To determine which of the given expressions demonstrates the associative property, it's important to understand what the associative property is. The associative property states that when adding or multiplying, the way in which numbers are grouped does not affect the sum or product.
Let’s analyze each option to check if it follows the associative property:
1. [tex]\( 2(5-3) = 10 - 6 \)[/tex]
- Here, the left-hand side simplifies to [tex]\( 2 \cdot 2 = 4 \)[/tex].
- The right-hand side simplifies to [tex]\( 4 \)[/tex].
- Both sides are equal, but this is not an example of the associative property because it involves subtraction and multiplication rather than just addition or multiplication.
2. [tex]\( \frac{1}{3} \cdot 3 = 1 \)[/tex]
- Simplifying the left-hand side, [tex]\( \frac{1}{3} \cdot 3 = 1 \)[/tex].
- The right-hand side is already [tex]\( 1 \)[/tex].
- Again, both sides are equal, but this expression represents the property of multiplicative inverses and not the associative property.
3. [tex]\( -7 + (19 + 5) = (-7 + 19) + 5 \)[/tex]
- Simplify inside the parentheses first:
[tex]\[
-7 + (19 + 5) = -7 + 24 = 17
\][/tex]
- Now for the right-hand side:
[tex]\[
(-7 + 19) + 5 = 12 + 5 = 17
\][/tex]
- Both sides are equal, and this indeed demonstrates the associative property of addition, which shows that the grouping of additions does not affect the result.
4. [tex]\( 11 + (6 + 8) = 11 + (28 + 6) \)[/tex]
- Simplify the expressions inside both sets of parentheses:
[tex]\[
11 + (6 + 8) = 11 + 14 = 25
\][/tex]
[tex]\[
11 + (28 + 6) = 11 + 34 = 45
\][/tex]
- The two sides are not equal, and this is not an example of any known mathematical property.
After reviewing all the options, the expression that correctly demonstrates the associative property is:
[tex]\[
-7 + (19 + 5) = (-7 + 19) + 5
\][/tex]
So, the correct choice is the third option.
