Question

How do you multiply \[( - 7 - 4i)( - 6 - 6i)\] ?

Answer 1

Hint: To solve this question, we need to multiply these terms in the same way we multiply algebraic expressions. We have to multiply every part of a term with every part of the other term, and write it down as a single expression. Next, we need to use the standard identities for solving exponents of iota, and obtain the final expression.

Complete step by step solution:
To multiply \[( - 7 - 4i)\] with \[( - 6 - 6i)\] , we will follow the same procedure that is performed for two algebraic expressions, multiply every term in a bracket with every term of the other bracket.
Starting with \[ - 7\] , we have.
  \[
   - 7 \times - 6 = 42 \\
   - 7 \times - 6i = 42i \;
 \]
Now, moving on to \[ - 4i\] , we have,
  \[
   - 4i \times - 6 = 24i \\
   - 4i \times - 6i = 24{i^2} = - 24 \;
 \]
Here, \[24{i^2} = - 24\] because \[{i^2} = {\left( {\sqrt { - 1} } \right)^2} = - 1\] .
Thus, now that we have obtained all possible products, we will now add them up to obtain the final expression.
On adding these terms, we get,
  \[42 + 42i + 24i - 24\]
On Subtracting \[24\] from \[42\] , we will get \[18\] .
On adding \[42i\] and \[24i\] , we will obtain \[66i\] .
Thus, the final expression obtained is \[18 + 66i\] . This is the product of \[( - 7 - 4i)\] with \[( - 6 - 6i)\] .
So, the correct answer is “\[18 + 66i\]”.

Note: Complex numbers are the numbers which have two parts — a real number and an imaginary number. These are the building blocks of more intricate math, such as algebra, and can be applied to many aspects of real life.
The standard format for complex numbers is a + bi, where a is the real number first and b is the imaginary number. Since either a or b or both could be 0, technically any real number or imaginary number can be considered a complex number.
Real numbers are the values that can be plotted on a horizontal number line, such as fractions, decimals or integers. Imaginary numbers are abstract concepts which are used when you need the square root of a negative number.