Question

How do you write the following expression in standard form \[\dfrac{{1 + i}}{i} - \dfrac{3}{{4 - i}}\]?

Answer 1

Hint: Here we will firstly take the LCM of both the fraction and solve it to get a simplified fraction. Then we will simply multiply and divide the simplified fraction with the conjugate of the denominator of the fraction and solve it to get the standard form of the given equation.

Complete Step by Step Solution:
Given equation is \[\dfrac{{1 + i}}{i} - \dfrac{3}{{4 - i}}\].
Firstly we will take the LCM of the both the fraction in the given equation. Therefore, we get
\[ \dfrac{{\left( {1 + i} \right)\left( {4 - i} \right) - 3i}}{{i\left( {4 - i} \right)}}\]
Now we will simplify the above equation, we get
\[ \dfrac{{4 - i + 4i - {i^2} - 3i}}{{4i - {i^2}}}\]
\[ \dfrac{{4 - {i^2}}}{{4i - {i^2}}}\]
We know that the value of \[{i^2}\] is equal to \[ - 1\]. Therefore, we will put the value in the above equation, we get
\[ \dfrac{{4 - \left( { - 1} \right)}}{{4i - \left( { - 1} \right)}}\]
\[ \dfrac{5}{{1 + 4i}}\]
Now we will simply multiply and divide above the equation with the conjugate of the denominator of the fraction i.e. \[1 - 4i\]. Therefore, we get
\[ \dfrac{5}{{1 + 4i}} \times \dfrac{{1 - 4i}}{{1 - 4i}}\]
We know this algebraic property i.e. \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\]. Therefore, by using this we get
\[ \dfrac{{5 - 20i}}{{{1^2} - {{\left( {4i} \right)}^2}}}\]
\[ \dfrac{{5 - 20i}}{{1 - 16{i^2}}}\]
Now we again put the value of \[{i^2}\] in the above equation, we get
\[ \dfrac{{5 - 20i}}{{1 + 16}}\]
\[ \dfrac{{5 - 20i}}{{17}}\]
Now we will write it in the split form. Therefore, we get
\[ \dfrac{{1 + i}}{i} - \dfrac{3}{{4 - i}} = \dfrac{5}{{17}} - \dfrac{{20i}}{{17}}\]

Hence, the expression \[\dfrac{{1 + i}}{i} - \dfrac{3}{{4 - i}}\] in the standard form is written as \[\dfrac{5}{{17}} - \dfrac{{20i}}{{17}}\].

Note:
We should know the standard form of writing an imaginary number. Imaginary numbers are generally written in the form of \[a + ib\] where \[a,b\] are the real numbers. In the imaginary number, the first term is the real part of the number and the second term is the imaginary part of the number.
Real value is a number which has some real or discrete or possible value. But imaginary value is the number with a real number multiplied with an imaginary part \[i\].