# Find the Square Root of the Following Complex Number: 1 + 4 √ − 3 - Mathematics

Find the square root of the following complex number:

$1 + 4\sqrt{- 3}$

#### Solution

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

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

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

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

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

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

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

Concept: Concept of Complex Numbers - Square Root of a Complex Number
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.3 | Q 1.7 | Page 39