Question

Find the square root of the following complex number:

−7 − 24i

Answer 1

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

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

\[ z = - 7 - 24i, Re\left( z \right) = - 7, \left| z \right| = \sqrt{49 + 576} = 25 \]

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

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

\[ = \pm \left[ \sqrt{\frac{25 - 7}{2}} - i\sqrt{\frac{25 + 7}{2}} \right]\]

\[ = \pm \left( 3 - 4i \right)\]