91Ó°ÊÓ

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$z=r{e}^{i\mathrm{θ}}$

r=1

$\mathrm{θ}=\mathrm{π},or3\mathrm{π},5\mathrm{π},7\mathrm{π},9\mathrm{π},11\mathrm{π},....$

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

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

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

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

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

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

$$\begin{gathered} r=1 \\ \mathrm{θ}=\frac{\mathrm{π}}{5},\frac{3\mathrm{π}}{5},\mathrm{π},\frac{7\mathrm{π}}{5},\frac{9\mathrm{π}}{5} \end{gathered}$$

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