Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Find the Square Root of the Following Complex Number: −7 − 24i - Mathematics

Find the square root of the following complex number:

−7 − 24i

#### Solution

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

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

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

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

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

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

$= \pm \left( 3 - 4i \right)$

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 1.2 | Page 39