Question

How do you simplify \[\left( {6 + 2i} \right)\left( {5 - 3i} \right)\] ?

Answer 1

Hint: In this question we are asked to simplify the expression which is of complex numbers, first we make use of the distributive property, i.e., Multiply each term in the first binomial with each term in the second binomial using Foil method, and simplify the expression and remember that, then combine all like terms i.e., combine real numbers with real numbers and imaginary numbers with imaginary numbers.

Complete step-by-step answer:
Given expression is \[\left( {6 + 2i} \right)\left( {5 - 3i} \right)\],
Now using the FOIL method, in foil method we first multiply the first terms, then the outer terms, then the inner terms and finally the last terms.
Now applying foil method we get,
$\Rightarrow 6 \times 5+6 \times (-3i)+2i \times 5+2i \times (3i)$
Now multiplying the terms we get,
\[ \Rightarrow 30 - 18i + 10i - 6{i^2}\],
 We know that\[{i^2} = - 1\], and substituting the value of, we get,
\[ \Rightarrow 30 - 8i - 6\left( { - 1} \right)\],
Now adding the like terms we get, i.e., add real numbers with real numbers and imaginary numbers with imaginary numbers and simplifying we get,
\[ \Rightarrow 30 - 8i + 6\],
So, further simplification we get,
\[ \Rightarrow 36 - 8i\].
So, the simplified form of the given expression is\[36 - 8i\].

\[\therefore \]The simplified form of the given expression \[\left( {6 + 2i} \right)\left( {5 - 3i} \right)\],

is \[36 - 8i\].

Note:
Complex numbers are combination of two types of numbers i.e., real numbers and imaginary numbers, and they are defined by the inclusion of the term, the general form for a complex number is defined by,
\[z = a + ib\], where \[z\] is the complex number, a is any real number and \[b\] is the imaginary part of the complex number, both of which can be positive or negative. The complex numbers cannot be marked on a number line.