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Introduction to Exponents

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Today we're going to focus a little bit more on exponents. Exponents are used when you're talking about things that get big really quickly, like the spread of disease, or get small very quickly, like the value of your car. So we'll talk a little bit about what exactly an exponent is, and then we'll do a few examples to see how they work.

So let's look at exactly what an exponent is used for. Behind me I've got this long expression-- 3 times 3 times 3 times 3 times 3 times 3-- I've got eight 3's multiplied together. So we can use an exponent to represent this type of repeated multiplication. And the exponent is going to tell us how many times I am multiplying by a number. So the exponent of 8 means that I am multiplying 3 eight times.

There's a few different ways that you could say this. 3 to the eighth is probably the most common, but you could also say 3 to the eighth power, 3 raised to the eighth, and 3 raised to a power of 8. All of those are perfectly fine ways of saying this.

Now this might be something that you want to get into your notes. We've got a general form, or a formula, for how we write something with an exponent. The b, or the number that's being multiplied, is called our base. And the smaller number on top of it, as we know, is called our exponent. And this tells us, again, how many times are we multiplying by our base. So in case you're wondering, 3 to the eighth power is 6,561. So as I was saying before, using an exponent or repeating multiplication makes things get big or small very quickly.

So let's look at some examples of how we use exponents. So let's do some examples. The first one is 4 to the fifth power. Our base is 4, which means that's the number we're multiplying by, and the exponent is 5, which means we're going to multiply five times. So five 4's multiplied together is going to give me 1,024. Let's look at what it would be if we flipped it. 5 to the fourth power means the number I'm multiplying is 5-- that's my base-- and the exponent is 4, which means I'm going to multiply it four times. 5 times 5 times 5 times 5 will give me 625. 6 to the third power means I'm multiplying 6 three times, which is going to give me 216.

And just a special note-- anything that has an exponent of three, we have some common language for that or some slang. Instead of saying 6 to the third power, we could say 6 cubed. So anytime you see an exponent of three, sometimes that is called cubed. 7 to the second power would be 7 times 7, which is 49. And we also have some slang, which you've probably heard, for an exponent of 2, and that is just saying squared. So instead of saying 7 to the second power, we could say 7 squared.

Now we've got two special exponents-- an exponent of 1 and then an exponent of zero. An exponent of 1 means that I'm only multiplying my base-- in this case 8-- one time. So I only write the number 8 one time. 8 is just going to be equal to 8, so an exponent of 1 does not actually change anything. We're not multiplying 8 by anything, so our answer just remains our base.

The last special case is an exponent of 0. Anything to the exponent of 0 is going to be equal to 1. And that's kind of a weird concept. Most of the time you would think that an exponent of 0 we make something equal to 0. So I'm going to show you using a base of 2 why it's equal to 1. If I start with 2 to the third power, that means I'm multiplying by 2 three times. 2 times 2 times 2 is going to give me 8. If I decrease that exponent by one, I'm going to have 2 to the second power. 2 times 2 is 4. And if I decrease that exponent by one again, I'm going to have 2 to the first power, which, as we just talked about, is just going to give me 2.

So let's stop and look at this pattern. As I decrease the exponent by one, my answer is just dividing by 2, which is my base. So if I do this one more time and look at an exponent of 0, I'm going to continue my pattern of dividing by 2. 2 divided by 2 is going to give me 1. And I used 2 as an example because it's easy to see the pattern, but this works for any base. So again, 9 to the zero power is going to give me 1.

So today we talked a little bit more about exponents and we did some examples using those exponents. Let's take a look at our key notes. Exponents are used to denote how many times a number is being multiplied by itself or repeated multiplication. And in the form b to the x, our b is the number that we're being multiplied, which we call our base. And the x is the exponent, which again is telling us how many times we're multiplying by that base.

And then we had a couple of specific special cases. For any base, or for any number, b, with an exponent of 1 is just going to be equal to that number, whatever you had for your base. And if we have any number, b, to an exponent of 0, it's just going to be equal to 1.

So make sure that you get these key points down into your notes, keep looking at your examples, and keep on practicing, and soon you'll be a pro. Thanks for watching.