The associative property is stating that when we add or multiply a series of numbers, it actually doesn’t matter how these terms are ordered.
Keep in mind that the first step when we use the order of operations is to simplify within the parentheses.
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The associative property of addition is stating that numbers in any addition may be grouped in any different way without changing their sum.
Below you can see two different ways of simplifying the same addition. In our first example, 4 (four) is grouped with 5 (five), and 4 and 5 = 9.
4 + 5 + 6 = 9 + 6 = 15
Then, the same problem has been worked by grouping first 5 and 6, and 5 + 6 = 11.
4 + 5 + 6 = 4 + 11 = 15
In both ways, we come up with the same sum. This illustrating that when we’re adding and change the grouping of numbers, we’ll end up with the same sum.
Mathematicians are often using parentheses for the indication of which operation should come first in algebraic equations. Look at the rewritten addition problems listed above. This time, we’ve used parentheses to indicate associative grouping:
(4 + 5) + 6 = 9 + 6 = 15
4 + (5 + 6) = 4 + 11 = 15
Obviously, the parentheses are not affecting the sum. Our sum will be the same regardless of where we place the parentheses.
Associative Property of Addition:
For real numbers a, b, c counts that (a + b) + c equals a + (b + c).
The associative property of multiplication is working exactly like with addition. With multiplication, the associative property states that numbers in multiplication expressions may be regrouped by using parentheses.
Associative Property of Multiplication:
For real numbers a, b, c counts that (a • b) • c equals a • (b • c).
We will find that associative and commutative properties are very helpful tools when we do algebra, especially when evaluating expressions.
When we use the commutative and associative properties, we can reorder terms in expressions in a way that compatible numbers are placed next to each other and are grouped together.