Question

Find the square root of the following complex number:

\[- 11 - 60\sqrt{- 1}\]

Answer 1

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

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

\[- 11 - 60\sqrt{- 1} = - 11 - 60i, Re\left( z \right) = - 11, \left| z \right| = \sqrt{121 + 3600} = 61\]

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

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

\[ = \pm \left[ \sqrt{\frac{61 - 11}{2}} - i\sqrt{\frac{61 + 11}{2}} \right]\]

\[ = \pm \left( 5 - 6i \right)\]