Question

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

Answer 1

Let $z=\sqrt{5}+3i$

Then, $\overline{z}=\sqrt{5}-3i$ and ${|z|}^2={\left(\sqrt{5}\right)}^2+3^2=5+9=14$

Therefore, the multiplicative inverse of $\sqrt{5}+3i$ is given by

$z^{-1}=\frac{\overline{z}}{{|z|}^2}=\frac{\sqrt{5}-3i}{14}=\frac{\sqrt{5}}{14}-\frac{3i}{14}$