What Are Exponents?
Question
An exponent is a number that tells you how many times you're going to multiply the
Hi, I'm Rebecca Pierce for About.com and today we're going to be talking about exponents.
Exponents Involve Multiplication of the Base

So as an example, the 2 is what you'd call your base and the four that appears in the top right hand corner, or as a superscript, is what you would call your exponent. What an exponent actually is is a number that tells you how many times you're going to multiply the base. And every time you want to multiply the base by itself.
Examples of Exponents

Let's say that you have two to the first power. So that's one two which is equal to two. If you have two to the second power, that's equal to two times two, or two, two times, which is equal to four.

And if you have two to the third power, that's equal to two times two times two, or two three times, which is equal to eight.

Your base could also be a variable, as you might see in algebra. So, if you had x to the fourth, you would simply have x times x times x times x.
Rules for Multiplying Exponents

So now let's look at what you have to do if you have two terms with exponents that are multiplied or divided by each other. Let's start with two to the third power times two to the second power.

Let's start with the two to the third, which is equal to two times two times two. So we put that on the right hand side of our equation. Now we bring over the multiplication sign on the right hand. And then we have to look at the two squared, which we know to be two times two, so we put that at the end on the right hand side.

Now, on this whole side, we have two times two times two times two times two, which is equal to two to the fifth. So if we go back to two to the third power times two to the second power, we can see that two to the fifth is simply two to the three plus two.

From here we get our general term, which is two to the a times two to the b equals two to the a plus b.
Rules for Dividing Exponents

For our division example, let's use the same numbers. Let's say that we have two to the third divided by two to the second. And let's write this as a fraction. We're again going to start with two to the third, which we know to be two times two times two.

So we go ahead and put that on the right hand side of our equation. We bring over the division sign as a fraction. And then we'll put this two squared as two times two on the bottom of the fraction. Now we have three twos on the top and two twos on the bottom, so we can go ahead and cancel some of those twos. We can cancel these and we can cancel these.

And we're left with this one two over here, which means that two to the third divided by two squared is simply two; which is also equal to to two to the three minus two. So for our general term we get that two to the a divided by two to the b is equal to two to the a minus b.
Rules for Multiple Exponents

Now we're going to look at what to do if you have multiple exponents. Let's say that we have two to the third power all to the second power, which means that three is our first exponent and two is our second exponent.

So we say that this is equal to two to the three times two. And we know that three times two is equal to six, so this whole term is equal to two to the sixth. For our general term we'll get two to the a all to the b is equal to two to the a times b.