 This video talks about exponent properties, and it will be 1 of 3. So b and n means n factors of the base, and the base is going to be defined as what's directly below the exponent. So if I look at this problem of 2 squared, the base is just the 2. That's pretty obvious. When I look at this one, my base is negative 2 because the parenthesis is directly below that exponent, so everything inside that parenthesis is my base. And then finally, this one, the 2 is the only thing that is directly below my exponent. So the base is 2. This really means the opposite of 2. That's what this means, or the opposite of 2 squared. So how many factors of x do we have here? One, two, three, four, five factors of x. So we would write that as x to the fifth. Now I want to go the other way and see if I can figure out some rules here. So I have four factors of x, so I have to write x four times. And then I want to multiply that times two factors of x. And if I count them, I have one, two, three, four, five, six factors of x. And if I do this one, I have three factors of x. And I'm going to multiply that times five factors of x. And when I add them, I have one, two, three, four, five, six, seven, eight factors of x. But what if I'd had x to the 30 times x to the 50? And then I don't want to multiply those all out. So we have these properties. And the property, if you look closely, I have an exponent of 4 and 2. And I have my final answer of 6, which is the sum of those exponents. Same thing here, 3 and 5 give me a total of 8. A to the m times A to the n is going to be the same base, one base, but I will just add my exponents. So m plus n, and again, you add your exponents. Well, let's look and see what we know what the base is here. Because notice I have more than one exponent going on. But the base for this one is going to be 3 squared. Remember we said that the parentheses, everything inside the parentheses would be directly below an exponent on the outside? So the base here would be x to the sixth. And if I multiply this one out, it really means 3 squared times 3 squared times 3 squared. And if we use the property that we just learned that we have the same base of 3, we can add our exponents, then this is going to be 3 to the 2, 4, 6. And if I did the same thing here, I'd have x to the sixth, times x to the sixth, times x to the sixth, times x to the sixth. And that's going to be equal to x to the, well, add them up. 6 plus 6 is 12, plus 6 more would be 18, plus 6 more would be 24. So I have x to the 24. So if you come back, I had a 2 and a 3 and a final answer of an exponent of 6. A 6 and a 4 exponents and a final answer of 24. Looks like we took a, and we took our exponents and multiplied them. Now be very careful. The first one and the second one, we get them mixed up very easily. But if you think about n, you have to kind of take, distribute it in. It's kind of the way I think about it, and so when you distribute, you multiply. In the case before, we weren't distributing because they were right next to each other, so we could add them. What do you suppose you would do with r times t to the 2? Well, if we multiply it out, we'd have r t times r t. And we know that we can multiply it in the order, so that would be the same thing as r times r times t times t. So I would have r squared and I would have t squared, two factors of r, two factors of t. If I have a product AB raised to the M, it looks like r got raised to the 2 and t got raised to the 2, so maybe A gets raised to the M and B gets raised to the M. So this is where we say you distribute the exponent to everything inside. If I look at these closely then, I'm going to practice everything that I just looked at. So I have a product that's being raised to a power, so that means 7 to the 3rd times p to the 3rd. And if we had a calculator, we would figure out what 7 to the 3rd was, I think I have one here. I have 7, caret 3, so that's actually 343 p cubed. And now we have, again, a product that's raised to a power, so we have negative 11. That number is going to be squared, just like the variable. And x to the 4th is going to be squared. And negative 11 times negative 11 is a positive 121. And x to the 4th squared, remember, we're going to multiply those exponents. So 4 times 2 will be 8. So 121 x to the 8. And then finally, we have everything inside being raised to the 5th. So again, the number gets raised. So 2 to the 5th, m squared is going to be raised to the 5th, and so is n cubed. Now we come back through and say, okay, 2 to the 5th happens to be 32. m squared raised to the 5th, now I have to use that power raised to a power rule. That said, multiply, 2 times 5 will give me m to the 10. And then this will be n to the 3 times 5 or 15. We're going to do some practice here of what we learned earlier. We have y cubed times y to the 4th times y to the 6th. So remember that's the same base here, so we can just add these exponents. And 3 plus 4 is 7, and 7 plus 6 is going to be y to the 13. Here we have numbers and same bases, all kinds of things happening here. If you would rather, you could rewrite it with the like things together. Negative 2 times negative 8 times x to the 4th times x to the 3rd. If that's helpful, some people will just say, well, negative 2 times negative 8, and then x to the 4th times x cubed. Either way, I'm going to end up with negative 2 times negative 8 being positive 16. And x to the 4th times x cubed means I'm going to add those exponents. So I have x to the 7. And finally, we have a real simple p to the 8 squared. That's the simplest one we have on here. And that just means that we have to multiply our exponents. So 8 times 2 would give us p to the 16.