Understanding Nth Roots and Rational Exponents

So today we're studying nth roots and rational exponents. And before we get started, I have a list of powers here that I'd like you to jot down. So if you could pause the video and jot these down, please do that now. It's going to be important to recognize these numbers. They come up a lot when you're doing these. And to be able to know that 125 is 5 to the 3rd power, or for example, 27 is 3 to the 3rd, or 81 is 3 to the 4th, it's going to be very helpful for for us. So go ahead, pause the video, jot these down, and then we'll get started. Now when we're talking about rational exponents, what we're talking about is a fraction in your exponents. So for example, a raised to the 1 over n, and n could be any number here. If n was 2, this would be a to the 1 half power, or if n was 3, it'd be a to the 1 half to the one-third power, and if a was four, it would be a to the one-fourth, and so on down the line. What you can do is you can turn this into a radical expression, which we call this the nth root of a. You've most often seen this. square root. But there are other roots as well. There's the cube root, the fourth root, the fifth root, and so on. Now on the left hand side, this is what we call rational exponent form. Rational exponent, a rational number is a fraction, a ratio of two integers, and it's an exponent. On the right hand side, it's what we call radical form. And you're going to have to be able to go back and forth between these two. All right, the letter A. n is what we call the index. So for example, if it was a to the 1 third, which is the third root of a, the index would be 3 in that case. Now here's what we're going to have to do today. We're going to have to rewrite things in radical form. If you see radical form, you're going to have to rewrite in rational exponent form. And here's what I want to do. I want to rewrite a to the 1 third using some radicals. And when I say radical, I really am talking about that. that kind of check mark over the A, or that symbol right there, that is what we call a radical. Now before we do that, let's look a little bit at some more rational exponents. So we're going to leave that alone for a second, and we'll be coming back to it. For example, x to the 1 half power can be written as the square root of x. Now mathematicians, we write the square root so often that they decided, hey, we don't need that 2 right there. We're just going to leave it blank. But if you see the square root of the x, as I've written it there, we know that is x to the 1 half power. So you have to be able to work from left to right, but also from right to left. Similarly, if you see x to the 1 3rd power, that can be rewritten as the cubed root of x. The index is 3 in this case. And if you see x to the 1 4th power, that can be rewritten as the 4th root of x. Continuing on here, we know that the square root of 16 is 4. Think about why that is. It's because 4 times 4 is 16, or 4 squared is 16. So 16 can be rewritten as the 4 squared. root of 4 squared is 4. So if you want to figure out what is the cubed root of the number, you've got to think to yourself what times itself 3 times makes 27. And in this case it's 3. 3 to the third is 27, so the cubed root of 27 is 3. If you want to know what the fourth root of 625 is, think to yourself, and you can see the answer is 5 there, but think to yourself, what times itself, 5 or 4 times, is 625? And in this case, that's 5. 5 to the 4th. is 625 and the fourth root of 5 to the fourth power is 5. So again, square root of 16. Well, 4 times 4 is 16, so the square root of 16 is 4. Cube root of 27. Well, 3 times 3 is 4. times 3 is 27 so the cube root of 27 is 3. Oh, fourth root of 625, well 5 times 5 times 5 times 5 is 625 and so the fourth root of 625 is 5. And that's why I wanted you to write down the list at the beginning of this video because you can see that 5 to the fourth is 625, 3 to the third is 27. So if you want to know what is the fourth root of 16, you're going to look at what raised to the fourth. power is 16. And if you're looking in your list, you'll see that's 2. 2 times 2 times 2 times 2 is 16. So let's go back now over here. We were looking at to rewrite each of these in radical form. form and then we were going to evaluate. So 8 to the 1 third. What I want to do is I want to rewrite that as a radical and I know that the 1 third power is the same as hey who is the cubed root of 8. And so if you look in your list you're looking for a number cubed that equals 8. And that would be 2. 2 times 2 times. times 2 is 8. Now with negative 27 to the 1 third power, you're still going to rewrite as the cubed root of negative 27, and then what times itself 3 times makes negative 27? What cubed is negative 27? And in this case it's negative 3. 81 to the 1 fourth power, we'll rewrite as the fourth root of 81. And then what raised to the fourth is 81? It's 3. Negative 64 to the 1 third, we will rewrite as the cube root of negative 64. What raised to the third power is negative 64? It would be negative 4. And then 64 to the 1 sixth power. What raised to the 6th power is 64? That would be 2. So being able to rewrite in radical form is very important. And then having that list right there is going to be important as well. Okay, now you'll notice in all the other examples we've done, the numerator in your exponents has been 1, but that's not always going to be the case. For example, 16 to the 3 halves. The first thing we're going to do is we're going to rewrite this using radical notation. Then we're going to figure out what it is. To do that, we're going to think of two numbers that multiply to give us 3 halves. And it's a lot easier than you think. We're going to make 3 halves in that exponent be 1 half times 3. Then what we're going to do is we're going to use a property of exponents. We're going to rewrite this as 16 to the 1 half raised to the 3rd. Because if you remember, the 1 half raised to the 3rd power, you're multiplying your exponents there. Then we rewrite in radical notation. So that would be the square root of 16 raised to the 3rd power. Then we know who the square root of 16 is the square root of 16 is 4 Then we raise 4 to the 3rd to get 64 So do the same thing with the 27 to the 2 thirds. Just like we did with 16 and 3 halves, you're going to split up that 2 thirds. In this case, we'll rewrite it as 27 to the 1 third times 2. We will use our power of a power property to make it 27 to the 1 third raised to the second. Then you rewrite in exponential form. not exponential form, I apologize, radical notation, and then you take the cubed root of 27, which is 3, square that number, and we get 9. Now the difference in the last two examples is the negative. And so what we're going to do is when we rewrite, we're going to make the numerator the negative number. So negative 3 fifths gets rewritten as 1 fifth times negative 3. Then we use our power of power. 32 to the 1 5th raised to the negative 3rd. Rewrite this and using radical notation, the 1 5th power becomes the 5th root of 32 raised to the negative 3. Figure out who is the 5th root of 32. Look at your list. And then with the 2 to the negative third, again, we don't like leaving negative exponents, so it's time to move that to the denominator, make the exponent positive, and then 2 to the third is the 8, so our answer is 1 eighth. Now do the same thing, pause the video, do the same thing with 81 to the negative 3 fourths. We're going to rewrite the negative 3 fourths as 1 fourth times negative 3. Then we're going to use our power of a power. Then we will rewrite the 1 to the 1 fourth power as the fourth root of 81. Evaluate the fourth root of 81. We don't like to leave negative exponents, so we move it to the bottom, and then we evaluate three to the third, so our answer is one twenty-seventh. Now the reason we're going to need these is we're going to need them to solve equations. When we were solving quadratic equations we had things like x squared equals 4 and we take the square root of each side. But if we have x cubed or x to the fourth we can't just take a square root we need a cubed root or a fourth. So in this case, what we're going to do is, I gave you a couple hints here, what we're going to have to do is get the variable all by itself. We've been doing that since, you know, Algebra 1. Then you're going to take the nth root of each side. So, for example, if it was x squared, you take the square root. If it was x to the third, you take the cube root. x to the fourth, take the fourth root. x to the fifth, take the fifth root. So the first one is 4x to the 5th is equal to 128. Now we have to get the variable all by itself. So I see a 4 times, and that means we're going to be dividing by 4 on each side, just like we've been doing for a long, long time. We'll be left with x to the 5th is 32. Now again, since we have an x to the 5th power, we need to take the 5th root of both sides, and the 5th root of 32 is 2. The next one is 3x to the fourth is 48. Go ahead, try this one. Pause the video, try this one. And then see if you get it correct. Now we're going to divide each side by 3. And we'll get x to the fourth is 16. And then we're going to take the fourth root of each side. And we're going to actually get x equals plus or minus 2. And the reason this is, is because if you do 2 times 2 times 2 times 2, you get 16. But negative 2 times negative 2 times negative 2 times negative 2 is also 16. And so that's why we're getting plus or minus 16. The easiest way to think about it is if the index is even, you need to put a plus or minus. We have 1 half x to the 5th equals 512. In this case, I don't want to divide by a half, so I'm going to multiply by 2 on each side. I'm going to get x to the 5th equals 1024. Then when we take the fifth root of each side, we end up with x equals 4. Last two examples are very similar, just a little bit different. They involve some parentheses here, and you need to get the parentheses all by itself first. And then if the parentheses is cubed, you take the cubed root. If it's raised to the fourth, take the fourth root, and on down the line. So in this one, we have x, y, and z. 3 to the fourth power is equal to 21. The parentheses is all by itself in this case, so we're just going to go ahead and take the fourth root of each side. Now the fourth root of x minus 3 to the fourth power is x minus 3. and 21 is not a perfect fourth. You'll notice that there's no number in your list where if you raise it to the fourth power, you're going to get 21. So what we'll do is we'll just write it as plus or minus the fourth root of 21. Now we're going to add 3 to both sides, and we get x equals 3 plus or minus the fourth root of 21. Now we'll talk about tomorrow how to use our calculator to estimate what these decimals are. In each case... where in this case we get 5.14 or 0.86. The last one is 3 times x plus 5 raised to the third equals 81, and this is what I talk about when I say get the parentheses all by itself. That 3 on the outside is being multiplied right now. We need to do the opposite. Divide by 3 on each side. And so we'll... have x plus 5 raised to the third is 27. Now we want to take the cube root of each side. And since my index was odd, I don't need to put plus or minus 3. Subtract your 5, and you get x equals negative 2.

They come up a lot when you're doing these. And to be able to know that 125 is 5 to the 3rd power, or for example, 27 is 3 to the 3rd, or 81 is 3 to the 4th, it's going to be very helpful for for us. So go ahead, pause the video, jot these down, and then we'll get started.

Now when we're talking about rational exponents, what we're talking about is a fraction in your exponents. So for example, a raised to the 1 over n, and n could be any number here. If n was 2, this would be a to the 1 half power, or if n was 3, it'd be a to the 1 half to the one-third power, and if a was four, it would be a to the one-fourth, and so on down the line. What you can do is you can turn this into a radical expression, which we call this the nth root of a.

You've most often seen this. square root. But there are other roots as well. There's the cube root, the fourth root, the fifth root, and so on.

Now on the left hand side, this is what we call rational exponent form. Rational exponent, a rational number is a fraction, a ratio of two integers, and it's an exponent. On the right hand side, it's what we call radical form.

And you're going to have to be able to go back and forth between these two. All right, the letter A. n is what we call the index.

So for example, if it was a to the 1 third, which is the third root of a, the index would be 3 in that case. Now here's what we're going to have to do today. We're going to have to rewrite things in radical form.

If you see radical form, you're going to have to rewrite in rational exponent form. And here's what I want to do. I want to rewrite a to the 1 third using some radicals.

And when I say radical, I really am talking about that. that kind of check mark over the A, or that symbol right there, that is what we call a radical. Now before we do that, let's look a little bit at some more rational exponents.

So we're going to leave that alone for a second, and we'll be coming back to it. For example, x to the 1 half power can be written as the square root of x. Now mathematicians, we write the square root so often that they decided, hey, we don't need that 2 right there.

We're just going to leave it blank. But if you see the square root of the x, as I've written it there, we know that is x to the 1 half power. So you have to be able to work from left to right, but also from right to left. Similarly, if you see x to the 1 3rd power, that can be rewritten as the cubed root of x. The index is 3 in this case.

And if you see x to the 1 4th power, that can be rewritten as the 4th root of x. Continuing on here, we know that the square root of 16 is 4. Think about why that is. It's because 4 times 4 is 16, or 4 squared is 16. So 16 can be rewritten as the 4 squared.

root of 4 squared is 4. So if you want to figure out what is the cubed root of the number, you've got to think to yourself what times itself 3 times makes 27. And in this case it's 3. 3 to the third is 27, so the cubed root of 27 is 3. If you want to know what the fourth root of 625 is, think to yourself, and you can see the answer is 5 there, but think to yourself, what times itself, 5 or 4 times, is 625? And in this case, that's 5. 5 to the 4th. is 625 and the fourth root of 5 to the fourth power is 5. So again, square root of 16. Well, 4 times 4 is 16, so the square root of 16 is 4. Cube root of 27. Well, 3 times 3 is 4. times 3 is 27 so the cube root of 27 is 3. Oh, fourth root of 625, well 5 times 5 times 5 times 5 is 625 and so the fourth root of 625 is 5. And that's why I wanted you to write down the list at the beginning of this video because you can see that 5 to the fourth is 625, 3 to the third is 27. So if you want to know what is the fourth root of 16, you're going to look at what raised to the fourth.

power is 16. And if you're looking in your list, you'll see that's 2. 2 times 2 times 2 times 2 is 16. So let's go back now over here. We were looking at to rewrite each of these in radical form. form and then we were going to evaluate. So 8 to the 1 third. What I want to do is I want to rewrite that as a radical and I know that the 1 third power is the same as hey who is the cubed root of 8. And so if you look in your list you're looking for a number cubed that equals 8. And that would be 2. 2 times 2 times.

times 2 is 8. Now with negative 27 to the 1 third power, you're still going to rewrite as the cubed root of negative 27, and then what times itself 3 times makes negative 27? What cubed is negative 27? And in this case it's negative 3. 81 to the 1 fourth power, we'll rewrite as the fourth root of 81. And then what raised to the fourth is 81? It's 3. Negative 64 to the 1 third, we will rewrite as the cube root of negative 64. What raised to the third power is negative 64? It would be negative 4. And then 64 to the 1 sixth power.

What raised to the 6th power is 64? That would be 2. So being able to rewrite in radical form is very important. And then having that list right there is going to be important as well.

Okay, now you'll notice in all the other examples we've done, the numerator in your exponents has been 1, but that's not always going to be the case. For example, 16 to the 3 halves. The first thing we're going to do is we're going to rewrite this using radical notation.

Then we're going to figure out what it is. To do that, we're going to think of two numbers that multiply to give us 3 halves. And it's a lot easier than you think. We're going to make 3 halves in that exponent be 1 half times 3. Then what we're going to do is we're going to use a property of exponents. We're going to rewrite this as 16 to the 1 half raised to the 3rd.

Because if you remember, the 1 half raised to the 3rd power, you're multiplying your exponents there. Then we rewrite in radical notation. So that would be the square root of 16 raised to the 3rd power. Then we know who the square root of 16 is the square root of 16 is 4 Then we raise 4 to the 3rd to get 64 So do the same thing with the 27 to the 2 thirds.

Just like we did with 16 and 3 halves, you're going to split up that 2 thirds. In this case, we'll rewrite it as 27 to the 1 third times 2. We will use our power of a power property to make it 27 to the 1 third raised to the second. Then you rewrite in exponential form. not exponential form, I apologize, radical notation, and then you take the cubed root of 27, which is 3, square that number, and we get 9. Now the difference in the last two examples is the negative.

And so what we're going to do is when we rewrite, we're going to make the numerator the negative number. So negative 3 fifths gets rewritten as 1 fifth times negative 3. Then we use our power of power. 32 to the 1 5th raised to the negative 3rd.

Rewrite this and using radical notation, the 1 5th power becomes the 5th root of 32 raised to the negative 3. Figure out who is the 5th root of 32. Look at your list. And then with the 2 to the negative third, again, we don't like leaving negative exponents, so it's time to move that to the denominator, make the exponent positive, and then 2 to the third is the 8, so our answer is 1 eighth. Now do the same thing, pause the video, do the same thing with 81 to the negative 3 fourths. We're going to rewrite the negative 3 fourths as 1 fourth times negative 3. Then we're going to use our power of a power. Then we will rewrite the 1 to the 1 fourth power as the fourth root of 81. Evaluate the fourth root of 81. We don't like to leave negative exponents, so we move it to the bottom, and then we evaluate three to the third, so our answer is one twenty-seventh.

Now the reason we're going to need these is we're going to need them to solve equations. When we were solving quadratic equations we had things like x squared equals 4 and we take the square root of each side. But if we have x cubed or x to the fourth we can't just take a square root we need a cubed root or a fourth. So in this case, what we're going to do is, I gave you a couple hints here, what we're going to have to do is get the variable all by itself.

We've been doing that since, you know, Algebra 1. Then you're going to take the nth root of each side. So, for example, if it was x squared, you take the square root. If it was x to the third, you take the cube root.

x to the fourth, take the fourth root. x to the fifth, take the fifth root. So the first one is 4x to the 5th is equal to 128. Now we have to get the variable all by itself.

So I see a 4 times, and that means we're going to be dividing by 4 on each side, just like we've been doing for a long, long time. We'll be left with x to the 5th is 32. Now again, since we have an x to the 5th power, we need to take the 5th root of both sides, and the 5th root of 32 is 2. The next one is 3x to the fourth is 48. Go ahead, try this one. Pause the video, try this one. And then see if you get it correct. Now we're going to divide each side by 3. And we'll get x to the fourth is 16. And then we're going to take the fourth root of each side.

And we're going to actually get x equals plus or minus 2. And the reason this is, is because if you do 2 times 2 times 2 times 2, you get 16. But negative 2 times negative 2 times negative 2 times negative 2 is also 16. And so that's why we're getting plus or minus 16. The easiest way to think about it is if the index is even, you need to put a plus or minus. We have 1 half x to the 5th equals 512. In this case, I don't want to divide by a half, so I'm going to multiply by 2 on each side. I'm going to get x to the 5th equals 1024. Then when we take the fifth root of each side, we end up with x equals 4. Last two examples are very similar, just a little bit different.

They involve some parentheses here, and you need to get the parentheses all by itself first. And then if the parentheses is cubed, you take the cubed root. If it's raised to the fourth, take the fourth root, and on down the line.

So in this one, we have x, y, and z. 3 to the fourth power is equal to 21. The parentheses is all by itself in this case, so we're just going to go ahead and take the fourth root of each side. Now the fourth root of x minus 3 to the fourth power is x minus 3. and 21 is not a perfect fourth. You'll notice that there's no number in your list where if you raise it to the fourth power, you're going to get 21. So what we'll do is we'll just write it as plus or minus the fourth root of 21. Now we're going to add 3 to both sides, and we get x equals 3 plus or minus the fourth root of 21. Now we'll talk about tomorrow how to use our calculator to estimate what these decimals are. In each case...

where in this case we get 5.14 or 0.86. The last one is 3 times x plus 5 raised to the third equals 81, and this is what I talk about when I say get the parentheses all by itself. That 3 on the outside is being multiplied right now.

We need to do the opposite. Divide by 3 on each side. And so we'll... have x plus 5 raised to the third is 27. Now we want to take the cube root of each side.

And since my index was odd, I don't need to put plus or minus 3. Subtract your 5, and you get x equals negative 2.

Transcript for:Understanding Nth Roots and Rational Exponents

Transcript for:
Understanding Nth Roots and Rational Exponents