Question

How do you simplify $\left( {5 - 3i} \right)\left( {2 + 4i} \right)$?

Answer 1

Hint: In this question, we are given a complex expression and we have been asked to simplify the given expression. So, first simply open the brackets by multiplying the terms outside the bracket with the terms inside the bracket. Then, put the desired values and add or subtract the like terms. On simplification, you will get your answer.

Complete step-by-step solution:
We have been given a complex expression $\left( {5 - 3i} \right)\left( {2 + 4i} \right)$. Let us open the brackets and use distributive property.
$ \Rightarrow \left( {5 - 3i} \right)\left( {2 + 4i} \right)$ …...…. (given)
Opening the first bracket and using distributive property,
$ \Rightarrow 5\left( {2 + 4i} \right) - 3i\left( {2 + 4i} \right)$
Next step involves multiplying the terms outside the bracket with the terms inside the bracket.
$ \Rightarrow 10 + 20i - 6i - 12{i^2}$
Now, we know that $i = \sqrt { - 1} $
If we square this expression, we get, ${i^2} = {\left( {\sqrt { - 1} } \right)^2} = - 1$
Hence, putting ${i^2} = - 1$ in the expression,
$ \Rightarrow 10 + 20i - 6i - 12\left( { - 1} \right)$
Simplifying the expression by adding or subtracting the like terms,
$ \Rightarrow 10 + 20i - 6i + 12$
$ \Rightarrow 14i + 22$
Arranging the terms according to the standard form, which is ${\rm Z} = x + iy$.
$ \Rightarrow 22 + 14i$

Hence, $22 + 14i$ is the simplified expression.

Note: 1) Complex numbers are expressions in the form $x + iy$, where $x$ is the real part and $y$ is the imaginary part. These numbers cannot be marked on the number line. (note that the imaginary part is $y$, and not $iy$.)
2) What are like terms? Like terms are those terms which have the same variables or those terms which can be added or subtracted with each other. For example: i) $4x$ and $75x$ are like terms.
ii) $7xy$ and $9xy$ are like terms.
iii) $6y$ and $8x$ are not like terms as they cannot be added with each other. Such terms are also called unlike terms.
Basically, these terms are “like” each other.