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# Find the multiplicative inverse of each of the complex numbers. (12) sqrt 5 + 3i

Find the multiplicative inverse of each of the complex numbers.

Q: 12        $\sqrt{5}+3i$

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Let    $z = \sqrt{5}+3i$
Then,
$\bar z = \sqrt{5}-3i$
And
$|z|^2 = (\sqrt5)^2+(3)^2 = 5+9 =14$
Now, the multiplicative inverse is given by
$z^{-1}= \frac{\bar z}{|z|^2}= \frac{\sqrt5-3i}{14}= \frac{\sqrt5}{14}-i\frac{3}{14}$

Therefore, the multiplicative inverse is   $\frac{\sqrt5}{14}-i\frac{3}{14}$

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