Follow steps (a), (b), (c) above to find all the values of the indicate droots $\sqrt[\mathbf{3}]{\mathbf{i}}$.

The given expression is$\sqrt[3]{i}$.

Complex numbers have both real numbers and imaginary numbers in them; a complex can be written in the form of:

z=a+ib

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

The Complex number is in the form 0+i ..

x=0,y=1

The polar coordinates of the point are in the form of $z=r{e}^{i\theta }$.

$$\begin{gathered} r=1 \\ \theta =\frac{\pi }{2},or\frac{5\pi }{2},\frac{9\pi }{2},\frac{13\pi }{2},... \end{gathered}$$

The equation $z=r{e}^{i\theta }$can also be written in another form.

$$\begin{gathered} z^{\frac{1}{n}}={\left(r{e}^{i\theta }\right)}^{\frac{1}{n}} \\ {z}^{\frac{1}{n}}=r^{\frac{1}{n}}{e}^{\frac{i\theta }{n}} \\ {z}^{\frac{1}{n}}=\sqrt[n]{r}\left(\cos \frac{\theta }{n}+i\sin \frac{\theta }{n}\right)\left(1\right) \end{gathered}$$

When n=3, the equation becomes 3rd the root of the complex number.

$$\begin{gathered} z^{\frac{1}{3}}=r^{\frac{1}{3}}{e}^{\frac{i\theta }{3}} \\ r=1 \\ \theta =\frac{\pi }{6},\frac{5\pi }{6},\frac{9\pi }{6},\frac{13\pi }{6},...... \\ =\frac{\pi }{6},\frac{5\pi }{6},\frac{3\pi }{2},\frac{13\pi }{6},..... \end{gathered}$$

It is clear from the above graph that the points $\left(1,\frac{\pi }{6}\right)$and the point $\left(1,\frac{13\pi }{6}\right)$are the same.

The radius of the circle is and equally spaced $\frac{2\pi }{3}$apart.

$$\begin{gathered} r=1 \\ \theta =\frac{\pi }{6},\frac{5\pi }{6},\frac{3\pi }{2} \end{gathered}$$

Hence, the value of $\sqrt[3]{i}=-i,\frac{\sqrt{\left(\pm \sqrt{3}+i\right)}}{2}$.