In this unit, we'll explore radicals and rational exponents. It's likely that you've seen radicals before, but before we get into all these symbols and different definitions, let's just look at some squares. We can think of a square root as the side length of one of these squares. This square has an area of 4, this square has an area of 25, And this square has an area of 81. Knowing each of these areas, what do you think would be the square root, or the side length of each of these squares? Write the side length for each of these sides here and the same for the other squares.
Well, we square, or multiply the two side lengths together to get the area. So this first side length would be 2, since 2 times 2 equals 4. The second square would have a side length of 5, since 5 times 5 equals 25. And the last square would have a side length of 9, since 9 times 9 equals 81.
To find the side length of each of these squares, we actually took the square root of each of the areas. We can rewrite 4 as 2 squared. So we have the square root of 2 squared. This square root undoes the square power, leaving us with just two. The same thing happens for the square root of 25. We rewrite 25 as 5 squared, and then we take the square root of 5 squared, which equals 5. And the same is true for this third case. We have the square root of 81, which gives us actually square roots. It's likely that you've seen these before, but you might not have known there's an index of two here. The index indicates the route we are taking since we don't always have to take a square root. We'll see other types of roots later but for now just know that all square roots have an index of two, but it's really rare that you'll ever see written. You'll usually just see it like this. This is a list of some perfect squares that you run into often. You want to make sure that your comfortable recognizing these perfect squaresm since they help us simplify radicals. Remember, this is the radical symbol and in this case, the index is at 2 so we have a square root. We don't list the 2 for square roots but we know it's there.
I said we wouldn't always be working with square roots, sometimes we can work with a cube root. When we take a cube root, we're really trying to find triples of a factor that are underneath the radical sign. In other words, we need a factor raised to the 3rd power underneath here in order to remove it from the radical symbol since it's a cube root. We can think of cube roots as a side length of cube with a given volume. For example, if this cube has a volume of us 8. So, what do you think are the side lengths for each of these cubes? Keep in mind that the side lengths of a cube are equal, and the volume can be found by multiplying the same number together three times.
The first cube has a side length of 2, since 2 times 2 times 2 equals 8. The second cube has a side length of 5, since these three factors multiply together to give us 125. And finally, this last one has a side length of 7, since 7 cubed equals 343.
To find the side link of a cube, we actually take a cube root of the volume. So the cube root of eight would be the cube root of two cube. We need this factor of two of third in order for us to take it out. Since this index is a three, we need to group the three factors to take out the two from the radical symbol. A cube root really just send us a third power. We can think of roots and powers and inverse operations of one another. Pretty neat, huh? We can apply the same concept for taking the cubed root of 125. We see that a factor of five appears three times underneath the radical, so we just have five. And for the cubed root of 343, we just have seven. Here is the list of some perfect cubes. These numbers are considered to be perfect cubes since we can take the cube root of them to get an integer. Like we saw before, 8 was the same thing as 2 cubed. So the cubed root of 8, or 2 cubed is just 2. You'll want to keep a list of these perfect squares and perfect cubes close by since they come up frequently. Make a list on a separate sheet of paper or use and index card but either way you want to be really comfortable with your perfect squares and your perfect cubes.
We can also write each of these square roots and each of these cubed roots in another way. We can use a fractional exponent or a rational exponent to write them. Remember that the index here for these square roots is a 2. So using this index of 2 and this power of 1, we can rewrite the square root of 4 as 4 to the square root of 81 can be written as 81 to the 1 half. So notice that this fractional exponent really represents a radical, and in this case a square root. We can write the cube root of 8 as 8 to the 1 3rd, and we can rewrite the cubed root of 125 as 125 to the 1 3rd power. The same is true for 343. So, what does each part of the fractional exponent represent? Does the numerator represent a power, the index of the root, a square root, or a reciprocal? And what does the denominator represent? Is it a power, an index of the root, a square root, or a reciprocal? Use these same words for each of the answer choices over here, and then pick one for this blank, and one for this blank. Good luck.
The numerator of our fractional exponents represents a power, since this one comes from a power that's inside the radical or 4 to the 1. The same is true for our cubed roots. Each of these is raised to the first power, so we see that one and each of the fractional exponents. And the denominator actually represents the index of the root. The 2 is the index of a square root, and a 3 is the index of a cubed root. It tells us what type of root we're taking. The denominators of the fractional powers on the left indicate that we're taking a square root. Whereas the denominators in the fractional powers on the right indicate that we're taking a cubed root.
When we see a fractional exponent, we want to think about power divided by root. The numerator is the power of our base, and the denominator indicates the root we're taking, a square root here. For the second example, we'll still have a power of 1 here, and then 3 as the root, the cubed root. Let's see if you can put this knowledge of fractional exponents to use.
What do you think would be the answers for each of these roots? Keep in mind what we just learned, power divided by root. When you think you're ready put your answers in each of these boxes. Good luck.
The answers are 8, 4 and 16. Nice work if you got those three correct. Now, we haven't seen something like this third one before, so don't worry if they gave you trouble. For the first one, I'm going to have power divided by root. So, 64 to the 1 for my power, and I take a square root since the denominator is a 2. So here, I really have the square root of 64, which we know is 8. And if I think about rewriting this, I can see all the intermediate steps we can take to get there. We rewrite 64 as 8 squared and now, since we have a square root in a square, we can use power divided by root. We have 8 to the 2 divided by 2, which is really just 8 to the 1 or 8. Now, these steps here are not necessary in order to get to our final answer. If we understand that the square of 64 is power divided by root to simplify. We use similar reasoning for the second problem. We know that we have power divided by root, so 64 to the 1 for our power, and 3 for our root, a cube root. We know the cube root of 64 is 4, and we can rewrite 64 as 4 cubed and use power divided by root to simplify. Again, this part isn't necessary if we know the cube root of 64 is 4. This is one of the ones on our list of perfect cubes. For this third one, we're going to have the cube root of 64 squared. This is our power of 2, and this is our root of 3. We know the cube root of 64 is simply 4. So we're going to have 4 squared. This gives us a final result of 16. Now, you might have done this problem differently. You might of squared the 64 first. We could have also written our power inside of the radical first, and then taken the cube root. If we square So, in terms of doing a root first or an exponent first, it doesn't matter. Usually, it's easiest to take the root first since we can get a smaller number here, and then square it. This is much harder since I wouldn't really know the cube root of 4,096 off the top of my head. And, in fact, I definitely don't.
In general, a fractional exponent represents a power devided by a root. X is the radicand of what's underneath the radical symbol. A is going to be the power of the radicand. So if the numerator are here, we'll see that as a power inside of our radical sign. And then b, is the index of the radical or it tells us which type of root we're taking. A square, a cube or maybe something else. So if we see something like x to the 3 5ths, we really know that's the 5th root of x cubed. Here's our power and here's our index. Power divided by root.
For the rest of the unit you'll be entering in routes. Let's get familiar with how to enter these different types of roots using our awesome tool Math Quill. To get the square root symbol we simply want to type a backslash and then the letters sqrt for square route. When you type this you won't see anything quite yet, you'll just see your backslash and the letters. You'll need to hit space in order for the square root symbol to appear. You'll see this, and your cursor bar will be underneath the radical symbol for you to type in whatever is inside. And for this first one, you would want to type in a 2. But we haven't just seen square roots. We've also seen cube roots. For a cube root or the nth root, we simply want to type in a \ and then these letters. Then of course we'll need to hit the Space. This will create a radical sign with our cursor bar in the index position for the radical. Here our index is a 3, so we want to type the number 3 to get that same index. Your cursor will still be in the position of the index, so we need to hit the Right Arrow key once. To move it back down underneath the radical, the cursor bar should appear here now and were free to type in whatever expression is inside radical. In this case we would just type x. Now see few have the hang of it. Try entering these two expressions and using the keystrokes on the screen then try tackling these to challenge problems. This one combines fractions, exponents, and radicals. And be careful on this last one there's really a 4 for the index of this group.
This would be the keystrokes to enter this third expression, and these would be the keystrokes to enter this last expression. We can start entering this expression by typing in a fraction bar, that's the forward slash. It'll create a spot for a numerator and a denominator. After we type in the numerator, we want to hit the right arrow key twice to move the cursor from the exponent position down to the denominator. Then, we enter in our denominator using what we know about the square root. After we enter this under this underneath our radical, our cursor will appear here and we need to hit our right arrow key twice to move back to the main line. Then, we just type in negative 4. For this last expression, we just want to be careful with the parentheses. If we start with an open parenthesis, another one is automatically created on the other side. So, notice that when I get to the 5, I don't need to type in another parenthesis, I just the right arrow key and the carrot key and a 2 for the exponent. If you got all of these correct, you're really becoming a Math Quill pro.
Let's use our new knowledge of fractional exponents to express these powers as a group. So I want you to convert each of these to a Radical Form. You'll want to use your knowledge of MathQuill and the idea of power divided by root. Good luck.
We know a fractional exponent represents power divided by root, so this first one will be the cube root of r squared, we have r squared here since the power is 2 and we take the third root since our denominator is 3. For our second example, we take the 3rd root of 4x, we don't need to write this exponent of 1 in here since anything raised to the 1st power is the same expression, 4x to the 1 really equals 4x. For this third one we also have power divided by root. So we take the third root of this expression raised to the 4th power. You might have distributed this 4th power to each of the factors inside of a radical. And you would wind up with this radical. Either one is correct. It turns out that we can simplify this radical but weren't getting into that yet. We're do that in the next lesson. The other way you could have written the answer is like this, you could have take the cube root of 5ab squared c and then raise that quantity to the 4th power. We can take the root first and then raise the result to the 4th power or we can raise this expression to the 4th power and then take the 3rd root. This is true because the exponent of 4 and the exponent of one third have the same priority in the order of operations. A root really represents a type of exponent so we could do either exponent first.
So we've looked at radical form throughout this lesson. We know that this really represents a power divided by a root. We can also write radical form in exponential form using a fractional exponent. So lets work in the reverse direction this time. Lets go from radical form to the exponential form. Try expressing each of these radicals with one fractional exponent. Be sure you raise the entire radical to the exponent. Keep in mind that when you have something than just a variable and an exponent inside a radical, you'd erase this entire radical to the fractional exponent. Here we can also simplify further since the 4 m squared is the same thing as 2m raised to the second power. 2 squared is 4 and m squared is m squared. We're raising the power to a power so we really multiply these powers together to get 2m to the 2 3rds. These two should connect from what we know since. 2 m squared is 4 m squared, and then we have a root in our denominator, or a cube root. Take some time on these, and when you're ready, enter your answers here.
For the first one we just have p to the 3 5ths, power divided by root or the index. For the second one we'll have the quantity of 3, 2 to the 4th raised to the 1, 6th. The power of this radican is really a 1, so we have our power of 1 divided by our index, our root, which is 6. For the third one we'll have the quantity 5x squared raised to the 1 3rd power. We're taking the cubed root of this expression, and again we don't use the 2 as a power because the square is not the power of the 5 as well. It's only the power of the x. We have this quantity raised to the 1, so there's our power and then the 3 or the cube for our root. And for out last one we have 25 x squared raised to the one third power. We have power divided by a root. You might have simplified this, which is great. We could have rewritten 25 x squared as the quantity 5x squared. Then, since we're raising a power to a power we can multiply 2 times 1 3rd. This would give us 5x raised to the 2 3rds, another acceptable answer. Great work if you got at least two of those correct. I know these were pretty tricky.