Write the values of the square root of i.

Let the square root of $i$ be $x+i y$.

$\Rightarrow \sqrt{i}=x+i y$

$\Rightarrow i=x^{2}+y^{2} i^{2}+2 i x y$ 

$\Rightarrow i=x^{2}-y^{2}+2 i x y$ (Squaring both the sides)

Comparing both the sides:

$x^{2}-y^{2}=0$     ...(i)

and $2 x y=1$   ...(ii)

By equation (ii), we find that $x$ and $y$ are of the same sign.From equation (i),

$x=\pm y$

$\therefore x y=\frac{1}{2}, x^{2}=\frac{1}{2}$

$x=\pm \frac{1}{\sqrt{2}}, y=\pm \frac{1}{\sqrt{2}}$

$\therefore \sqrt{i}=\pm \frac{1}{\sqrt{2}}(1+i)$