Question

Find the whole root of 3- 3 i

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{z}=3-3\mathrm{i} \\ \mathrm{Now},\mathrm{Re}\left(\mathrm{z}\right)=3 \\ \mathrm{Now},\left|\mathrm{z}\right|=\sqrt{{\left(3\right)}^2+{\left(-3\right)}^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2} \\ \mathrm{Now},\sqrt{3-3\mathrm{i}}=\pm \left\{\sqrt{\frac{\left|\mathrm{z}\right|+\mathrm{Re}\left(\mathrm{z}\right)}{2}}-\mathrm{i}\sqrt{\frac{\left|\mathrm{z}\right|-\mathrm{Re}\left(\mathrm{z}\right)}{2}}\right\} \\ \Rightarrow \sqrt{3-3\mathrm{i}}=\pm \left\{\sqrt{\frac{3\sqrt{2}+3}{2}}-\mathrm{i}\sqrt{\frac{3\sqrt{2}-3}{2}}\right\} \end{gathered}$$