# Find the multiplicative inverse of the following complex number: ( 1 + i √ 3 ) 2 - Mathematics

Find the multiplicative inverse of the following complex number:

$(1 + i\sqrt{3} )^2$

#### Solution

$z = \left( 1 + \sqrt{3}i \right)^2$

$= 1 + 3 i^2 + 2\sqrt{3}i$

$= - 2 + 2\sqrt{3}i$

$\text { Then }, \frac{1}{z} = \frac{1}{- 2 + 2\sqrt{3}i} \times \frac{- 2 - 2\sqrt{3}i}{- 2 - 2\sqrt{3}i}$

$= \frac{- 2 - 2\sqrt{3}i}{4 - 12 i^2}$

$= \frac{- 2 - 2\sqrt{3}i}{16}$

$= \frac{- 1}{8} - \frac{\sqrt{3}}{8}i$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.2 | Q 4.2 | Page 32