Using the any-order property of multiplication

Together, the commutative and associative laws give the any-order property of multiplication.

A list of whole numbers can be multiplied two at a time in any order to give the same result.

The any-order property of multiplication can be used to simplify calculations by changing the order in which we perform multiplications. For example:

2 × 2 × 2 × 5 × 5 × 5	= (2 × 5) × (2 × 5) × (2 × 5)
	= 10 × 10 × 10
2 × 17 × 5 = 10 × 17
25 × 7 × 4 = 25 × 4 × 7 = 100 × 7.

Sometimes this reordering happens after we decompose one of the factors. For example:

• 17 × 4 = (17 × 2) × 2 = 34 × 2 = 68
• 36 × 5 = 18 × (2 × 5) = 180