How to Apply the Associative Property
Order of Operations in Algebra
Permutations When Order Matters

How to Reorder Operations with the Commutative Property

The commutative property makes working with algebraic expressions easier. The commutative property changes the order of some numbers in an operation to make the work tidier or more convenient — all without affecting the result.

You can use the commutative property with addition and multiplication operations, but not subtraction or division (with a few exceptions):

  • Addition: a + b = b + a

    Example: 4 + 5 = 9 and 5 + 4 = 9, so 4 + 5 = 5 + 4

    Reordering the numbers doesn't affect the result. Regardless of the order of the numbers, the answer is still 9.

  • Multiplication: a × b = b × a

    Example: 3 × (–7) = –21 and (–7) × 3 = –21, so 3 × (–7) = (–7) × 3

    Just as with the addition example, reordering the numbers when multiplying doesn't affect the result. Regardless of the order of the numbers, the answer is still –21.

  • Subtraction: abba (except in a few special cases)

    Example: (–5) – (+2) = (–7) and (+2) – (–5) = +7, so (–5) – (+2) ≠ (+2) – (–5)

    Here, you see how subtraction doesn't follow the commutative property.

    Exception: If a and b are the same number, then the subtraction appears to be commutative because switching the order doesn’t change the answer.

    Example: 2 – 2 = 0 and –2 + 2 = 0, so 2 – 2 = –2 + 2

  • Division: a ÷ bb ÷ a (except in a few special cases)

    Example: (−6) ÷ (+1) = −6 and (+1) ÷ (−6) = −1/6, so (−6) ÷ (+1) ≠ (+1) ÷ (−6)

    Division also doesn't follow the commutative property.

    Exception: If a and b are opposites, then you get –1 no matter which order you divide them in.

    Example: 2 ÷ (–2) = –1 and –2 ÷ 2 = –1, so 2 ÷ (–2) = –2 ÷ 2

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