Common Core Algebra II.Unit 4.Lesson 2.Rational Exponents

Let's jump right into it. Rational or fractional exponents, right? I mean, we all know that 5 to third means 5 times 5 times 5. 5 to the zero equals one eh, maybe I'm not totally comfortable with that, but at least I've used it enough that I'm kind of aware of it. And 5 to the negative three, well, that means dividing one by 5, three times, or dividing one by 5 to the third. But again, today, what we're going to be dealing with is something like that. What does it mean to have 5 to the one third? Does that mean multiply by 5, one third times? Why don't I don't really know what that even means? So let's jump into it. Now strangely enough, what we're going to need more than anything else is the exponent property from the last lesson, the product property of exponents. Okay? 

Now in this problem, it says use that property to rewrite each of the following as a simplified exponential expression. There is no need to find a final numerical value. All right? So try these out, pause the video after the last lesson you should be able to do all four of these rather quickly. Go ahead. All right. Well, if you recall from our last lesson, when we have a base raised to a power raised to another power, we simply multiply the two powers together. So two to the third to the fourth is two to the three times four. Which is two to the 12. Now that is some number, but it's rather large and we're not going to we're not going to worry about it. 5 to the negative two to the 5th, remember this we extended this to negative exponents as well. Would be 5 to the negative two times 5 or 5 to the negative ten. Three to the 7th to the zero. Well, that's going to be three to the 7 times zero. Which is going to be three to the zero. Or if you couldn't help yourself and you wrote down the number one, that's great too. 

Now this one looks horrible. But keep in mind, we could always sort of work our way out, right? For the second to the negative two would be four to the negative four. And then we would have four to the negative 8. All right. So I just wanted to review that law because strangely enough, strangely enough, that is going to be how we are going to try to first understand fractional exponents. Anyhow, pause the video now if you need to and then we'll move on and start to tackle these things. All right. Let's do it. Fractional exponents here we come. All right, so we're going to start with the easiest fraction. The one that you learned first, a fraction of one half. And we're going to ask ourselves, what does it mean to have 16 to the one half? It certainly doesn't mean that we have the number 8. That would be 16 times one half. But let's take a look at letter a, it says apply the product property to simplify 16 to the one half squared. 

Then it says what other number squared yields 16. So think about this for a second. If we have 16 to the one half. Squared, then that property that we just had said that it should be 16 to the one half times two. Which is 16 to the first, and that's just 16. All right. So what? Right? Well, so what is that this follow-up question? What other number squared yield 16? So what other number? When we squared equals 16, right? Because I mean, that's what we're doing here. We're taking a number. We're squaring it, and we're getting 16. Well, four squared is also 16. So that means that 16 to the one half must be four. 16 to the one half must be four. But that means that 16 to the one half must be the square root of 16. There it is. Right. And there we have it. Fractional exponents are roots. 

Fractional exponents. Are roots. Isn't that cool? Specifically, the one half power is the square root. All right, we're going to extend this to cube roots and four roots and things like that. But isn't that cool? A power of one half just indicates a square root. That's a lot to absorb. So what I want you to do is pause the video right now and think about this a bit. If it seems strange and you're not really quite comfortable with it, don't worry about it. We'll get more comfortable as the lesson moves on. So pause the video now. All right, let me clear out the text. Then let's try to reinforce this. Test the equivalence of the one half exponent to the square root by using your calculator to evaluate each of the following. 

Be careful how you enter each expression. All right, well, it's been a little bit since we've opened up the TI 84 plus, but let's do it right now. All right, there's my TI 84 plus. Awesome. Okay, so all I really want to do is just some button pushing on here. Nothing fancy. I'm not graphing equations or anything like this. I just want to look at 25 to the one half. And see whether it's really equal to the square root of 25. Is this really the case? So let's do it. Let's type in 25. Then let's use that carrot button right here. Do that to the one half. Now my calculator and many of yours have kind of a modern operating system. But you do have to watch out because on some calculators, you absolutely positively must enter it like this. With that one half in parentheses. On my calculator, I don't have to do that. 

In fact, all I have to do now, since I have 25 to the one half, is hit enter. So let me do that. And take a look. Look at that. We get 5. Which is obviously the square root of 25. Notice we're not getting a negative 5 as well, because 25 to the one half is by definition. The positive square root of 25. The positive square root. All right? Now, I've picked perfect squares here, 81 and 100. Why don't you on your calculator verify that those also give you square roots? Pause the video now. All right. Well, hey, let's do them both real quick. All right, let me type in 81, carrot button. One half, enter. There we go. We get 9, right? Just the square root of 81. Let's try a hundred. Let's do 100. Raised, so the one half equals, there it is again. I love fractional exponents. They're so cool. Because they indicate roots fractional exponents indicate roots. All right. 

I don't think we're going to need the TI 84 for the rest of this lesson. But you may want to keep your own calculator handy in order to do numerical work. Okay? So keep your calculator handy, but I'm going to put mine away now. All right. I'm also going to clear out the screen now. So pause the video if you need to. Kind of doubt you will, but positive you have to. All right, let's clear it out. So let's summarize. Make sure that we really have this. Unit fraction exponents. For N, given as a positive integer, B to the one over N is simply its n-th root. Now all we've done is worked with square root so far. But pretty soon we'll be working with cubed roots and other ones as well. All right? But that's the key. So if I have a number to the one third, that's the third root, a number to the one fourth. That's the fourth root, the number to the one 5th, the 5th root. I think I'll stop there. You probably get the picture. Let's work with it a little bit more. All right, it says rewrite each of the following using roots instead of fractional exponents. Then, if necessary, evaluate using your calculator to guess and check to find the roots, don't use the generic root function. 

Check with your calculator. All right, so in other words, 125 to the one third is the same as the third root of 125. Now, of course, what we're looking for is the number that we multiply by itself three times to get 25, and that happens to be 5. 16 to the one fourth, that happens to be the fourth root of 16. Again, we're looking for a number when we multiply it by itself four times gives us 16. And that's two. Now, let her see in letter D, we have negative exponents, and there are no negative roots, I mean, you can have a negative square root of 9. It's not that. But no, here we're just going to combine both the idea of a negative exponent being division. And then a fractional exponent being a root. To end up with an answer of one third. So we get rid of the negative by placing it in the denominator. And then we get rid of the fractional exponent by placing it under a root. 

So same thing here, 32 to the negative one 5th would be equal to one over 32 to the one 5th. Which would be one over the 5th root of 32, 5th through to 32 is two. So that's one half. All right. So fractional exponents, when we have what's known as a unit fraction, one 8th, one 9th, one half, negative one fourth, et cetera. Those indicate pretty simple roots. But now it boils down to a really fascinating question. What would it mean to have something like this? 8 to the three halves. What does that mean exactly? So that's what we're going to explore in the next exercise. Pause the video now, write down anything you have to before we move on. All right, we're doing it. Cleared. And moving on. Okay, let's take a look at four to the three halves. So now we're going to deal with a fractional exponent that isn't just one half, one third, one fourth, one 5th. 

We've got three halves. Let array, fill in the missing blank, and then evaluate this expression. So four to the three halves is equal to something to the one half. I'll never forget this basic exponent property. If I've got X to the a to the B, I get X to the a times B but keep in mind that three halves is equal to three times one half. And it's also equal to one half times three. So four to the three halves would be the same as four to the third to the one half. But then four to the third is four times four, which is 16. Times four is 64. So I'd have 64 to the one half, which would be the square root of 64 based on what we just did. Which would be a. Or you could look at it the other way. We could say four to the three halves is the same as four to the one half to the third. Well for the one half is the square root of four. Which is two, and then two to the third is 8. All right, so it's pretty easy, right? When we have something like four to the three halves, we've got two pieces. We've got the numerator, and we've got the denominator. 

This really represents the power, and this represents the root. And the cool thing is we can do them in either order. Either order. So what I'd like you to do, I'm not going to break out the TI 84 plus right now. But I'd like you to verify on your calculator that four to the three halves is equal to 8. Take a moment to do that. All right, great. Well, I hope it worked out all well for you. Letter D says evaluate 27 to the two thirds without your calculator. Show your thinking and then verify with your calculator. So go ahead and do this pause the video now. And try to figure out what 27 to the two thirds is without your calculator. This could be a little challenging. All right. Let me explain why I said this could be a little bit challenging. You see? When you're doing 27 to the two thirds, you have to decide. Are you going to do the power first or are you going to do the root first? Are you going to look at it as 27 squared? To the one third, or are you going to look at it as 27 to the one third? Squared. Well, this is much more challenging without a calculator. Number one, what's 27 squared, right? I have to think about that pretty tough. And then I'd have to take the cube root. Oh, I don't want to do that. 

This is going to be the much easier one without a calculator. Because we're doing the cube root of 27. And then we're squaring it. So the cubed root of 27 is three. And we square, and we get 9. Hopefully, you verify that on your calculator. If not, pause the video now, take in everything we've got and check that one on your calculator. All right. Let's clear out the screen. Summarize, and then we'll finish up the lesson. Rational exponent connection to roots. For the rational number M divided by N, we define B to the M divided by N to be the n-th root of B to the M or to be the n-th root of B raised to the N so in other words, if I have 5 to the two fifths, I can think about this as the 5th root of 5 squared or I can think of it as 5 squared and then I've got the 5th root. Either way, if I have three to the two sevenths, I can either think about that as the 7th root of three. 

That I then square, or I can think of that as three squared, and then I take the 7 through. That's it. That's the whole deal with fractional exponents. They simply allow us to express a root square root cube root four through root, et cetera, as an exponent. And vice versa. All right, copy down what you need to, and then we'll finish up the lesson. All right, here we go. Oh, we've got one more exercise. What am I thinking? Sorry about that. One more exercise. It says evaluate each of the following exponential expressions involving rational exponents without the use of your calculator. Show your work, then check your final answers with the calculator. All right, so see how much you understand about this by pausing the video now and working through these three problems. All right, let's do it. 16 to the three fourths. 

Again, when it's numerical, there's no variables involved. The first thing that I'm always going to do is do the root. So I've got 16 to the fourth, I'm sorry, 16 to the one fourth that I'm cubing it. So that's the fourth root of 16, which is two, and then I need to cube it. So that's a 16 to the three fourths is 8. 25 to the three halves. Well, that's going to be the square root of 25. Cubed. So this indicates a square root, this indicates a Cuban. The square root of 25 is 5. 5 cubed is a 125. Finally, 8 to the negative two thirds, again, like before, what I first want to do is deal with that negative exponent. That'll be one divided by 8 to the two thirds. Then I'm going to deal with the root first. That's the cube root of 8. That I then square. The cube root of 8 is two, two squared is four. So I get one fourth. All right, now that I'm fairly certain is the last exercise. 

So pause the video now and copy down what you need to. All right. So in this lesson, we really did look at something quite new and quite different. And that's the idea of a fractional exponent. What does it mean to raise a number to the one half? To the one third, then by extension, what does it mean to raise it to things like the four fifths to the 7 halves, et cetera it's can be tricky. And we're going to work with it a lot throughout the course. And of course, as you move on to pre capitalist next year, hopefully you'll be working with fractional exponents a lot too. For now though, I want to thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking. And keep solving problems.