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Negative Exponents

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Today we're going to talk about negative exponents. We already know that an exponent tells you how many times you are multiplying by a number. So what does that mean if that exponent is negative? So we'll look more closely about exactly what a negative exponent means, a general form for writing with a negative exponent, and then we'll do some examples.

So to figure out what happens when we have a negative exponent, let's start by evaluating some numbers with positive exponents. And we'll look for a pattern between what happens when you decrease your exponent by 1. And then we'll see what happens when we have a negative exponent.

So 2 to the third power means 2 multiplied by itself three times. 2 times 2 times 2 is going to give me an 8. 2 to the second power means 2 times 2, is going to give me 4. And 2 to the first power just means 2 by itself, which is going to give me 2.

So if I look for a pattern between these three numbers, I can see that every time I decrease my exponent by one, I divide my answer by 2. 8 divided by 2 is 4. And 4 divided by 2 is 2. And this 2 comes from our base. Every time I decrease my exponent by 1 I'm multiplying by 2 one less time, which is the same thing as dividing by 2. So if I keep on decreasing my exponent by 1, I'm going to follow the same pattern of dividing by 2.

So 2 divided by 2 is going to give me 1. And that matches up with what we know about a 0 exponent. Anything to a 0 exponent is just going to be 1. And now I can see what happens with my negative exponent. 1 divided by 2 is going to give me 1 over 2 or 1/2. And 1/2 divided by 2 is going to give me 1 over 4, or 1/4.

So I can see that I'm starting to get some fractions. And there's even a pattern with the fractions. This 2 in the denominator for 2 to the negative 1 is the same thing as 1/2 to the positive 1 exponent. And 1/4 is the same as 1/2 to the positive 2 exponent.

So now I can see my relationship between negative exponents and positive exponents. Instead of 2 to the negative 1, I have 1/2 to the positive 1. And instead of 2 to the negative 2, I have 1/2 to the positive 2.

So we can summarize this relationship with a property. And that property is any base b to a negative exponent n can also be written as 1 over the same base b to the positive exponent n. So you're going to want to make sure that you get this property into your notes so that you can use it while we do our examples.

So we're going to do six quick and easy examples using our property. If you're feeling pretty confident, go ahead and pause this, write down these examples and try them on your own, and then check back and see how you did.

All right, so let's start with 4 to the negative third. Using our property, we know that this is just going to be the same as 1 over 4 to the positive third. Same thing with 5 to the negative 2 exponent. That's going to be 1 over 5 to the positive 2. So all you need to do is make your exponent from negative to positive, and then flip it so that it's in the denominator of the fraction with a 1 in the numerator.

Let's try the reverse way. 1 over 8 to the positive 4 using our property is going to be 8 to the negative 4. Same thing here. 1 over 3 to the positive 2 exponent is going to give us a 3 to the negative 2 exponent.

And then the last two examples have a negative exponent and are already in the denominator of the fraction. But you're still going to do the same thing. We're going to make the exponent from negative to positive, so the opposite sign. And we're going to flip it so that it's not in the denominator of the fraction.

So that's just going to become 7 to the positive 3. Same thing here. Instead of negative 4, we'll have a positive 4 exponent. And we'll flip it so that the 6 is not in the denominator of the fraction. So a 6 to the positive 4 exponent.

So let's go over our key points from today. Make sure that you get these into your notes if you don't have them already so you can refer to them later.

So we talked about the fact that negative exponents and positive exponents have a relationship. And you can rewrite any negative exponent as positive using one of these two properties. So any base to a negative exponent n can be written as 1 over the same base b to a positive exponent n. So the exponent goes from negative to positive. And we have our base and the exponent in the denominator of the fraction. So we kind of flip the fraction.

Same idea, if you have a fraction 1 over some base b to a negative exponent n, we can write it as the same base b to the positive exponent n. So again, our exponent goes from negative to positive. And instead of the base and exponent being in the denominator of the fraction, we have it written by itself.

So I hope that these key points and these examples helped you understand a little bit more about negative exponents. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.