Question

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

Answer 1

Hint: To solve the given question, we need to know the expansion of the expression \[\left( a+b \right)\left( c+d \right)\]. The expression \[\left( a+b \right)\left( c+d \right)\] is expanded by multiplying each term of the first bracket with each term of the second bracket and then adding their products. Algebraically it is expressed as, \[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\]. We will simplify the given expression using this expansion, to do this we need to substitute the values of the variables in the expansion, and evaluate the expression.

Complete step by step solution:
We are asked to simplify the expression \[\left( 5-7i \right)\left( -4-3i \right)\]. This expression of the form \[\left( a+b \right)\left( c+d \right)\]. We know that this expression is expanded by multiplying each term of the first bracket with each term of the second bracket and then adding their products. Algebraically it is expressed as, \[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\].
Here, we have \[a=5,c=-4,b=-7i\And d=-3i\]
Substituting these values in the expansion of the general expression, we get
\[\begin{align}
  & \Rightarrow \left( 5-7i \right)\left( -4-3i \right) \\
 & \Rightarrow -20-15i+28i+21{{i}^{2}} \\
\end{align}\]
Simplifying the above expression, we get
\[\Rightarrow -20+13i+21{{i}^{2}}\]
Here \[i\] is a complex number, and \[i=\sqrt{-1}\]. Hence, its square equals \[-1\]. Substituting this value in the above expression, and simplifying the expression we get
\[\begin{align}
  & \Rightarrow -20+13i-21 \\
 & \Rightarrow -41+13i \\
\end{align}\]
Hence, the simplification of the expression \[\left( 5-7i \right)\left( -4-3i \right)\] is \[-41+13i\].

Note: To solve these types of questions, we must know different types of expressions and their expansions. Here, we used the expansion \[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\], and substitute the values for each variable. One should avoid calculation mistakes in these questions, while simplifying these expressions.
The other expressions we might need are \[{{\left( a+b \right)}^{2}}\] its expansion is \[{{a}^{2}}+{{b}^{2}}+2ab\], \[\left( a+b \right)\left( a-b \right)\] its expansion are \[{{a}^{2}}-{{b}^{2}}\].