gathered}$$

Solve for $z_2$.

$$\begin{gathered} z_2=\left[\cos \left(\frac{9\mathrm{Ï€}}{10}\right)+i\sin \left(\frac{9\mathrm{Ï€}}{10}\right)\right] \\ =-0.95+0.31i \end{gathered}$$$$\begin{gathered} z_2=\left[\cos \left(\frac{9\mathrm{Ï€}}{10}\right)+i\sin \left(\frac{9\mathrm{Ï€}}{10}\right)\right] \\ =0.95+0.31i \end{gathered}$$

Solve for $z_3$.

role="math" $$\begin{gathered} z_3=\left[\cos \left(\frac{13\mathrm{Ï€}}{10}\right)+i\sin \left(\frac{13\mathrm{Ï€}}{10}\right)\right] \\ =-0.588-0.81i \end{gathered}$$

Solve for$z_4$.

$$\begin{gathered} z_4=\left[\cos \left(\frac{17\mathrm{Ï€}}{10}\right)+i\sin \left(\frac{17\mathrm{Ï€}}{10}\right)\right] \\ =-0.588-0.81i \end{gathered}$$

Hence, the values of the complex number $\sqrt[5]{i}$are;

$$\begin{gathered} z_0=0.95+0.31i \\ {z}_1=i \\ {z}_2=-0.95+0.31i \\ {z}_3=-0.588-0.81i \\ {z}_4=0.588-0.81i \end{gathered}$$