Question

Square root of 4i - 3

Answer 1

Dear student
$$\begin{gathered} \mathrm{Let}\mathrm{z}=-3+4\mathrm{i} \\ \mathrm{Then},\mathrm{Re}\left(\mathrm{z}\right)=-3\mathrm{and}\left|\mathrm{z}\right|=\sqrt{9+16}=5 \\ \therefore \sqrt{-3+4\mathrm{i}}=\pm \left\{\sqrt{\frac{\left|\mathrm{z}\right|+\mathrm{Re}\left(\mathrm{z}\right)}{2}}+\mathrm{i}\sqrt{\frac{\left|\mathrm{z}\right|-\mathrm{Re}\left(\mathrm{z}\right)}{2}}\right\} \\ =\pm \left\{\sqrt{\frac{5-3}{2}}+\mathrm{i}\sqrt{\frac{5+3}{2}}\right\} \\ =\pm \left\{1+2\mathrm{i}\right\} \end{gathered}$$
Regards