Question

Find the square root of the following complex number:

8 −15i

Answer 1

\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} + i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) > 0\]

\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} - i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) < 0\]

\[ z = 8 - 15i, Re\left( z \right) = 8, \left| z \right| = \sqrt{64 + 225} = 17\]

\[\text { Here, Im }(z) < 0\]

\[ \therefore \sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} - i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right]\]

\[ = \pm \left[ \sqrt{\frac{17 + 8}{2}} - i\sqrt{\frac{17 + 8}{2}} \right]\]

\[ = \pm \frac{1}{\sqrt{2}}\left( 5 - 3i \right)\]