Question

How do you simplify \[6( - 7 + 6i)( - 4 + 2i)\]?

Answer 1

Hint: Here this question we have to multiply these two terms. The both terms are in the form of complex numbers. The complex number is a combination of real part and complex part. If the terms in the braces then we multiply the terms. Hence we can determine a solution for the given question.

Complete step-by-step solution:
The numbers which are present in the question are in the form of complex numbers. The complex number is a combination of real part and imaginary part. The complex number is represented as \[(a \pm bi)\], here is a real part and the term involving i, is an imaginary part.

Now consider the given term \[6( - 7 + 6i)( - 4 + 2i)\]
Multiply first term that is present in the bracket with the number 6. So we have
 \[ \Rightarrow \left( {6( - 7) + 6(6i)} \right)( - 4 + 2i)\]
On multiplying we get
\[ \Rightarrow \left( { - 42 + 36i} \right)( - 4 + 2i)\]
On multiplying the two terms which are in the braces we get
\[ \Rightarrow \left( { - 42( - 4 + 2i) + 36i( - 4 + 2i)} \right)\]
On simplifying we get
\[ \Rightarrow 168 - 84i - 144i + 72{i^2}\]
In the complex number we know that \[{i^2} = - 1\]
Substituting this value in the above equation we have
\[ \Rightarrow 168 - 84i - 144i + 72( - 1)\]
On simplifying we get
\[ \Rightarrow 168 - 84i - 144i - 72\]
Adding the constant terms and the imaginary terms we get
\[ \Rightarrow 96 - 228i\]
Dividing the above equation by the number 12 we get
\[ \Rightarrow 8 - 19i\]
Hence we have multiplied the two terms and obtained the solution and it is in the form of a complex number.

Note: In mathematics we have different forms of numbers namely, natural number, whole number, integers, rational numbers, irrational numbers, real numbers, complex numbers. The arithmetic operations like addition, subtraction, multiplication and division are applied to these numbers. The numbers which are in the braces we apply the multiplication operation.