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In this resource, you will be simplifying expressions. A typical question you will see is something like this.

Simplify the expression 6y − 3(5x + 2y) − 10x.

The official expectation in mathematics would read:

The student uses the Interactive popup. Assistance may be required. commutative, Interactive popup. Assistance may be required. associative, and Interactive popup. Assistance may be required. distributive properties to simplify algebraic expressions.

For more information about these properties, click each link above.

Commutative Property

Is 3 plus 2 the same as 2 plus 3?
Is 8 times 5 the same as 5 times 8?
How about a + b? Is this the same as b + a?
Of course, they are!

This is the Commutative Property! The Commutative Property works for addition and multiplication. Numbers and variables in expressions work like commuters in a car going to work, they all go to the same place so it does not matter who sits in front and who sits in back because they all go to work at the same time.

However, Commutative Property does not work for subtraction and division. Think about it. Does 7 minus 3 equal 3 minus 7? NO. How about 15 divided by 3? Is that the same as 3 divided by 15?
NO.

Associative Property

Is 4 + (1 + 3) the same as (4 + 1) + 3?
Is 6 × (7 × 2) the same as (6 × 7) × 2?
Is 2a + (3a + a) the same as (2a + 3a) + a?
Of course, they are!

These are examples of the Associative Property. Numbers and variables are like associates at a work place. It does not matter which numbers and variables are associated in the group first, they are all still added or multiplied.

Just like the Commutative Property, the Associative Property does not work for subtraction and division. Think about it.
Does 7 − (5 − 4) equal (7 − 5) − 4?
NO.

Distributive Property

Is 4(3 + 2) equal to 4(3) + 4(2)?
Is -2(a + 3) equal to -2(a) + -2 (3)?
Of course, they are!

These are examples of the Distributive Property. Numbers and variables on the outside of a parentheses have to be distributed to the numbers or variables inside the parentheses. Sometimes it is said that the Distribution Property distributes multiplication over addition.

The signs of the numbers and variables can be negative or positive so it works with subtraction, also. And since dividing by a number or a variable is the same as multiplying by the reciprocal, it also works for division. Think about it.

Is 5(6 − 2) equal to 5(6) − 5(2)? YES
Is (12 + 4) divided by 2 the same as 1 over 2 1 2 (12 + 4)?
And is that the same as 1 over 2 1 2 (12) + 1 over 2 1 2 (4)? YES