Question

In the complex numbers, where ${i^2} = - 1,$ what is the value of $5 + 6i$ multiplied by $3 - 2i?$
A) $27$
B) $27i$
C) $27 + 8i$
D) $15 + 8i$
E) $15 - 8i$

Answer 1

First of all we will find the product of two binomials and then apply ${i^2} = - 1,$ and then will simplify the expression for the resultant required value.

Complete step by step solution:
Given Expression: $5 + 6i$ multiplied by $3 - 2i$
Mathematically Expressed as $= (5 + 6i)(3 - 2i)$
Simplify the above expression finding the product of the terms in the above expression.
$= 5(3 - 2i) + 6i(3 - 2i)$
When there is a positive sign outside the bracket, then there is no change in the signs of the terms in the bracket.
$= 15 - 10i + 18i - 12{i^2}$
Now place ${i^2} = - 1,$ in the above equation –
$= 15 - \underline {10i + 18i} - 12( - 1)$
Simplify the like terms, when you subtract a smaller number from a bigger number the resultant value is positive. Also, the product of two negative terms gives the resultant value as the positive.
$= 15 + 8i + 12$
Find the simplified form of the two like terms.
$= 27 + 8i$
Hence, option (C) is the correct answer.

Note:
The complex number is defined as a number which consists of the real part and an imaginary part and is denoted by “Z”. It can be stated as $z = a + ib$ where “a” is the real part and “b” is the imaginary part. Also, be good in multiples and simplifications of the equation. Remembering the square of the negative terms also gives the positive values.

Also, go through the properties of the complex numbers which contents the following properties under addition and multiplication –

Closure Property
The commutative Property
The associative Property
The additive property
The additive inverse
The multiplicative inverse
Distributive property (Multiplication distributes over the addition)