Question:
If $z=(\sqrt{5}+3 i)$, find $z^{-1}$.
Solution:
We have, $z=(\sqrt{5}+3 i)$
$\Rightarrow \overline{\mathrm{z}}=(\sqrt{5}-3 \mathrm{i})$
$\Rightarrow|\mathrm{z}|^{2}=(\sqrt{5})^{2}+(3)^{2}$
$=5+9=14$
$\therefore$ The multiplicative inverse of $(\sqrt{5}+3 \mathrm{i})$,
$\mathrm{z}^{-1}=\frac{\overline{\mathrm{z}}}{|\mathrm{z}|^{2}}=\frac{\sqrt{5}-3 \mathrm{i}}{14}$
$\mathrm{z}^{-1}=\frac{\sqrt{5}}{14}+\frac{3}{14} \mathrm{i}$