Question

How do you simplify $\dfrac{{2 - i}}{{3 - 4i}}$?

Answer 1

Hint: Here we will take the conjugate of the number and will multiply and divide it with the given expression and then will simplify it for the required value.

Complete step-by-step solution:
Take the given expression: $\dfrac{{2 - i}}{{3 - 4i}}$
Conjugate of the number can be found by changing the sign of the term. Positive is changed to negative and negative is changed to positive. So, conjugate of $3 - 4i$is $3 + 4i$
$\Rightarrow \dfrac{{2 - i}}{{3 - 4i}} \times \dfrac{{3 + 4i}}{{3 + 4i}}$
Now, the numerator is multiplied with the numerator of the second fraction and the denominator of the first fraction is multiplied with the denominator of the second term.
$\Rightarrow \dfrac{{(2 - i)(3 + 4i)}}{{(3 - 4i)(3 + 4i)}}$
The above denominators show the identity and so using the identity of difference of two squares${x^2} - {y^2} = (x - y)(x + y)$ and open the brackets on the numerator.
$\Rightarrow \dfrac{{2(3 + 4i) - i(3 + 4i)}}{{{{(3)}^2} - {{(4i)}^2}}}$
Simplify the above expression finding the squares of the terms-
$= \dfrac{{6 + 8i - 3i - 4{i^2}}}{{9 - 16{i^2}}}$
Now, by the value of imaginary term “I” its value is ${i^2} = - 1$
$\Rightarrow \dfrac{{6 + 5i - 4( - 1)}}{{9 - 16( - 1)}}$
Product of two negative terms is always positive
\[= \dfrac{{6 + 5i + 4}}{{9 + 16}}\]
Simplify the above terms, finding the like terms and then combine it
\[= \dfrac{{10 + 5i}}{{25}}\]
Find the common factors in the numerator and the denominator.
\[= \dfrac{{5(2 + i)}}{{5 \times 5}}\]
Common factors from the numerator and the denominator cancels each other.
\[= \dfrac{{(2 + i)}}{5}\]
This is the required solution.

Thus the required answer is \[\dfrac{{(2 + i)}}{5}\].

Note: The complex number consists of the real part and an imaginary part and is denoted by “Z”. It can be expressed as $z = a + ib$ where “a” is the real part and “b” is the imaginary part. An imaginary number is the complex number which can be written as the real number multiplied by the imaginary unit “i”. Be careful about the sign convention and remember that the product of two negative terms gives a positive term.