 This video is going to talk about compound interest with exponential functions. So exponential functions are needed when we're talking about savings or we're borrowing money that has compound interest. And there are a couple of equations or formulas that we use to do these. And here's the first one. A is equal to P times 1 plus R over N to the NT, where A is the amount in the account. P is the original amount that we invested. R is the interest rate, and we remember to change the percent into a decimal. N is the number of times that this is being compounded per year, and then T is the number of years that it's in there. So if we look at this, it says, what is the balance after two years of an account that contains $1,500 that earns an interest rate of 8% compounded annually? So let's look at the parts. We have our principal, which is our $1,500, and then we've got our interest rate, which is 8%, and then it tells us that we have N would be the number of times per year. That means annually, which is just 1. And then finally, we have the number of years that is invested, and this problem told us it was two years. So if we rewrote this problem, we have A is equal to that principal of $1,500, in fact let's just put some color in here, $1,500, I didn't have a red highlighter, times 1 plus my rate, which was .08, and then we're going to divide that by the annually, which is 1, and times or raised to the N times our T, which is two years. And if we wanted to simplify that a little bit, we could say that that's 1,500 times 1 plus .08 to the 2, in fact we could just call this 1.08. So over here in my calculator, I'm going to put 1,500 times 1.08, 1 plus 0.8, and then caret 2 tells us that after two years, we have in our account $1,749.60 after two years. Okay, what happens if we were to change this to monthly, compounded monthly? That means that our N is going to be equal to 12 times a year. So now we have an equation that says A is equal to our 1,500 again, times 1 plus our rate, which is .08, but then it's divided by 12, not 1 this time, because it's 12 times per year, and then it's 12 times 2, because it's N times 2. So if I want to simplify this a little, the only thing I would simplify would to be say 1 plus, and then I'm going to put this in parentheses, so I know exactly how to put it in my calculator. I'm going to need a double parentheses here, and then to the 24. So 1,500, then 1 plus, but then I've got a fraction, so I have to put it in parentheses .08 divided by 12, close my fraction, close my quantity, and then care at 24. So after two years, when it's compounded monthly, we have $1,759.33, so we made a little bit more money when we compounded it monthly, so we have this much money after two years of monthly compounding. I need to talk about the last type, that is continuous compounding. That means every second of every minute of every hour of every day of every week and so on. Forever. It's always compounding. In our case then, N isn't going to be something that we can put a number to, because it's always happening, and if you let N get bigger and bigger and bigger, you eventually end up with this formula. For A, again, is the amount in the count. P is still the principal, or the amount invested. R is still our rate, and T is still the number of years, but you'll notice there's no N in this one. But there's this little E. So what is E? Well, E is a mathematical quantity, and I learned this is Euler's, not everybody says that, but I learned it is Euler's number, and that's approximately 2.718. It's a naturally occurring number, and there is a button on your calculator to do that, and the second LN, that gives you E to the X, but you could just put one. Here we have our formula, and it says, going back to that same problem, what is the balance after two years, okay, so that's our T. After two years, actually that should have been green, probably, two years, and $1,500 is our principal, and it's compounded at 8%, that's our rate, but compounded continuously is going to give us E. So A is equal to our 1,500 times E, and then to the R, which is 0.08 times T, which is two years. So in my calculator, I'm going to put 1,500, and then you can do it one of two ways. You can do second divide, and that gives you an E, and then you'll have to put your carat in here, and then since we're multiplying, you need parentheses, 0.08 times 2, and close your parentheses. And we find out that we get even more money if we do it continuously, $1,760.27, and just to show you the other way you could do it, start again with our 1,500, but you could use second LN here, if you see in the blue it says E to the X, so it automatically gives you the carat, and then you put in the parentheses, so 0.08 times 2. Some people like to have it automatically come up, some people don't like it automatically come up, so I'll show you both ways. But in both cases we get the same amount of money, so we have this much money after two years compounding continuously.