Follow steps (a), (b), (c) above to find all the values of the indicated roots $\sqrt[\mathbf{3}]{\mathbf{-}\mathbf{1}}$.

The given expression is $\sqrt[3]{-1}$.

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

$\mathbf{z}\mathbf{=}\mathbf{a}\mathbf{+}\mathbf{ib}$

Here a and b are real numbers, andi is the imaginary number which is known as iota, whose value is role="math" localid="1658727470365" $\sqrt{\mathbf{-}\mathbf{1}\mathbf{}}$.

The Complex number is in the form -1+0i .

x=-1, y=0

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

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

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 the root of the complex number as:

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

It is clear from the above graph that the points$\left(1,\frac{\pi }{3}\right)$ and the point $\left(1,\frac{7\pi }{3}\right)$are the same.

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

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

Hence, the final answer is $\sqrt[3]{-1}=-1,\left(\frac{1\pm i\sqrt{3}}{2}\right)$.