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Simplifying Radical Expressions

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Today, we're going to talk about simplifying radical expressions. A radical expression is just an algebraic expression-- so some combinations of numbers and variables. And that expression also has a radical sign in it. So we're going to talk about two different properties that we can use to simplify radical expressions, and then we'll do some examples.

So before we get into our two properties, let's quick look at how radicals and exponents are related to each other. So if you remember, an exponent that is a fraction-- so like some base 5 to an exponent 1/3-- is equivalent to writing that as a radical, where 5 is underneath the radical, or is the radicand, and our index would be 3, which is the denominator of our exponent.

So because we have the ability to write exponents as radicals and vice-versa, we can use properties to simplify radicals just as we did with simplifying exponents. So properties involving products and quotients. So our properties today, we'll be looking at how we can break down this radicand underneath the radical sign using multiplication and division.

Let's look at our product property of radicals. This tells us that the n-th root of a times the n-th root of b is equal to the n-th root of a times b. And this is provided that your index of your original radicals are the same. So let's see what that would be with some numbers as an example.

So I could have the square root of 20 multiplied by the square root of 3. Now, these two radicals have the same index. They both have an index of 2 because they are square roots, so I can use my product property of radicals to combine these.

20 times 3 is just going to give me 60. So this will simplify to be the square root of 60. Let's do an example going the other way.

So let's say I start with the cubed root of 320. I can break down 320 as the product of 64 and 5. So using my product property of radicals, I know I can rewrite this as the cubed root of 64 times 5. Or, the cubed root of 64 times the cubed root of 5.

Now, this is useful because I know that the cubed root of 64 is going to evaluate to be an integer, which is just 4. So the cubed root of 64 is going to give me 4. And then I can just bring over my cubed root of 5. So I can see that the cubed root of 320, using my product property, can be evaluated to just be 4 times the cubed root of 5. Let's see how we can simplify this same cubed root of 320 by breaking it down into its prime factors.

So to break it down into its prime factors, we're just going to start by breaking it down into factors. So 320. I'm going to start by breaking that down to be 160 times 2. And 2 already is a prime number, so I already have one prime factor.

I'm going to continue breaking down 160 to be 40 times 4. And then I can break down 4 to be 2 times 2. So now I have two more prime factors.

I can break down 40 to be 4 times 10. Again, I can break down this 4 to be 2 times 2. And break down this 10 to be 2 times 5. So now I've broken down 320 to factors which are all prime. So I'm going to use this to rewrite my cubed root of 320 using my factors, which are all prime.

So I've got one, two, three, four, five, six 2's. So I'm going to write that out as the cubed root of 2 times 2 times 2 times 2 times 2 times 2. And then I have one 5. So times 5.

Now, since I'm taking the cubed root, any time I have a product of three of the same numbers, I know that that will simplify to be just that number. Because 2 times 2 times 2 is the same as 2 to the third power. And we know that a 3 exponent and a cubed root will cancel each other out. And it will simplify to just be the base, or just 2.

So I actually have two groups of this 2 to the third power. So I can simplify this as bringing both of these groups of 2 being multiplied by itself outside of my radical. And so that will just be 2 from this group times 2 from this group times my cubed root of 5.

I can simplify one step farther by multiplying 2 times 2, which I know is just going to give me 4. And then again, I have times this cubed root of 5.

So let's look at our quotient property for radicals. I've got the n-th root of a divided by the n-th root of b. And that is equal to or the same as the n-th root of a divided by b. So let's look at an example of that using numbers.

And again, to be able to use this property we need to have our index be the same. So I have the square root of 80 divided by the square root of 5. They're both square roots, so they both have an index of 2. So we can use our quotient property.

I'm going to combine them to be the square root of 80/5. And 80 divided by 5 gives me 16, so I know that this is just equal to the square root of 16. And the square root of 16 evaluates to be 4. So using my quotient property of radicals, I simplified this to be 4. Let's try this the other way with an example with numbers.

So let's say I have the cubed root of 125/8. Using my quotient property of radicals, I can rewrite that as the cubed root of 125 over the cubed root of 8.

Now, the cubed root of 125 I know is just the same as 5. And the cubed root of 8 I know is just 2. So this, using my quotient property of radicals, simplifies to be 5/2.

Here are our key points from today. Make sure you get them in your notes if you don't have them already.

So we can use the product or quotient properties to combine or break down radicands through multiplication or division. And this will help us simplify radical expressions. When we're combining two radicals into one using the product or quotient property, the index of the radicals must be the same.

So I hope that these key points and examples helped you understand a little bit more about simplifying radical expressions. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Product Property of Radicals
- Product Property of Square Roots
- Quotient Property of Radicals