Question

How do you divide \[\dfrac{4+3i}{7+i}\]?

Answer 1

Hint: In this problem, we have to divide the given fraction which is in complex form. We know that we should not have any imaginary terms in the denominator, i.e. The conjugate of the denominator \[7+i\] is \[7-i\], so we can take the complex conjugate of the denominator to be multiplied to the both numerator and the denominator to get a simplified form.

Complete step-by-step solution:
We know that the given fraction is,
\[\dfrac{4+3i}{7+i}\]
We can now find the complex conjugate of the denominator by changing the sign in the imaginary part.
The conjugate of the denominator \[7+i\] is \[7-i\].
We can now multiply the complex conjugate in both the numerator and the denominator, we get
\[\Rightarrow \dfrac{4+3i}{7+i}\times \dfrac{\left( 7-i \right)}{\left( 7-i \right)}\]
We can now multiply every term in both the numerator and the denominator using the FOIL method, we get
\[\Rightarrow \dfrac{28+21i-4i-3{{i}^{2}}}{49+7i-7i-{{i}^{2}}}\]
Now we can simplify the above step by cancelling and adding or subtracting similar terms,
\[\Rightarrow \dfrac{28+17i-3{{i}^{2}}}{49-{{i}^{2}}}\]
 We also know that in complex numbers,
\[{{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1\]
We can substitute the above value in the above step, we get
\[\Rightarrow \dfrac{28+17i-\left( -3 \right)}{49-\left( -1 \right)}\]
Now we can simplify the above step, we get
\[\Rightarrow \dfrac{31+17i}{50}\]
We can see that we have two terms in the numerator with one denominator, we can separate it, we get
\[\Rightarrow \dfrac{31}{50}+\dfrac{17}{50}i\]
Therefore, the answer is \[\dfrac{31}{50}+\dfrac{17}{50}i\].

Note: Students make mistakes While taking complex conjugate for the complex number. We should always remember that the complex conjugate of the number is changing the sign to its opposite one in the imaginary part. We should always try to provide the simplest form result by simplifying it.