Question

How do you simplify \[\left( -5-4i \right)\left( 3-7i \right)\]?

Answer 1

Hint: This question belongs to the topic of complex numbers. In solving this question, we are going to use a foil method. In this question, we will know about the term iota. We will also know about the value of the square of iota. We will first use the foil method. After using that method, we will solve the further equation. After solving, we will get our answer.

Complete step by step solution:
Let us solve this question.
In this question, we have asked to simplify the term \[\left( -5-4i \right)\left( 3-7i \right)\].
Before solving this question, let us first know about the term iota.
The term iota is written as the square root of negative one. The symbol of iota is \[i\]. We can write iota in mathematical form as:
\[i=\sqrt{-1}\]
The square of iota can be written as
\[{{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1\]
Now, let us know about the foil method. The foil method says that the term \[\left( a+b \right)\left( c+d \right)\] can also be written as \[ac+ad+bc+bd\].
Now, let us simplify the term \[\left( -5-4i \right)\left( 3-7i \right)\].
Using foil method, we can write the term \[\left( -5-4i \right)\left( 3-7i \right)\] as
\[\left( -5-4i \right)\left( 3-7i \right)=\left( -5 \right)\times 3+\left( -5 \right)\times \left( -7i \right)+\left( -4i \right)\times 3+\left( -4i \right)\times \left( -7i \right)\]
The above equation can also be written as
\[\Rightarrow \left( -5-4i \right)\left( 3-7i \right)=-15+35i-12i+28{{i}^{2}}\]
The above equation can also be written as
\[\Rightarrow \left( -5-4i \right)\left( 3-7i \right)=-15+23i+28{{i}^{2}}\]
As we know that the square of iota is -1. So, we can write
\[\Rightarrow \left( -5-4i \right)\left( 3-7i \right)=-15+23i-28\]
The above equation can also be written as
\[\Rightarrow \left( -5-4i \right)\left( 3-7i \right)=-43+23i\]

Hence, we have solved this question. And, we get that the simplified value of \[\left( -5-4i \right)\left( 3-7i \right)\] is \[-43+23i\].

Note: As we can see that this question is from the topic of complex numbers, so we should have a better knowledge in the topic of pre-calculus and complex numbers. We should know about the term iota. Remember the following values of iota:
\[i=\sqrt{-1}\]
\[{{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1\]
And also remember about the foil method. The foil method says that \[\left( a+b \right)\left( c+d \right)\] can also be written as \[ac+ad+bc+bd\], or we can write this in mathematical form as
\[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\]