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Q)

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

$(a)\;\normalsize \large\frac{\sqrt{5}}{14}- \normalsize \large\frac{3}{14} \normalsize i\qquad(b)\;\normalsize \large\frac{3}{14}- \normalsize \large\frac{\sqrt{5}}{14} \normalsize i\qquad(c)\;\normalsize \large\frac{\sqrt{3}}{14}- \normalsize \large\frac{5}{14} \normalsize i\qquad(d)\;\normalsize \large\frac{5}{14}- \normalsize \large\frac{\sqrt{3}}{14} \normalsize i$

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A)
Answer : $\;\normalsize \large\frac{\sqrt{5}}{14}- \normalsize \large\frac{3}{14} \normalsize i$
Explanation :
Let$\;z=\sqrt{5} +3 \normalsize i\; $
$\overline{z} = \sqrt{5} -3 \normalsize i\; and\; |z|^{2} = (\sqrt{5})^2+(3)^{2}$
$= 5 +9 = 14$
Therefore , the multiplicative inverse of $\;\sqrt{5} +3 \normalsize i\;$ is given by
$z^{-1} =\large\frac{\overline{z}}{|z|^{2}} =\large\frac{\sqrt{5} -3 \normalsize i}{14}$
$=\normalsize \large\frac{\sqrt{5}}{14}- \normalsize \large\frac{3}{14} \normalsize i\;.$
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