What is the difference between the commutative distributive and associative properties when multiplying?

The associative rule is a rule about when it's safe to move parentheses around. You can remember that because the parentheses determine which expressions you have to do first--which numbers can associate with each other. It looks like this:

For addition: (a + b) + c = a + (b + c)
For multiplication: (ab)c = a(bc)

The commutative property is about which operations you can do backward and forward. You can remember this by thinking of people commuting to work: they go to work every morning, then they repeat the same operation backward when they commute home. It looks like this:

For addition: a + b = b + a
For multiplication: ab = ba

Finally, the distributive property tells you what happens when you distribute one operation against another kind in parentheses. It looks like this:

a * (b + c) = ab + ac

In other words, the a is "distributed" over the b and c.

Of course, you can make these work together:

a * (b + (c + d))
= a * ((b + c) + d) (by the associative property)
= a * (d + (b + c)) (by the commutative property)
= ad + a (b + c) (by the distributive property)
= ad + ab + ac (by the distributive property again).

Hope this helps.

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