Question

Find the square root of the following complex number:

 \[1 + 4\sqrt{- 3}\]

Answer 1

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

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

\[ z = 1 + 4\sqrt{3}\sqrt{- 1} = 1 + 4\sqrt{3}i, Re\left( z \right) = 1, \left| z \right| = \sqrt{1 + 16 \times 3} = 7\]

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

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

\[ = \pm \left[ \sqrt{\frac{7 + 1}{2}} + i\sqrt{\frac{7 - 1}{2}} \right]\]

\[ = \pm \left( 2 + \sqrt{3}i \right)\]