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Properties of Fractional and Negative Exponents

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Today we're going to talk about the properties of exponents when the exponent is a fraction or negative. So we'll go ahead and review our five properties of exponents that we've talked about before. But first we'll review the relationship between a fractional exponent and a radical, and a negative exponent and a positive exponent.

If you remember, the property for fractional exponents tells us that if we have some number a, or our base, to an exponent that's a fraction, m over n. We can also write that as a radical, where the denominator of our fraction becomes the root of our radical, and the numerator of our fraction comes underneath the radical, staying with the base. So a couple of quick examples just to make sure that's clear.

Let's say I have 4 to the 3 over 2. If I want to write that as a radical, my denominator becomes the root of my radical. So I could put a 2 there. But we also know that that's just a square root, and normally we don't put a 2. And then my base in the numerator of my fraction comes underneath the radical. So 4 to the power of 3 over 2, can also be written as the square root of 4 to the 3rd.

OK. If we looked at it in the flip order, we could start with the 5th root of 6 to the 7th power. And if we wanted to write that as a fractional exponent, our base would stay 6. Our numerator is the exponent underneath the radical that stayed with the base. And our denominator of the fraction is the root of our radical. So the 5th root of 6 to the 7th power can also be written as 6 to the 7 over 5.

If I have a negative exponent, the base a to a negative exponent m, I can rewrite that as 1 over the same base a with a positive exponent m. So an example of that would be 3 to the negative 5th power could be written as 1 over 3 to the positive 5th power.

We could do an example of that in the flip order. Let's say we had 1 over 2 to the 7th, a positive 7. I could rewrite that as 2 to the negative 7.

So let's look at our five properties of exponents. These properties involve products, quotients, and powers. And we're going to see how these five properties still hold when your exponent is fractional or a negative number. So we'll go through each of the five properties, and do an example of each, again when the exponent is either a fraction or a negative number.

So for the first property, it's called the Product of Powers, and this tells us that some base a to the exponent n, multiplied by the same base a to the exponent m, can be rewritten as a to the n plus m. So for example, if I have 4 to the negative 12, and I multiply that by 4 to the negative 3rd, my property tells me I can just add my exponents. So this will be 4 to the negative 12, plus negative 3. And that would give me 4 to the negative 15.

My second property is called the Power of a Product property. And that tells me that some basic a, multiplied by another base b, both to the exponent n, can be rewritten as a to the n times b to the n. So for example, if I have 8 times 2 to the 4/7 power, I can rewrite that by separating the exponent. So 8 to the 4/7 multiply by 2 the 4/7.

Our third property is called the Quotient of Powers property. And that tells us that some base a to the exponent n, divided by the same base a to the exponent m, can be rewritten as a to the n minus m. So for example, if I have 9 to the negative 3/5 power, divided by 9 to the negative 1/5 power, I can rewrite that as nine to the negative 3/5 minus negative 1/5. Subtracting my two exponents, I see that I will have 9 to the negative 2/5 power.

My fourth property is called the Power of a Quotient property. And that tells us that a over b like a fraction, to some exponent n, can be rewritten as a to the n over b to the n. So for example, if I have 12 over 11 to the power of negative 3, I can rewrite that as 12 to the negative 3, over 11 to the negative 3.

So our last property is called the Power of a Power property. And that tells us that some base a to the exponent n raised to another exponent m, can be rewritten as a to the n times m. So for example, if I have 3 to the 1/2 power, and then raise to the 5/6 power, I can rewrite that as 3 to the power of 1/2 multiplied by 5/6. And when I multiply my fractions together, I multiply numerator numerator, denominator times denominator. So that's going to give me three to the 5 over 12.

So let's go over our key points from today. Make sure you get these points into your notes if you don't have them already, so you can refer to them later. So we first talked about the relationship between a fractional exponent and a radical, being that some base a to a fractional exponent m over n can be written as a radical, the nth root of 8 to the m.

We then looked at the relationship between positive and negative exponents, and that is that some base a to a negative exponent n can be written as 1 over the same base a to the positive exponent n.

And then we looked at our five properties of exponents, and we saw that those properties can be used for exponents that our fraction and/or negative. I'm not going to rewrite those properties. So if you didn't get them down from the table the first time, make sure that you go back and get those into your notes

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

Formulas to Know

- Power of a Power Property of Exponents
- Power of a Product Property of Exponents
- Power of a Quotient Property of Exponents
- Product Property of Exponents
- Property of Fractional Exponents
- Quotient Property of Exponents