Exponent properties with quotients

Let's do some exponent examples that involve division. Let's say I were to ask you what 5 to the sixth power divided by 5 to the second power is? Well, we can just go to the basic definition of what an exponent represents and say 5 to the sixth power, that's going to be 5 times 5 times 5 times 5 times 5-- one more 5-- times 5. 5 times itself six times. And 5 squared, that's just 5 times itself two times, so it's just going to be 5 times 5. Well, we know how to simplify a fraction or a rational expression like this. We can divide the numerator and the denominator by one 5, and then these will cancel out, and then we can do it by another 5, or this 5 and this 5 will cancel out. And what are we going to be left with? 5 times 5 times 5 times 5 over 1, or you could say that this is just 5 to the fourth power. Now, notice what happens. Essentially we started with six in the numerator, six 5's multiplied by themselves in the numerator, and then we subtracted out. We were able to cancel out the 2 in the denominator. So this really was equal to 5 to the sixth power minus 2. So we were able to subtract the exponent in the denominator from the exponent in the numerator. Let's remember how this relates to multiplication. If I had 5 to the-- let me do this in a different color. 5 to the sixth times 5 to the second power, we saw in the last video that this is equal to 5 to the 6 plus-- I'm trying to make it color coded for you-- 6 plus 2 power. Now, we see a new property. And in the next video, we're going see that these aren't really different properties. They're really kind of same sides of the same coin when we learn about negative exponents. But now in this video, we just saw that 5 to the sixth power divided by 5 to the second power-- let me do it in a different color-- is going to be equal to 5 to the-- it's time consuming to make it color coded for you-- 6 minus 2 power or 5 to the fourth power. Here it's going to be 5 to the eighth. So when you multiply exponents with the same base, you add the exponents. When you divide with the same base, you subtract the denominator exponent from the numerator exponent. Let's do a bunch more of these examples right here. What is 6 to the seventh power divided by 6 to the third power? Well, once again, we can just use this property. This going to be 6 to the 7 minus 3 power, which is equal to 6 to the fourth power. And you can multiply it out this way like we did in the first problem and verify that it indeed will be 6 to the fourth power. Now let's try something interesting. This will be a good segue into the next video. Let's say we have 3 to the fourth power divided by 3 to the tenth power. Well, if we just go from basic principles, this would be 3 times 3 times 3 times 3, all of that over 3 times 3-- we're going to have ten of these-- 3 times 3 times 3 times 3 times 3 times 3. How many is that? One, two, three, four, five, six, seven, eight, nine, ten. Well, if we do what we did in the last video, this 3 cancels with that 3. Those 3's cancel. Those 3's cancel. Those 3's cancel. And we're left with 1 over-- one, two, three, four, five, six 3's. So 1 over 3 to the sixth power, right? We have 1 over all of these 3's down here. But that property that I just told you, would have told you that this should also be equal to 3 to the 4 minus 10 power. Well. What's 4 minus 10? Well, you're going to get a negative number. This is 3 to the negative sixth power. So using the property we just saw, you'd get 3 to the negative sixth power. Just multiplying them out, you get 1 over 3 to the sixth power. And the fun part about all of this is these are the same quantity. So now you're learning a little bit about what it means to take a negative exponent. 3 to the negative sixth power is equal to 1 over 3 to the sixth power. And I'm going do many, many more examples of this in the next video. But if you take anything to the negative power, so a to the negative b power is equal to 1 over a to the b. That's one thing that we just established just now. And earlier in this video, we saw that if I have a to the b over a to the c, that this is equal to a to the b minus c. That's the other property we've been using. Now, using what we've just learned and what we learned in the last video, let's do some more complicated problems. Let's say I have a to the third, b to the fourth power over a squared b, and all of that to the third power. Well, we can use the property we just learned to simplify the inside. This is going to be equal to-- a to the third divided by a squared. That's a to the 3 minus 2 power, right? So this would simplify to just an a. And you could imagine, this is a times a times a divided by a times a. You'll just have an a on top. And then the b, b to the fourth divided by b, well, that's just going to be b to the third, right? This is b to the first power. 4 minus 1 is 3, and then all of that in parentheses to the third power. We don't want to forget about this third power out here. This third power is this one. Let me color code it. That third power is that one right there, and then this a in orange is that a right there. I think we understand what maps to what. And now we can use the property that when we multiply something and take it to the third power, this is equal to a to the third power times b to the third to the third power. And then this is going to be equal to a to the third power. times b to the 3 times 3 power, times b to the ninth. And we would have simplified this about as far as you can go. Let's do one more of these. I think they're good practice and super-valuable experience later on. Let's say I have 25xy to the sixth over 20y to the fifth x squared. So once again, we can rearrange the numerators and the denominators. So this you could rewrite as 25 over 20 times x over x squared, right? We could have made this bottom 20x squared y to the fifth-- it doesn't matter the order we do it in-- times y to the sixth over y to the fifth. And let's use our newly learned exponent properties in actually just simplify fractions. 25 over 20, if you divide them both by 5, this is equal to 5 over 4. x divided by x squared-- well, there's two ways you could think about it. That you could view as x to the negative 1. You have a first power here. 1 minus 2 is negative 1. So this right here is equal to x to the negative 1 power. Or it could also be equal to 1 over x. These are equivalent. So let's say that this is equal into 1 over x, just like that. And it would be. x over x times x. One of those sets of x's would cancel out and you're just left with 1 over x. And then finally, y to the sixth over y to the fifth, that's y to the 6 minus 5 power, which is just y to the first power, or just y, so times y. So if you want to write it all out as just one combined rational expression, you have 5 times 1 times y, which would be 5y, all of that over 4 times x, right? This is y over 1, so 4 times x times 1, all of that over 4x, and we have successfully simplified it.