To add the given complex numbers [tex]\((7 + 2i) + (4 + 3i)\)[/tex] and determine which properties are used, let's break down the process step by step:
1. Definition of Complex Numbers:
Each complex number is composed of a real part and an imaginary part. So, [tex]\(7 + 2i\)[/tex] is a complex number where 7 is the real part and [tex]\(2i\)[/tex] is the imaginary part. Similarly, [tex]\(4 + 3i\)[/tex] consists of a real part 4 and an imaginary part [tex]\(3i\)[/tex].
2. Grouping Real and Imaginary Parts:
The expression [tex]\((7 + 2i) + (4 + 3i)\)[/tex] can be rewritten by grouping the real parts together and the imaginary parts together:
[tex]\[
(7 + 4) + (2i + 3i)
\][/tex]
3. Simplifying Each Group:
Adding the real parts together:
[tex]\[
7 + 4 = 11
\][/tex]
Adding the imaginary parts together:
[tex]\[
2i + 3i = 5i
\][/tex]
Therefore, the sum of the complex numbers is:
[tex]\[
11 + 5i
\][/tex]
Next, let’s analyze which properties are used during this addition:
- Commutative Property:
This property states that the order in which numbers are added does not change the sum. In this context, it assures us that grouping [tex]\((7 + 4) + (2i + 3i)\)[/tex] is valid regardless of the order in which we combine terms.
- Associative Property:
This property states that the way in which numbers are grouped in addition does not change the sum. In this context, it allows us to regroup the addition inside the parentheses: [tex]\((7 + 4) + (2i + 3i)\)[/tex].
Therefore, the properties used in adding these complex numbers are the Commutative Property and the Associative Property. The correct answers are:
- A. Commutative Property
- C. Associative Property
Hence, the answer is:
[tex]\[
[\text{1, 3}]
\][/tex]
