Question

Find the multiplicative inverse of the complex numbers given the following:
$\sqrt{5}+3i$

Answer 1

Let multiplicative inverse of $\sqrt{5}+3i$ be $x$

$\Rightarrow x=\frac{1}{\sqrt{5}+3i}$

Rationalize the denominator

$\Rightarrow x=\frac{1}{\sqrt{5}+3i}\times \frac{\sqrt{5}-3i}{\sqrt{5}-3i}$

$\Rightarrow x=\frac{\sqrt{5}-3i}{(\sqrt{5}{)}^2-(3i{)}^2}$

$\Rightarrow x=\frac{\sqrt{5}-3i}{5-(-9)}=\frac{\sqrt{5}-3i}{5+9}$

$\Rightarrow x=\frac{\sqrt{5}-3i}{14}$

The multiplicative inverse of $\sqrt{5}+3i$ is $\frac{\sqrt{5}-3i}{14}$