 In this video, I'm going to talk about the properties of exponents. I'm going to talk about three properties, these aren't all of them, just three of them. I'm going to talk about the product of powers, the quotient of powers, and the power of powers. And I'm going to do this in two different ways. The first thing I'm going to do is I'm going to talk about the algebra behind this. I'm going to use a bunch of variables to kind of explain in general how these work. And then I'm actually going to do examples with numbers. So that's kind of how I'm going to do this. But first what I want to do is I'm going to go over just a little bit of vocabulary with exponents. Whenever you do exponents, you've got to kind of remind yourself of some of the vocabulary. So if I have an exponent, if I have some number, we'll call it B, some number to some power is always going to be equal to an answer. So these are the variables that I use when explaining exponents. If I have a number to some power, it's going to be equal to an answer. So this could be 5 to the second power is equal to 25. Something to that effect. Now the reason I use BX and A is because B is the base. B, that number is what we call the base. So in this example down here, 5 is the base. This X up here, that's the base. This X up here we call the exponent. Or we call it the power. You call it exponent, you call it power. Either one of those works. But in this case with this number, example 2 is your exponent. And then I use A for answer. A for answer. And in this case the 25 down here is your answer. So that's the vocabulary I'm going to use. Base, exponent or power. Either one and an answer here. And again it's always good to go over some of that vocabulary. It might have been a while since you've seen exponents. Okay. Now for the product of powers. For the product rule. Basically what the product rule says is that if you are multiplying either variables or numbers, it doesn't matter which, if you're multiplying with like bases you need to add the exponents. So let's give an algebraic example. So if I have some base to some power times some base to a different power. Notice I'm using Y for kind of a different power. Okay, so if I have a base and a base these two numbers are the same and I'm multiplying them together the product of power rules tells me that all I really need to do is just add the exponents together. So this just tells me that this is going to be B to the X plus Y. Notice that that X plus Y is raised up in the exponent position. Okay, now this explanation of it is kind of confusing because we have all sorts of variables in here so let's put some numbers in here. So for example let's do, let's have my base be a 3. So what if I have 3 to the second power times 3 to the fifth power. 3 to the second power and 3 to the fifth power. Okay, like bases, the rule is if I have like bases then I can take the exponents and add them together. So this is going to be 3 to the seventh power. 3 to the seventh power. Okay, so that's your product of powers rule. Now we can continue to evaluate that 3 to the seventh power. I don't know what that is or off the top of my head but you can continue to evaluate that but I'm just showing just the basics here. Anyway, let's move on to the quotient of powers. Now the first one product means we multiply. So notice that we multiply like bases. Quotient of powers you can imagine, quotient means divide. So I'm going to divide power. So I'm going to use some of the same variables. What's going to be just a little bit different. Quotient of powers, that rule tells us that if you are dividing like bases then you need to subtract the exponents. If you're dividing like bases you need to subtract the exponents. So for example, if I have some number, we'll call it B to some power, call it X. Again, it's kind of similar here but instead of multiplying I'm going to divide. So I'm going to use a fraction bar to denote that I am dividing. So some number to some power divided by that same number in the same base to a different power. If I'm dividing like bases if I'm dividing like bases then that means I need to subtract my exponents. Subtract my exponents. The product rule, we added the exponents quotient rules since I'm dividing these I'm going to subtract my exponents. So I'm going to use the same numbers that I did for my previous one. Okay, so if I have 3 to the second power divided by 3 to the fifth power I'm using kind of the same numbers that I used before. I could use different ones but I'm just going to keep it the same. So in this case if I'm dividing like bases I'm going to take the exponents and subtract them. Now notice that the base for both product and quotient notice that the base hasn't changed. The base doesn't change. That's something I should have pointed out earlier. Alright, so if I have 3 to the second power divided by 3 to the fifth power like bases and we're dividing so I need to subtract my powers is going to be 3 to the negative third. 3 to the negative third. Now in a later video I will explain a little bit more about negative exponents but I'm going to leave that right there so that we can see where the negative 3 comes from. Alright, last but not least is power of powers. So notice that we've multiplied like bases we have divided like bases now we're going to do power of a power. So this one is just a little bit different. So what if I have a base to some power again using some of the same exponents or excuse me, not exponents using some of the same variables here so if I have some base to some power but then I want to take that number to yet again a different power. So this is why we call it a power of a power. This power is being taken to a power that's why it's called power of powers. I know it sounds a little bit redundant but that's kind of how we categorize this one. This type of problem. Alright, so the rule for power of powers if you are doing if an exponent is being taken to another exponent we have a power being taken to another power we need to multiply the two exponents. So this would be, this would end up being b to the x times y. Okay, so notice that there's a little bit of a pattern here if you haven't recognized it yet. Notice product b to the x plus y quotient b to the x minus y and then power b to the x times y so we're using add subtract and multiply here as you can well imagine. A little bit of a pattern there to it, not too much. If you're going to remember patterns this is actually a good method to memorize the rules for these three but that's a different story. Alright, so let's do a number example. Again the rule for this power of powers if I'm doing a power of a power I need to multiply the exponents so let's use some of the same numbers we used here if I have three to the second power to the fifth power power of a power so that two is being taken to the fifth power so I'm going to take three to the tenth power this two times five I'm going to multiply these two numbers here to get ten. Okay, those are some of the properties of exponents those are some examples of them again that's the product of powers quotient of powers and the power of powers three of the examples of properties of exponents I will be doing another video on the rest of the properties.