91Ó°ÊÓ

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

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 -4+0i .

x=-4y, y=0

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

$$\begin{gathered} r=\sqrt{2} \\ \mathrm{θ}=\mathrm{π},or3\mathrm{π},5\mathrm{π},7\mathrm{π}9\mathrm{π},...... \end{gathered}$$

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

role="math" localid="1658730367872" $$\begin{gathered} z^{\frac{1}{n}}={\left(r{e}^{i\mathrm{θ}}\right)}^{\frac{1}{n}} \\ {z}^{\frac{1}{n}}=r^{\frac{1}{n}}{e}^{\frac{i\mathrm{θ}}{n}} \\ {z}^{\frac{1}{n}}=\sqrt[n]{r}\left(\cos \frac{\mathrm{θ}}{n}+i\sin \frac{\mathrm{θ}}{n}\right)\left(1\right) \end{gathered}$$

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

$$\begin{gathered} z^{\frac{1}{4}}=r^{\frac{1}{4}}{e}^{\frac{i\mathrm{θ}}{4}} \\ r=\sqrt{2} \\ \mathrm{θ}=\frac{\mathrm{π}}{4},\frac{3\mathrm{π}}{4},\frac{5\mathrm{π}}{4},\frac{7\mathrm{π}}{4},\frac{9\mathrm{π}}{4},....... \end{gathered}$$

It is clear from the above graph that the points$\left(\sqrt{2},\frac{\mathrm{Ï€}}{4}\right)$ and the point $\left(\sqrt{2},\frac{9\mathrm{Ï€}}{4}\right)$are the same.

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

$$\begin{gathered} r=\sqrt{2} \\ \mathrm{θ}=\frac{\mathrm{π}}{4},\frac{3\mathrm{π}}{4},\frac{5\mathrm{π}}{4},\frac{7\mathrm{π}}{4} \end{gathered}$$

Hence, the value of $\sqrt[4]{-4}=Â\pm 1,Â\pm \mathrm{i}$ .