Question

find the square root of -9-40i

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{z}=-9-40\mathrm{i} \\ \mathrm{Now},\mathrm{Re}\left(\mathrm{z}\right)=-9 \\ \left|\mathrm{z}\right|=\sqrt{{\left(-9\right)}^2+{\left(-40\right)}^2}=\sqrt{81+1600}=\sqrt{1681}=41 \\ \mathrm{Now},\sqrt{\mathrm{z}}=\pm \left[\sqrt{\frac{\left|\mathrm{z}\right|+\mathrm{Re}\left(\mathrm{z}\right)}{2}}-\mathrm{i}\sqrt{\frac{\left|\mathrm{z}\right|-\mathrm{Re}\left(\mathrm{z}\right)}{2}}\right] \\ \Rightarrow \sqrt{-9-40\mathrm{i}}=\pm \left[\sqrt{\frac{41+\left(-9\right)}{2}}-\mathrm{i}\sqrt{\frac{41-\left(-9\right)}{2}}\right] \\ \Rightarrow \sqrt{-9-40\mathrm{i}}=\pm \left(4-5\mathrm{i}\right) \end{gathered}$$