Two expressions are said to be *equivalent* if they produce the same number for any possible value of the variable. We'll bet you want to see some examples of this stuff in action. Don't ask how we knew. We had a feeling.

1. The expressions x and *x* + 0 are equivalent, since for any value of *x*, *x* and *x* + 0 are the same thing. Sorry, zero, but you're pretty worthless.

One way of expressing the fact that these two expressions are equivalent is to write an equation: *x* = *x* + 0. Any time you have an equation, the two expressions on either side of the equals sign are equivalent. That's kinda the equals sign's whole deal. It doesn't appreciate it when you write something like 1 + 1 = 3. Drives it *batty*.

2. The expressions *x*^{2} and (-*x*)^{2} are equivalent, since for any value of x we will get the same value out of either expression. If *x* is 2, we find that 2^{2} = 4 from the first expression and (- 2)^{2} = 4 from the second expression. If *x* is 34,792, we get...oh, well, you probably got it from the first example. No need to wear ourselves out.

The goal is to create a toolbox of allowable ways to modify an expression to an equivalent expression. We are not, of course, speaking of a literal toolbox. You will never receive an emergency call to hop over and fix a plumbing leak with your trusty ratcheting box variable. Rather, these *allowable* moves are the rules we have for working with algebraic expressions. Once we finish developing this (virtual) toolbox, we will use our tools to solve equations and more. Okay, maybe the occasional leaky faucet, but that is where we draw the line.

There are several different things we can do to rearrange expressions. We have divided them into the following list of allowable moves:

1. Commuting

2. Associating

3. Distribution

4. Factorization

5. Combining like terms

6. Getting rid of parentheses

Use any moves that are not contained within the above "toolbox," and you might draw a penalty flag. Incorrect Rearrangement of Expressions: 15 yards and loss of down.

Using combinations of the allowable moves together to rearrange an expression into an equivalent expression is called *simplification*. A sentence such as the preceding one, however, which contains 8 words of at least 8 letters in length, has probably *not* been simplified.

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