# Exponents in Common Core Math

Common Core math students start to work with exponents in eighth grade. In algebra, you can think of *exponentiation* as repeated multiplication. The following analogy will help you understand the significance of this.

You know that

because there are 12 things in 4 groups of 3. If you didn’t *know* the product

you could find it in several ways. You could lay out 4 groups of 3 things and count them one by one, for example. Or you could use the *associative property* of multiplication, which means that to find

you can either multiply *a* and *b* first, or you can multiply *b* and *c* first — the final product is the same either way. Using the associative property, you could think of

is twice as much as

Finally, you could think

That is, one way to compute products is by using repeated addition.

It’s the same with exponents:

From 4 and 3, you compute a third number, 64. Just as you can compute

by using repeated addition, you can compute

using repeated multiplication:

But there are other ways too, and these ways depend on properties of exponentiation as an operation. You can double

to get

using the associative property of multiplication, and properties of exponentiation allow you to relate

These properties are known as *rules for operating with exponents*.

Three major rules appear in eighth grade. In the following statements, A is presumed to be a positive number:

You can understand these rules better by way of examples. You can see the first rule, that

by thinking of

Six threes are multiplied. The second rule you can see by thinking about

Eight threes are multiplied together. The third rule is the logical consequence of the first rule, and of the fact that

when A is any positive number. Here’s why:

by the first rule. Then

has to be the reciprocal of

Each of these rules is useful going in both directions. You don’t have to view these equations as machines that transform the left-hand side into the right-hand side. Instead, each side of each equation has the same value as the other side. Sometimes you have something that looks like this:

and it’s useful to write it as

Sometimes it goes the other way around. What matters is the equivalence — or *sameness* — of both sides of each equation.