By Kathryn White | Updated Aug 30, 2022
Associative properties—alongside commutative and distributive properties—form the backbone of algebraic manipulation. They enable you to regroup terms without altering the result, making equations easier to solve and everyday calculations more intuitive.
The associative property of addition lets you regroup numbers in a sum. For example, (3 + 4 + 5) + (7 + 6) can be rewritten as (3 + 4) + (5 + 7 + 6). Calculating inside the parentheses first confirms both expressions equal 25.
Similarly, the associative property of multiplication allows you to regroup factors. (15 × 2)(3 × 4)(6 × 2) can become (15 × 2 × 3)(4 × 6 × 2) and still produce the same product. It also applies to variables: 4(3X) can be written as (4 × 3)X = 12X.
Strictly speaking, subtraction is not associative. However, by rewriting subtraction as addition of a negative number, you can apply the associative property of addition. For instance: (3X – 4X) + (13X – 2X – 6X) becomes (3X + (–4X)) + (13X + (–2X) + (–6X)), which can be regrouped to (3X + (–4X) + 13X) + ((–2X) + 6X). Note that this technique fails when the subtraction sign sits between parentheses—there, the distributive property is needed.
Division lacks an associative property. To regroup expressions, rewrite division as multiplication by a reciprocal. For example: (5 × 7/3)(3/4 × 6) becomes (5 × 7 × 1/3)(3 × 1/4 × 6), which can then be regrouped as (5 × 7)(1/3 × 3 × 1/4 × 6). This method also fails if a division sign lies between parentheses.
