difference between commutative and distributive property
Answers
Commutative Property
The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.
Use the Commutative Property to restate "3×4×x" in at least two ways.
They want me to move stuff around, not simplify. In other words, my answer should not be "12x"; the answer instead can be any two of the following:
4 × 3 × x
4 × x × 3
3 × x × 4
x × 3 × 4
x × 4 × 3
Distributive Property
The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, for example, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation used the Distributive Property.
Why is the following true? 2(x + y) = 2x + 2y
Since they distributed through the parentheses, this is true by the Distributive Property.
Use the Distributive Property to rearrange: 4x – 8
The Distributive Property either takes something through a parentheses or else factors something out. Since there aren't any parentheses to go into, you must need to factor out of. Then the answer is:
By the Distributive Property, 4x – 8 = 4(x – 2).
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"But wait!" I hear you cry; "the Distributive Property says multiplication distributes over addition, not over subtraction! What gives?" You make a good point. This is one of those times when it's best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number ("x – 2") or else as the addition of a negative number ("x + (–2)"). In the latter case, it's easy to see that the Distributive Property applies, because you're still adding; you're just adding a negative.
The other two properties come in two versions each: one for addition and the other for multiplication. (Yes, the Distributive Property refers to both addition and multiplication, too, but it refers to both of the operations within just the one rule.)