Find the multiplicative inverse of the complex number

Question:

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

Solution:

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

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

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

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

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