# Find the Square Root of the Following Complex Number: 4i - Mathematics

Find the square root of the following complex number:

4i

#### Solution

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

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

$z = 0 + 4i, Re\left( z \right) = 0, \left| z \right| = 4$

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

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

$= \pm \left[ \sqrt{\frac{4 + 0}{2}} + i\sqrt{\frac{4 - 0}{2}} \right]$

$= \pm \left( \sqrt{2} + i\sqrt{2} \right)$

$= \pm \sqrt{2}\left( 1 + i \right)$

Concept: Concept of Complex Numbers - Square Root of a Complex Number
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.3 | Q 1.8 | Page 39