Follow steps (a), (b), (c) above to find all the values of the indicate droots $\sqrt[\mathbf{5}]{\mathbf{-}\mathbf{1}}$

The given expression is $\sqrt[5]{-1}$.

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 -1+0i.

x=-1,y=0

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

r=1

$\theta =\pi ,or3\pi ,5\pi ,7\pi ,9\pi ,11\pi ,....$

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

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

When n=5 , the equation becomes the 5th root of the complex number:

$$\begin{gathered} z^{\frac{1}{5}}=r^{\frac{1}{5}}{e}^{\frac{i\theta }{5}} \\ r=1 \\ \theta =\frac{\pi }{5},\frac{3\pi }{5},\frac{5\pi }{5},\frac{7\pi }{5},\frac{9\pi }{5},\frac{11\pi }{5},..... \\ =\frac{\pi }{5},\frac{3\pi }{5},\pi ,\frac{7\pi }{5},\frac{9\pi }{5},\frac{11\pi }{5},..... \end{gathered}$$

It is clear from the above graph that the points $\left(1,\frac{\pi }{5}\right)$and the point $\left(1,\frac{11\pi }{5}\right)$are the same.

The radius of the circle 1 is and equally spaced$\frac{2\pi }{5}$ apart.

$$\begin{gathered} r=1 \\ \theta =\frac{\pi }{5},\frac{3\pi }{5},\pi ,\frac{7\pi }{5},\frac{9\pi }{5} \end{gathered}$$

Hence, the value of $\sqrt[5]{-1}=-1,0.809\pm 0.588i,-0.309\pm 0.951i$