91Ó°ÊÓ

The given expression is $\sqrt[4]{-1}$.

Complex numbers consist of real numbers and imaginary numbers; 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 $-1+0\mathrm{i}$.

x=-1

y=0

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

$$\begin{gathered} r=1 \\ \mathrm{θ}=\mathrm{π},or3\mathrm{π},5\mathrm{π},7\mathrm{π},9\mathrm{π},..... \end{gathered}$$

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

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

When n=4 , the equation becomes the 4th root of the complex number.

$$\begin{gathered} z^{\frac{1}{4}}=r^{\frac{1}{4}}{e}^{\frac{i\mathrm{θ}}{4}} \\ r=1 \\ \mathrm{θ}=\frac{\mathrm{π}}{4},\frac{3\mathrm{π}}{4},\frac{5\mathrm{π}}{4},\frac{7\mathrm{π}}{4},\frac{9\mathrm{π}}{4},....... \end{gathered}$$

It is clear from the above graph that the points $\left(1,\frac{\mathrm{Ï€}}{4}\right)$and the point $\left(1,\frac{9\mathrm{Ï€}}{4}\right)$are the same.

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

$$\begin{gathered} r=1 \\ \mathrm{θ}=\frac{\mathrm{π}}{4},\frac{3\mathrm{π}}{4},\frac{5\mathrm{π}}{4},\frac{7\mathrm{π}}{4} \end{gathered}$$

Hence, the final answer is $\sqrt[4]{-1}=\frac{\left(Â\pm 1Â\pm i\right)}{\sqrt{2}}$.