\sqrt{x\pm{iy}}=\sqrt{\frac{\sqrt{x^2+y^2}+x}{2}}\pm{i\sqrt{{\frac{\sqrt{x^2+y^2}-x}{2}}}}
例として\sqrt{3+4i}
\begin{align}
\sqrt{3+4i}&=\sqrt{\frac{\sqrt{3^2+4^2}+3}{2}}+i\sqrt{{\frac{\sqrt{3^2+4^2}-3}{2}}}\\
&=\sqrt{\frac{\sqrt{9+16}+3}{2}}+i\sqrt{{\frac{\sqrt{9+16}-3}{2}}}\\
&=\sqrt{\frac{\sqrt{25}+3}{2}}+i\sqrt{{\frac{\sqrt{25}-3}{2}}}\\
&=\sqrt{\frac{5+3}{2}}+i\sqrt{{\frac{5-3}{2}}}\\
&=\sqrt{\frac{8}{2}}+i\sqrt{{\frac{2}{2}}}\\
&=\sqrt{4}+i\sqrt{{1}}\\
&=2+i
\end{align}
\begin{align} \left( 2+i \right)^2&=2^2+4i+i^2\\ &=4+4i-1\\ &=3+4i \end{align}
