Question

Find the multiplicative inverse of $\sqrt{5}+3i$

Answer 1

Given data: $z=\sqrt{5}+3i$

Using the concept of multiplicative inverse.
$z^{-1}=\frac{1}{\sqrt{5}+3i}$

Now, using rationalization method.

$z^{-1}=\frac{1}{\sqrt{5}+3i}\times \frac{\sqrt{5}-3i}{\sqrt{5}-3i}$

$z^{-1}=\frac{\sqrt{5}-3i}{5+9}$

$z^{-1}=\frac{\sqrt{5}}{14}-\left(\frac{3}{14}\right)i\text{or}{z}^{-1}=\frac{\sqrt{5}}{14}+i\left(\frac{-3}{14}\right)$