Find the multiplicative inverse of each of the complex numbers

$\sqrt{5} + 3 i$

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Solution

M.I. of $\sqrt{5} + 3 i = \frac{1}{\sqrt{5} + 3 i}$

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

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

$= \frac{\sqrt{5} - 3 i}{5 + 9} = \frac{1}{14} (\sqrt{5} - 3 i)$

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