Question

How do you simplify $\dfrac{1}{{2 + 3i}}$ and write in $a + bi$ form?

Answer 1

For solving such types of questions you have to first see the denominator. Here, in the denominator, there is a complex number. So whenever you have a complex number in a denominator you have to multiply the number by its opposite number. So after that, you will get a real number in the denominator.

Complete step by step answer:
So, for simplifying this equation we have to follow the below-given steps.
Now, assume you have a number $ai$ so the opposite of this number will be $- ai$.
Now, assume you have a number $c + ai$ so the opposite of this number will be $c - ai$.
Here, in this question, the denominator is a complex number so we have to multiply the number by its opposite number.
The denominator of the given number is $2 + 3i$ so the opposite of these numbers is $2 - 3i$.
So we will multiply the denominator and numerator by $2 - 3i$.
After multiplying the denominator and numerator by $2 - 3i$ we get,
$\begin{gathered} \Rightarrow \dfrac{1}{{2 + 3i}} \times \dfrac{{2 - 3i}}{{2 - 3i}} \\\ \Rightarrow \dfrac{{2 - 3i}}{{{2^2} - {{(3i)}^2}}} \\\ \Rightarrow \dfrac{{2 - 3i}}{{4 - 9{i^2}}} \\\ \end{gathered}$
Here, ${i^2} = - 1$
So, after substituting the value of ${i^2}$we get,
$\Rightarrow \dfrac{{2 - 3i}}{{4 - 9( - 1)}} \\\ \Rightarrow \dfrac{{2 - 3i}}{{4 + 9}} \\\ \Rightarrow \dfrac{{2 - 3i}}{{13}} \\\ \Rightarrow \dfrac{2}{{13}} - i\dfrac{3}{{13}} \\\$

So after comparing $\dfrac{2}{{13}} - i\dfrac{3}{{13}}$ to $a + bi$we get
Value of a is $\dfrac{2}{{13}}$ and value of b is $\dfrac{3}{{13}}$.

Note:
For solving such types of questions you should remember the opposite of $c + ai$ is $c - ai$. Sometimes students make mistakes by misunderstanding the opposite of $c + ai$ is $- c - ai$. If you misunderstood this thing then you will get the wrong answer.