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

Find the square root of the following complex number:

 4i

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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
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APPEARS IN

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