By Christopher Danielson

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

image0.png

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

image1.png

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

image2.png

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

image3.png

is twice as much as

image4.png

Finally, you could think

image5.png

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

It’s the same with exponents:

image6.png

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

image7.png

by using repeated addition, you can compute

image8.png

using repeated multiplication:

image9.png

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

image10.png

to get

image11.png

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

image12.png

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:

image13.png

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

image14.png

by thinking of

image15.png

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

image16.png

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

image17.png

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

image18.png

by the first rule. Then

image19.png

has to be the reciprocal of

image20.png

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:

image21.png

and it’s useful to write it as

image22.png

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