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# Algebraic Expressions

# Commutative Properties

This rule for rearrangement is almost so obvious it isn't even worth mentioning. But, we are 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 in which you perform the summation or multiplication of terms in an expression, is one of the most basic ways we can rearrange an expression to find an equivalent 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 are not quite as easy-going and could probably stand to chillax a bit.

### Sample Problem

Consider the expression - 4*y*^{2}*x* + *x*^{3}. We can use the commutative property of addition to write - 4*y*^{2}*x* + *x*^{3} = *x*^{3} - 4*y*^{2}*x.*

We can then use the commutative property of multiplication to write - 4*y*^{2}*x* = - 4*xy*^{2}

Putting it together, we can rearrange - 4*y*^{2}*x* + *x*^{3} using commutativity to find that *x*^{3} = 4*xy*^{2}. We are 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.