91Ó°ÊÓ

The given expression is$\sqrt{i}$.

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$\sqrt{\mathbf{-}\mathbf{1}}$

The Complex number is in the form 0+i .

x=0,y=1

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

$$\begin{gathered} r=1 \\ \mathrm{θ}=\frac{\mathrm{π}}{2},or\frac{5\mathrm{π}}{2},\frac{9\mathrm{π}}{2},.... \end{gathered}$$

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

role="math" localid="1658744103977" $$\begin{gathered} z^{\frac{1}{n}}={\left(r{e}^{i\mathrm{θ}}\right)}^{\frac{1}{n}} \\ {z}^{\frac{1}{n}}=r^{\frac{1}{n}}{e}^{\frac{i\mathrm{θ}}{n}} \\ {z}^{\frac{1}{n}}=\sqrt[n]{r}\left(\cos \frac{\mathrm{θ}}{n}+i\sin \frac{\mathrm{θ}}{n}\right)\left(1\right) \end{gathered}$$

When n=2, the equation becomes the 2nd root of the complex number.

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

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 $\mathrm{Ï€}$apart.

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

Hence, the value of$\sqrt{i}=\frac{Â\pm \left(1+i\right)}{\sqrt{2}}$