To decide which properties are used in adding the given complex numbers [tex]\( (7 + 2i) + (4 + 3i) \)[/tex], let's look at the steps one by one.
1. Initial Expression:
[tex]\[
(7 + 2i) + (4 + 3i)
\][/tex]
2. Grouping Real and Imaginary Parts Together:
The expression can be rewritten by grouping the real parts together and the imaginary parts together:
[tex]\[
(7 + 4) + (2i + 3i)
\][/tex]
3. Adding the Real Parts and the Imaginary Parts:
Now, add the real parts and the imaginary parts:
[tex]\[
11 + 5i
\][/tex]
### Identifying the Properties Used:
#### Commutative Property:
The commutative property states that [tex]\( a + b = b + a \)[/tex]. In the context of complex numbers, it holds separately for both the real and imaginary parts. Here:
- [tex]\( 7 + 4 = 4 + 7 \)[/tex]
- [tex]\( 2i + 3i = 3i + 2i \)[/tex]
Since we freely rearranged the terms here, this indicates the use of the commutative property.
#### Associative Property:
The associative property states that [tex]\( (a + b) + c = a + (b + c) \)[/tex]. In our case, it means within the grouped parts, we can further group numbers:
For the real parts:
- [tex]\( (7 + 4) \)[/tex]
For the imaginary parts:
- [tex]\( (2i + 3i) \)[/tex]
By doing this grouping step without changing the order or the outcome, we apply the associative property.
### Conclusion:
The properties used to add the given complex numbers [tex]\( (7 + 2i) + (4 + 3i) \)[/tex] are the Commutative property and the Associative property:
- A. Commutative property
- C. Associative property
Thus, the correct answer is:
[tex]\[
(1, 3)
\][/tex]
