Question

Find the square root of the following complex number:

 −8 − 6i

Answer 1

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

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

\[ z = - 8 - 6i, Re\left( z \right) = - 8, \left| z \right| = \sqrt{64 + 36} = 10\]

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

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

\[ = \pm \left[ \sqrt{\frac{10 - 8}{2}} - i\sqrt{\frac{10 + 8}{2}} \right]\]

\[ = \pm \left( 1 - 3i \right)\]