Today wer'e going to talk about applying the properties of exponents. And so we're going to start by reviewing the properties of exponents. We'll talk about some common mistakes people make. And then we'll do some examples.
So let's start by reviewing our properties of exponents. My first property, the product property, says that a to the n times a to the m equals a to the n plus m. So when you're multiplying terms, you add the exponents.
The quotient property says that when you're dividing terms with the same base, then you can write it as one term, subtracting your exponents. The power property says that a to the n to the m raised to a second exponent is equal to a to the n times m. So you can multiply your exponents.
The power of a product says that a times b to the n can be written separately as a to the n times b to the n. The power of a quotient is similar. a over b to the n can be written separately as a to the n over b to the n. And finally, the property of negative exponents says that a to the negative n is equal to 1 over a to the positive n.
So let's go over some common mistakes people make when applying the properties of exponents. The first is that something like 4 squared times 2 to the 3rd would not be equal to 8 to the 5th power. In other words, we cannot apply the property that when you're multiplying terms you simply add the exponents if the bases are not the same. So we have to have the same base to be able to apply that; property of adding our exponents.
The second common mistake that people make is something like thinking x plus y squared is equal to x squared plus y squared. This is not true. You cannot apply the power property if you have terms in the parentheses instead of just factors.
So we can do an example to show why that's true. Let's say we have 2 plus 4 squared. We want to show that's not equal to 2 squared plus 4 squared. 2 plus 4 is 6. And 6 squared is 36. However, 2 squared is 4 and 4 squared is 16. 4 plus 16 is 20. So we can see that these two things are not the same, and so that this is not a property that holds true.
So finally, let's do two examples applying the properties of exponents. For my first example, I have a to the 3rd b squared squared over a to the 8th. So I'm going to start by distributing my 2 exponent by multiplying this 2 exponent by the exponents on the inside. So this is going to become a to the 6th b to the 4th over a to the 8th.
Then I'm going to simplify using the quotient property. I know that I can write this as a to the 6th minus 8. So this would be a to the negative 2. And then I'll bring my b to the 4th down. And then finally I know that with a negative exponent, I can write that as a positive exponent in the denominator. So this is the same thing as b to the 4th over a to the positive 2 for my final answer.
For our second example, I'm going to start by distributing this 4 exponent to all of the variables on the inside by multiplying the exponents together. This is going to become xy to the 12th times x to the 8th over x to 4th y to the 4th. Then I'm going to simplify this by separating out my x and y terms. And I'm going to do that because I can combine them with other variables that are the same.
So this will become x to the 12th y to the 12th x to the 8th in the numerator and still x to the 4th y to the 4th in the denominator. So now I can combine my two x variable exponents by adding. So x to the 12th times x to the 8th is going to give me x to the 20th. And then I'll have y to the 12th. And then I'll still have x to the 4th and y to the 4th at the bottom.
And then finally, I can subtract my exponents. x to the 20th over x to the 4th-- I would subtract my exponents, so this will become x to the 16th. And this will be y to the 8th. 12 minus 4 is 8.
So let's go over our key points from today. Use the properties of exponents to simplify expressions and solve equations. Avoid these common mistakes. You can only add or subtract the exponents in multiplication or division if the bases are the same. And you can only distribute an exponent across factors and not terms.
So I hope that these key points and examples helped you understand a little bit more about applying the properties of exponents. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.
Power of a Product:
Power of a Quotient: