Certainly! Let's go through the given equation step by step to determine which mathematical property it illustrates.
The equation provided is:
[tex]\[
3 + (5 + 7) = (3 + 5) + 7
\][/tex]
### Step-by-Step Breakdown:
1. Look at the left side of the equation:
[tex]\[
3 + (5 + 7)
\][/tex]
Here, you first add the numbers inside the parentheses:
[tex]\[
5 + 7 = 12
\][/tex]
Now the expression becomes:
[tex]\[
3 + 12 = 15
\][/tex]
2. Now look at the right side of the equation:
[tex]\[
(3 + 5) + 7
\][/tex]
Similarly, add the numbers inside the parentheses first:
[tex]\[
3 + 5 = 8
\][/tex]
Now the expression becomes:
[tex]\[
8 + 7 = 15
\][/tex]
Both sides of the equation result in 15, showing that:
[tex]\[
3 + (5 + 7) = (3 + 5) + 7
\][/tex]
This equation is true regardless of how the numbers are grouped.
### Associative Property
What we've demonstrated here is a key principle of the associative property of addition. The associative property states that how you group numbers when adding (or multiplying) them does not change their sum (or product). In other words:
[tex]\[
a + (b + c) = (a + b) + c
\][/tex]
This property is essential because it allows us to group numbers in different ways without affecting the final result.
### Other Properties (for clarification only)
- Commutative Property: This says that the order in which you add or multiply numbers does not change the result. For addition, [tex]\(a + b = b + a\)[/tex].
- Distributive Property: This indicates that multiplying a number by a sum is the same as doing each multiplication separately. For example, [tex]\(a \cdot (b + c) = a \cdot b + a \cdot c\)[/tex].
- Identity Property: For addition, this property states that any number plus zero is the number itself, [tex]\(a + 0 = a\)[/tex]. For multiplication, any number times one is the number itself, [tex]\(a \cdot 1 = a\)[/tex].
### Conclusion
The given equation:
[tex]\[
3 + (5 + 7) = (3 + 5) + 7
\][/tex]
illustrates the associative property of addition.
