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Properties of Exponents

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Today we're going to talk about the properties of exponents. When you're simplifying or writing equivalent expressions that involve exponents, you can use one of these five properties that we're going to talk about today. So we'll look at these different properties that involve multiplying, dividing, and raising exponents to other exponents. And we'll do some examples for each of these properties.

Behind me is the table for the five properties of exponents. All five of these properties involve using products, quotients, and powers. So we'll go through each property and do an example using numbers for each. So you're either going to want to go ahead and get this table down in your notes as we go through it or pause it at the end and copy it down then.

The examples that I'll use will be involving numbers that are positive. But just so you know, they'll also work for negative numbers too. All right.

So our first property is called the power of product. I have two numbers, a and b, that are multiplied by each other. And they're raised to some exponent n. This property says that I can also separate that exponent and write that as a to the n and b to the n.

So what would that look like with some numbers? Let's say I had 3 times 5 in parentheses with a 2 exponent. My property says that I can separate that out and write that as 3 squared times 5 squared. So let's see if that's true.

So 3 times 5 squared means 3 times 5 times 3 times 5. But if I have two threes multiplied together, I can write that as 3 squared. And if I have two fives multiplied together, I can write that as 5 squared. So it looks like our property holds true. Instead of writing 3 times 5 squared, I can write as 3 squared times 5 squared.

Let's look at our second property, power of power. So this says if I have some number a to an exponent n and then another exponent m on top of it, I can simply multiply those two exponents and write them as one exponent. Let's do an example to see what it looks like with numbers.

So let's say I have 8 to the second power and then raised to the third power. So 8 to the second power means I have 8 times 8. And if a to the second power is raised to the third power, that means I have three of these multiplied together, so 8 times 8 times 8 times 8 times 8 times 8.

Now if I want to write that as 8 to some exponent, I just need to count how many times I multiplied by eight. One, two, three, four, fix, six. So this would be 8 to the sixth power. So again it looks like our property holds true. I can multiply my two exponents together, 2 times 3, and just write it as 8 to the sixth power. All right.

Our third property, power of a quotient. So here I have a divided by b raised to some exponent n. And our property tells me that I can write that as a to the n divided by b to the n. So similar to our power of product, I can separate my exponent out and write it for each of my two bases.

So with some numbers-- let's do 7 over 2 to the second power. So if I have 7 over 2 squared, I know that means I have 7 over 2 times 7 over 2. And now when I simplify that, I know that when I'm multiplying fractions I can just multiply straight across.

So this is going to be 7 times 7 in my numerator and 2 times 2 in my denominator. And since I have 7 times 7, I know I can write that as 7 squared. And similarly at the bottom, this will just give me 2 squared. So again my property holds true. I can separate out my 2 exponent and write it for the base in the numerator and in the denominator.

For our fourth example, our fourth property, it's called the product of powers. And we've got some base a to the exponent of n times the same base a to the exponent of m. And it's saying that we can write that as the same base and adding our two exponents together to give us one exponent.

So with numbers, let's see what that looks like. So let's say I have 3 squared times 3 to the fourth. OK. So we see our base has to be the same. Well, 3 squared is just 3 times 3. And 3 to the fourth is 3 times 3 times 3 times 3.

So if I want to write this as a single exponent, I know I just need to count up how many times did I multiply by 3. One, two, three, four, fix, six. So this will give me 3 to the sixth. So we can see that our property holds true. I can just add my exponents, 2 plus 4, and that will give me 6.

And our last property is called the quotient of powers. And it tells us that some base a to the n exponent divided by the same base a to the m exponent-- we can combine that with the base a and simply subtract our two exponents from each other.

So with numbers, let's see. Let's have 4 to the fifth power divided by 4 to the third power. So our base is the same. So looking at our property-- or sorry, if I were to expand this out using our exponents, 4 to the fifth power means 4 multiplied by itself five times. And 4 to the third power means 4 multiplied by itself three times.

And now I can simplify this fraction, because since I'm multiplying on the top and on the bottom and my fraction means dividing, anything that I have on the top and on the bottom will cancel out. So these fours will cancel. These will cancel. And these will cancel. So all I'm left with is 4 times 4, which I know simplifies to 4 to the second power. So I can see that my property holds true. I could simply subtract my 5 minus 3, and I would get 4 to the second power.

So when you're simplifying using these properties of exponents, you don't need to do any of this stuff in the middle. This was just to show you how the property works. All you need to do is apply these properties. And if you started with this step-- using your property-- you could skip to this last step here.

So today we talked about five different properties of exponents that involve using the products, quotients, and powers of exponents. If you didn't already get this table down in your notes, make sure that you rewind, pause, and copy it down, because it's going to be useful for you as you move forward with your examples.

So I hope that these notes and examples helped you understand a little bit more about the properties of exponents. Keep using these notes. And keep on practicing. 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
- Quotient Property of Exponents