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# What are commutative, associative and distributive properties?

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The commutative, associative and distributive properties describe how basic mathematical operations work. The properties are helpful in finding efficient ways to solve equations and in simplifying algebraic expressions.

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Addition and multiplication have the commutative property, meaning that numbers can be added or multiplied together in any order without affecting the result. In other words, adding 7 + 3 is the same as adding 3 + 7. Multiplying 2 by 4.5 has the same outcome as multiplying 4.5 by 2. The commutative property for addition is expressed as a + b = b + a. The commutative property for multiplication is expressed as a * b = b * a.

Addition and multiplication also have the associative property, meaning that numbers can be added or multiplied in any grouping (or association) without affecting the result. For example, (3 + 2) + 7 has the same result as 3 + (2 + 7), while (4 * 2) * 5 has the same result as 4 * (2 * 5). The associative property for addition is expressed as (a + b) + c = a + (b + c). The associative property for multiplication is expressed as (a * b) * c = a * (b * c).

Only multiplication has the distributive property, which applies to expressions that multiply a number by a sum or difference. Multiplication distributes over addition because a(b + c) = ab + ac. For example, to multiply 2 by the sum of 9 + 4, the numbers 9 and 4 can be added first to find the sum of 13, and then 13 can be multiplied by 2 to return 26. A different way to achieve the same result is to distribute the 2 by first multiplying 2 by 9 (18) and 2 by 4 (8). The two results can be added (18 + 8) to return 26 as before. In the same way, multiplication distributes over subtraction because a(b – c) = ab – ac.

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