 This video is going to talk about power functions. A power function is just when x is raised to some power. So we have f of x is equal to x to the p. And it so happens that if p is a fraction, 1 over n, and n is 2 or greater, so we'd have like 1 half or 1 third or 1 fourth, so on, then f of x equal x to the p is actually a root function. So when you see x to the 1 half, that's going to be the same thing as the square root of x. That's what that's saying. Alright, so let's use these power functions. For a certain species of whale, the relationship between the length of the whale and the weight of the whale can be modeled by this function given here. Where l is the length of the whale in meters and its weight of the whale in metric tons. So estimate the weight of a newborn calf that is 6 meters long. Well, that means l is equal to 6. So it's just a plug and chug. So 0.03 times 6 raised to the 27 over 11. And if I put it in my calculator, I'm going to have to have parentheses. In fact, let's change the color on that. So you definitely see that you have to put parentheses around your fraction exponent since it's more than just one thing up there. And bringing up my calculator, I'm going to say 0.03. And then in parentheses, I'll put my 6, just like I see it on my paper. And then carrot, parentheses, 27 divided by 11, blows my parentheses. And I find out that the weight of a newborn calf whale that is 6 meters long is actually 2.43 metric ton. At 81 metric tons, how long is an average adult? So this time we have 81 on the other side. And 0.03 l to the 27 over 11. We know we can divide off the 0.03. And that's going to be equal to the l to the 27 over 11. So when you take the 81 and divide it by 0.03, you get 2700. And that's equal to l to the 27 over 11. Well, we don't have wide enough math yet. You might know how to do this if you try to do logs, but we haven't really studied those again. So I'm not going to do it that way because I want to go back to the idea that if you had x to the fraction, in this case it would actually be m over n, then that's really going to be the m over n root of x. And then we could break it up in a fancy way. But for the calculator purposes, this is really simple. We're just going to say that this is the 27 over 11. We're going to have to put that in our calculator first. And then we have to put the root of 2700. Now you have to use a special root. So let me show you how to do this. We're going to have 27 divided by 11. And then we come under math because if we're not using a square root, we come under here to find our root. And then here at 5 it says x root. So we had to put our x, our type of root in first. Then we can use 5 and then the 2700. And then we find out it's approximately 25. So it's approximately going to be 25 and I believe that was meters. All right. One last problem. That in the table can be used to study the relationship between the weight of a bird and its wingspan. So we want to use a regression model to look at this. So I've already come in and put that data in my calculator. So I've done my stat and my edit. And I actually would like to look at the scattergram for this just to show you what a typical power function looks like. So let me clear out everything I have here and go turn on my plot. And then I want to do zoom nine. And that is a typical power function. Where it looks like it's got this curve going like this. So we want to do the regression now. So stat. I'm going to calculate and we want to come down to once you know it again. You don't have to always arrow. But I want to show you the arrow that you recognize. Here at A we have what says a power regression. And that's what we're trying to do. So we wanted to use A and just press enter now. And so we find out that yes we want to do a power regression. And notice it's A times x to the B. So A is, and it's said to three decimal places, 1.653. And then it's going to be 0.557. So 1.653 times x. We'll put that in parentheses. And it was 0.557. He stopped and rewrote that so it looks a little nicer so we can use it easier. So now we want to use the equation to estimate the wingspan of a bald eagle, which is 16 pounds. Now x, if you look back at our table, x is our weight and y is our wingspan. X is in pounds, y is in feet. So we know that we're solving for the wingspan. And we can call this maybe B of L since it's pounds. So B of L is going to be equal to 1.653 times 16, which we know to be x, to the 0.557. And if we pull up the calculator and put that in there. And we find that the bald eagle has a wingspan of approximately 7.8 feet. Okay, now we want to use this equation to estimate the weight of a bobwhite quail with a wingspan of 0.9 feet. That's the y or the B of L. So now we say 0.9 is equal to this 1.653 times x to that decimal. And again, we're going to divide it off. And when you take that 0.1653 and divide it into the 0.9, we actually end up with 0.55 if it's going to be equal to x to the 0.557. And again, we're going to take, and I'm just going to write it as 0.557, and then we'll take the root. So it is the 0.557 root of 0.55. I've rounded, but that's approximately what it is. And if we look at that, we have a wingspan of 9 feet gives us a weight of a bird that's approximately 0.34 pounds.