Question

Find the multiplicative inverse of $\sqrt 5 + 3i$
(A)$\sqrt 5 - 3i$
(B) $\dfrac{{\sqrt 5 - 3i}}{{14}}$
(C) $ - \sqrt 5 + 3i$
(D) $\dfrac{{ - \sqrt 5 + 3i}}{{14}}$

Answer 1

Hint:Before dealing with the question we need to focus on the process of finding the multiplicative inverse of a complex number.


Complete step by step solution:
A multiplicative inverse is a number that, when multiplied by the given number, yields \[1\].
The multiplicative inverse of any complex number $a + ib$ is $\dfrac{1}{{a + ib}}$
In this question we have,$\sqrt 5 + 3i$.
The multiplicative inverse of $\sqrt 5 + 3i$ is $\dfrac{1}{{\sqrt 5 + 3i}}$.
Now in this multiplicative inverse, we cannot leave \[I\] in the denominator, So we will rationalize it.
Multiply numerator and denominator of $\dfrac{1}{{\sqrt 5 + 3i}}$ by $\sqrt 5 - 3i$.
$
  \dfrac{1}{{\sqrt 5 + 3i}} \times \left( {\dfrac{{\sqrt 5 - 3i}}{{\sqrt 5 - 3i}}} \right) = \dfrac{{\sqrt 5 - 3i}}{{\left( {\sqrt 5 + 3i} \right)\left( {\sqrt 5 - 3i} \right)}} \\
   = \dfrac{{\sqrt 5 - 3i}}{{{{\left( {\sqrt 5 } \right)}^2} - {{\left( {3i} \right)}^2}}} \\
   = \dfrac{{\sqrt 5 - 3i}}{{5 + 9}} \\
   = \dfrac{{\sqrt 5 - 3i}}{{14}} \\
   = \dfrac{1}{{14}}\left( {\sqrt 5 - 3i} \right) \\
 $
Thus, the multiplicative inverse of $\sqrt 5 + 3i$ is $\dfrac{1}{{14}}\left( {\sqrt 5 - 3i} \right)$(in simplified form).
Thus, the correct option is\[B\].


Note: Here if we leave our answer in first stage that is $\dfrac{1}{{\sqrt 5 + 3i}}$ we may lose a point, so we need to rationalize the denominator by multiply numerator and denominator by its conjugate.