Question

How do you simplify \[\dfrac{{7i}}{{8 + i}}\]?

Answer 1

Hint: In order to simplify the above complex number, we have to multiply both denominator and numerator with the complex conjugate of the denominator .Simplifying further and apply formula$(A + B)(A - B) = {A^2} - {B^2}$in the denominator part and expand the numerator with the help of distributive law of multiplication. Replacing ${i^2}\,$with $ - 1$ and separating out the denominator, you will get your desired simplification.

Formula Used\[{(A + B)^2} = {A^2} + {B^2} + 2 \times A \times B\]
$(A + B)(A - B) = {A^2} - {B^2}$

Complete step-by-step solution:
Given a complex fraction \[\dfrac{{7i}}{{8 + i}}\].let it be z
Here i is the imaginary number
In order to simplify the given complex number, first we’ll find out the complex conjugate of denominator of z i.e. \[8 + i\] which will be \[8 - i\]
Now multiplying and dividing the equation with the complex conjugate of denominator
\[
  \Rightarrow \dfrac{{7i}}{{8 + i}} \times \dfrac{{8 - i}}{{8 - i}} \\
    \Rightarrow \dfrac{{7i\left( {8 - i} \right)}}{{\left( {8 + i} \right)\left( {8 - i} \right)}} \\
 \]
Now Using the formula $(A + B)(A - B) = {A^2} - {B^2}$ by considering A as 8 and B as I and using the distributive law which state that $A\left( {B + C} \right) = AB + AC$in the numerator, we get
\[ = \dfrac{{56i - 7{i^2}}}{{{8^2} - {i^2}}}\]
As we know that ${i^2} = - 1$.So Replacing ${i^2}\,$with $ - 1$in the above equation
\[
    \Rightarrow \dfrac{{56i - 7\left( { - 1} \right)}}{{64 - \left( { - 1} \right)}} \\
    \Rightarrow \dfrac{{56i + 7}}{{64 + 1}} \\
    \Rightarrow \dfrac{{56i + 7}}{{65}} \\
 \]
Converting the above question into standard form by comparing it with standard form $a + ib$
\[ \Rightarrow \dfrac{7}{{65}} + \dfrac{{56i}}{{65}}\]
Where Real number is \[\dfrac{7}{{65}}\] and imaginary number is \[\dfrac{{56i}}{{65}}\]
Hence we have successfully simplified our complex function.
Therefore, our required answer is \[\dfrac{7}{{65}} + \dfrac{{56i}}{{65}}\]

Note:
1. Real Number: Any number which is available in a number system, for example, positive, negative, zero, whole number, discerning, unreasonable, parts, and so forth are Real numbers. For instance: 12, - 45, 0, 1/7, 2.8, √5, and so forth, are all the real numbers.
2. A Complex number is a number which are expressed in the form $a + ib$ where $ib$ is the imaginary part and $a$ is the real number .i is generally known by the name iota. \[\]
or in simple words complex numbers are the combination of a real number and an imaginary number .
3. Complex numbers are very useful in representing periodic motion like water waves, light waves and current and many more things which depend on sine or cosine waves.
4.The Addition or multiplication of any 2 conjugate complex numbers always gives an answer which is a real number.