Question

Find the multiplicative inverse of each of the complex numbers given in the Exercises $11$ to $13$.
$\sqrt{5}+3i$

Answer 1

$\Rightarrow$ Multiplicative inverse of $z=z^{-1}$

$\Rightarrow$ Multiplicative inverse of $\sqrt{5}+3i=\frac{1}{\sqrt{5}+3i}$

$=\frac{1}{\sqrt{5}+3i}\times \frac{\sqrt{5}-3i}{\sqrt{5}-3i}$ [ Rationalizing ]

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

$=\frac{\sqrt{5}-3i}{(\sqrt{5}{)}^2-(3i{)}^2}$ $\left[\right(a-b\left)\right(a+b)=a^2-b^2]$

$=\frac{\sqrt{5}-3i}{5-9i^2}$

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

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

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