Hi, and welcome. My name is Anthony Varela. And today in this tutorial, I'm going to walk through simplifying radical expressions. So what we're going to do is look at properties involving products. We'll look at properties involving quotients or division. And we're going to keep in mind our perfect squares and our perfect cubes because that's really going to help us while we're simplifying.
So to start, let's remember that any radical expression can be written as an exponential expression. So, for example, here I have the cubed root of 6. This can be rewritten equivalently as 6 to the 1/3 power. So what's underneath our radical becomes a base in an exponent and our cubed root becomes a fractional exponent, 1/3.
Similarly, here we have the fourth root of 5x. We can write this as 5x to the 1/4 power, where it is taking what's underneath a radical and becomes a base in an exponential expression. This fourth root indicates that our exponent is a fraction, 1/4.
And we can go the other way, too. We can write 7 to 1/2 power as a radical. So this would be the square root of 7. So here our base and our exponential expression becomes the radicand, or what's underneath the radical. And the power of 1/2 translates to a square root.
And one more example here. We have x plus 2 and are raising that to the 1/5 power. So, as a radical, x plus 2 goes underneath the radical sign. And then our exponent of 1/5 indicates a fifth root.
So why did I go through all of these examples? Well, perhaps you've learned about properties of exponents. In seeing that exponential expressions can be rewritten as radical expressions, there must be properties to radicals. And that's what we're going to walk through today.
We're going to look at how you can combine radicals through multiplication or break down radicals through division. And we're going to develop some properties or some rules for radicals. And these properties are going to involve products and quotients, so multiplying and dividing.
So let's talk about the product property of radicals. And the product property states that we can have the nth root of a and multiply that by the nth root of b. And we can rewrite this as one radical, the nth root of a times b.
So here's an example of this product property. The square root of 3 times the square root of 5. I can write this as a single radical. I just need to multiply 3 and 5. So this equals the square root of 15. Pretty simple.
Now notice that our roots must be the same. So what does that mean? That means I can't say that the square root of 3 times the cubed root of 5 equals the square root of 15. I can't say that the square root of 3 times the cubed root of 5 equals the cubed root of 15 because our roots are not the same in these examples. Notice, our roots must be the same.
Now we can do this in the other direction, too. Let's take the cubed root of 40. And so here we're looking at an expression that looks like this. And we can write this as the product of two radicals with the same root.
So I can choose any two numbers that multiply together to equal 40. So I'm going to say 8 and 5. So I know that the cubed root of 40 equals the cubed root of 8 times the cubed root of 5, thanks for our product property of radicals. Now the cube roots of 8 equals 2. So this is where remembering your perfect cubes comes in handy because I can simplify this even further as 2 times the cube root of 5.
All right. Well, taking one more look at our product property of radicals, even though we write our general rule as taking two radicals and putting them into one, we can do several factors, as well. I don't just need to do two. So I'm going to break 180 down into several different factors.
So, first, I know that 180 equals 18 times 10. So I can write the square root of 180 as the square root of 18 times the square root of 10. I can break down 18 and 10 even further. 18 equals 2 times 9. And 10 equals 2 times 5.
And now I can break down 9 even further. So 9 I'm breaking down into 3 and 3. So here I have a string of multiplication involving radicals. And the cool thing here is, since we're dealing with square roots, if I see two identical factors, they multiply to be just that number underneath the radical.
So root 2 times 2 equals normal 2. And root 3 times root 3 equals regular old integer 3. So I can simplify this as 2 times 3 times the square root of 5. And one step further, 6 times the square root of 5. So that's how we simplified the square root of 180. This is considered fully simplified.
Now let's talk about quotient properties of radicals. So this involves dividing. And so our quotient property says that if you have the nth root of a and you divide that by the nth root of b, you can write this as a single quotient, a/b, and then take the nth root of that. Notice again that our roots must be the same. So that's our quotient property of radicals.
So here's an example. The square root of 18 divided by the square root of 2. I can write this as a single fraction, 18 over 2 underneath a single radical sign. And now I know that 18 divided by 2 is 9. Remember, our roots must be the same. And the square root of 9 equals 3. So this looks very complicated right off the bat. But using our quotient property of radicals, it simply equals 3.
Let's try another example going in the other direction. Here, we have something that looks like this. And we can use this quotient property of radicals to write it something like this.
So I'm going to write this as the cubed root of 27 divided by the cubed root of 8, because I'm splitting this up into two radicals. Now 27 is a perfect cube. An 8 is a perfect cube, as well. The cubed root of 27 equals 3. The cubed root of 8 equals 2. So I can write this as simply 3/2 or 1 and 1/2. So the cubed root of 27/8 equals 3/2.
So what did we talk about today? We talked about two properties for radicals that help simplify our radical expressions-- the product property and the quotient property. Our product properties says that we can have the nth root of a times and the nth root of b and combine those into a single radical expression, the nth root of a times b.
Our quotient property says we can divide a radical by another radical by taking those two individual radicands, a and b, making a single quotient out of those, and taking the nth root of that fraction.
And that's it. Thanks for watching this video on simplifying radical expressions. Hope to see you next time.
