Question

How do you simplify $\dfrac{{3 + i}}{{3 - i}}$ and write the complex number in standard form?

Answer 1

Hint: Complex number is combination of real part and imaginary part. Complex numbers are written as $a + ib$ where $a$ is real part and $b$ is imaginary part. Here is $3 + i$ , 3 is the real part and $1 \cdot i$ is the imaginary part. To solve this question multiply the given by conjugate.
A conjugate of a complex number is the same number but with an opposite sign. For example, the conjugate of $a + ib$ is $a - ib$.

Complete step by step answer:
In this question, we multiply both the numerator and denominator by the conjugate of the denominator.
Here, the denominator is $3 - i$, Conjugate of denominator $3 - i$ is $3 + i$
Multiplying the numerator and denominator by conjugate $3 + i$
$ \Rightarrow \dfrac{{3 + i}}{{3 - i}} \times \dfrac{{3 + i}}{{3 + i}}$
Solving the above,
$ \Rightarrow \dfrac{{{{(3 + i)}^2}}}{{(3 - i)(3 + i)}}$
Using${(a + b)^2} = {a^2} + {b^2} + 2ab$ and $(a - b)(a + b) = {a^2} - {b^2}$
Take 3 as $a$ and b as $i$
$ \Rightarrow \dfrac{{{{(3)}^2} + {{(i)}^2} + 2 \cdot 3 \cdot i}}{{{{(3)}^2} - {{(i)}^2}}}$
We know, ${i^2} = - 1$
$ \Rightarrow \dfrac{{9 + ( - 1) + 6i}}{{9 - ( - 1)}}$
Add and subtract like terms in both numerator and denominator
$ \Rightarrow \dfrac{{8 + 6i}}{{10}}$
It can be written as $\dfrac{8}{{10}} + \dfrac{{6i}}{{10}}$ to make it in the form of $a + ib$
It can be further simplified as
$ \Rightarrow \dfrac{4}{5} + \dfrac{{3i}}{5}$

Thus, simplified $\dfrac{{3 + i}}{{3 - i}}$ as $\dfrac{4}{5} + \dfrac{{3i}}{5}$ in standard form.

Addition information:
Sometimes complex numbers are represented as $Z$ , $\operatorname{Re} ()$ for real part and $\operatorname{Im} ()$ for imaginary parts.
Like $Z = a + ib$ , here\[Re\left( z \right) = a\] and $\operatorname{Im} (z) = b$ .

Note: Remember the values ${i^2} = - 1$ , $i = \sqrt { - 1} $ and ${i^3} = - i$ .
Split the complex-valued answer into two separate terms that are a real part and an imaginary part to make it easier.
Always write the answer in a standard form that is $a + ib$.