Find the values of the indicated roots.

$\sqrt[\mathbf{5}]{\mathbf{i}}$

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 $z=r\times e^{\left({\theta }_i\right)}$.

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

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

And n = 5

Angle ${\theta }_k$is written as ${\theta }_k=\frac{{\displaystyle \frac{\mathrm{\pi }}{2}}+2\mathrm{\pi k}}{n}$.

Find the roots of the complex number z for different values of $\theta$.

Solve z and $\theta$for $k=0,1.$

$$\begin{gathered} {\theta }_0=\frac{\mathrm{\pi }}{10} \\ {z}_0=e^{\left(\mathrm{\pi i}/10\right)} \\ {\theta }_1=\frac{\mathrm{\pi }}{2} \\ {z}_1=e^{(\mathrm{\pi i}/2)} \end{gathered}$$

Solve z and $\theta$for k = 2,3 .

$$\begin{gathered} {\theta }_2=\frac{9\mathrm{\pi }}{10} \\ {z}_2=e^{(9\mathrm{\pi i}/10)} \\ {\mathrm{\theta }}_3=\frac{13\mathrm{\pi }}{10} \\ {\mathrm{z}}_3={\mathrm{e}}^{13\mathrm{\pi i}/10} \end{gathered}$$

Solve z and $\theta$for k = 4.

$$\begin{gathered} {\vartheta }_4=\frac{17\mathrm{\pi }}{10} \\ {z}_4=e^{17\mathrm{\pi i}/10} \end{gathered}$$

Solve for$z_0$

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

Solve for$z_1$.

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

Solve for $z_2$.

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

Solve for $z_3$.

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

Solve for$z_4$.

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

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

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