Write the values of the square root of −i.

Let $\sqrt{-i}=x+i y$

Squaring both the sides

$-i=x^{2}+y^{2} i^{2}+2 i x y$

$\Rightarrow 2 x y=-1 \quad \ldots$ (i)

and $x^{2}-y^{2}=0 \quad \ldots$ (ii)

Equation (ii) shows that $x$ and $y$ are of opposite sign. From (ii),

$x=\pm y$

From (i)

$2(x)(-x)=\frac{-1}{2}$

$\Rightarrow x^{2}=\frac{1}{2}$

$\Rightarrow x=\pm \frac{1}{\sqrt{2}}$[Since $x$ and $y$ have opposite signs, $y=-\frac{1}{\sqrt{2}}$ when $x=\frac{1}{\sqrt{2}}$ and vice versa $]$

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