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Commutative Property

This rule for rearrangement is almost so obvious it isn't even worth mentioning. But we're suckers for overkill, so here goes. The commutative property says that the order in which we add or multiply numbers doesn't matter.

x + y = y + x

xy = yx

This move, changing the order of our terms when we're adding or multiplying, is one of the most basic ways we can rearrange an expression. Note that we don't say you can also do this move with subtraction or division. That's because you can't. Addition and multiplication are a little more go-with-the-flow; subtraction and division aren't quite as easygoing and could probably stand to chillax a bit.

Sample Problem

Consider the expression -4y2x + x3. We can use the commutative property of addition to rewrite the whole thing as x3 – 4y2x.

We can also use the commutative property of multiplication to rewrite -4y2x as -4xy2.

Putting it together, we can rearrange -4y2x + x3 using commutativity (and yes, we did make that word up) to get x3 – 4xy2. We're left with an expression that doesn't feature any fewer terms than the original, but at least it doesn't start out with a negative sign. That really chaps our patootie.

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