Find the multiplicative inverse of the complex number \[\sqrt 5 + {\rm{3i}}\].

Hint:
Here in this question we have to find the multiplicative inverse of the complex number. Multiplicative inverse equal to its inverse. So, by expanding the inverse and rationalizing it will give you the value of the multiplicative inverse of the complex number.

Complete step by step solution:
We all know that multiplicative inverse of \[{\rm{z = }}{{\rm{z}}^{ - 1}}\] and multiplicative inverse of \[{\rm{z = }}\dfrac{1}{{\rm{z}}}\].
According to the question \[{\rm{z = }}\sqrt 5 + {\rm{3i}}\]
Therefore, multiplicative inverse of \[\sqrt 5 + {\rm{3i}} = \dfrac{1}{{\sqrt 5 + {\rm{3i}}}}\]
Now, we have to rationalize it.
\[\sqrt 5 + 3{\rm{i}} = \dfrac{1}{{\sqrt 5 + 3{\rm{i}}}} \times \dfrac{{\sqrt 5 - 3{\rm{i}}}}{{\sqrt 5 - 3{\rm{i}}}} = \dfrac{{\sqrt 5 - 3{\rm{i}}}}{{(\sqrt 5 + 3{\rm{i)}} \times (\sqrt 5 - 3{\rm{i}})}}\]

By simply using \[({\rm{a}} + {\rm{b)}} \times {\rm{(a}} - {\rm{b)}} = {{\rm{a}}^2} - {{\rm{b}}^2}\] formula, we get
\[\sqrt 5 + 3{\rm{i}} = \dfrac{{\sqrt 5 - 3{\rm{i}}}}{{{{(\sqrt 5 )}^2} - {{(3{\rm{i)}}}^2}}} = \dfrac{{\sqrt 5 - 3{\rm{i}}}}{{5 - 9{{\rm{i}}^2}}}\]

As we all know that the value of \[{{\rm{i}}^2} = - 1\] so, equation became
\[\begin{array}{l}\sqrt 5 + 3{\rm{i}} = \dfrac{{\sqrt 5 - 3{\rm{i}}}}{{5 - 9 \times ( - 1)}}\\ = \dfrac{{\sqrt 5 - 3{\rm{i}}}}{{5 + 9}}\\ = \dfrac{{\sqrt 5 - 3{\rm{i}}}}{{14}}\end{array}\]

Therefore, multiplicative inverse of \[\sqrt 5 + {\rm{3i}} = \dfrac{{\sqrt 5 - 3{\rm{i}}}}{{14}} = \dfrac{{\sqrt 5 }}{{14}} - \dfrac{{3{\rm{i}}}}{{14}}\]

Note:
Alternate way of finding the multiplicative inverse of z is by using the direct formula of multiplicative inverse of \[{\rm{z = }}{{\rm{z}}^{ - 1}} = \dfrac{{\overline {\rm{z}} }}{{{{\left| z \right|}^2}}}\].
According to the question \[{\rm{z = }}\sqrt 5 + 3{\rm{i}}\]
Then, \[\overline {\rm{z}} = \sqrt 5 - 3{\rm{i}}\] and \[{\left| z \right|^2} = {(\sqrt 5 )^2} + {(3)^2} = 5 + 9 = 14\].
By putting the values in the formula of multiplicative inverse, we get
Multiplicative inverse of \[\sqrt 5 + {\rm{3i}} = \dfrac{{\sqrt 5 - 3{\rm{i}}}}{{14}} = \dfrac{{\sqrt 5 }}{{14}} - \dfrac{{3{\rm{i}}}}{{14}}\]