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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how to apply the product rule to your radical expressions. Learn how this product rule actually helps you to simplify your radicals as well.

Your **radical expressions** are your mathematical expressions with the radical symbol in them. The radical symbol is the symbol that you use for your square root. Your square root is actually a radical where the root is 2. That little number that you sometimes see to the top left of the symbol tells you which root you are working with. If you see a number 3, it is the third root. If you see a number 4, it is the fourth root.

If you recall, taking the square root of something means finding the number that gives you the number underneath the symbol when squared, or multiplied by itself twice. Likewise, when taking the third root, it means finding the number that gives you the number underneath the symbol when cubed, or multiplied by itself three times.

We will be talking about these radicals in this video lesson. Specifically, we are talking about how to simplify our radicals by factoring. As with all things in math, we try to simplify in any which way we can. So let's get going and see how we can simplify our radicals.

What we are going to do is to make use of the **product rule for radicals** to help us simplify. This rule tells us that one radical multiplied by another radical will equal the radical of the product of the numbers inside the two radicals.

So if I had the square root of 3 times the square root of 5, this would equal the square root of 15. I just multiply the numbers underneath our radical symbol.

One thing you have to remember here, though, is that your radicals must be the same root. You can't apply this rule if you are multiplying a square root and a third root. But you can apply it if you are multiplying a fourth root with another fourth root.

You can remember this rule by simply thinking of the word 'product' and what it means. Ask yourself: what does the word 'product' usually mean in math? Why, it means 'multiplication.' So, applying the product rule means you multiply. Now, let's see how we can use this rule to help us.

Let's try simplifying the square root of 32. We begin first by figuring out the different ways that we can factor 32. We can factor 32 as 2 * 16 or 8 * 4.

We can apply the product rule for radicals and rewrite our square root of 32 as the square root of 2 times the square root of 16 and also as the square root of 8 times the square root of 4.

Now, our job is to pick one of the rewritten expressions and see if we can simplify or evaluate one of the square roots. We look for square roots that we know. We see the square root of 16 and realize that we know that it equals 4.

So I can rewrite this expression as 4 times the square root of 2. When we rewrite in this form, we don't need to put in a multiplication symbol; we can simply write our number in front of the radical symbol.

Now we look at the radical that we are still left with and we repeat what we just did to see if we can simplify that even further. If we can't, then we stop and we are done simplifying our radical.

Let's look at another example. Let's simplify the third root of 24. We first find the different ways that we can factor 24. We have 6 * 4, 2 * 12, and 8 * 3.

We apply the product rule for radicals and we rewrite our third root of 24 using our factors. Now we go through and look at which one of the rewritten expressions has a third root that we know. We see the third root of 8, which we know. So we will use this rewritten expression.

We know that the third root of 8 is 2, so we can rewrite this expression as 2 times the third root of 3. Now we repeat our process with the third root of 3 to see if we can simplify this any further. We can't, so we are done.

Now let's see what we've learned. We've learned that **radical expressions** are your mathematical expressions with the radical symbol in them. Factoring them is one way to simplify the expressions.

When we factor them, we make use of the **product rule for radicals**, which tells us that one radical multiplied by another radical will equal the radical of the product of the numbers inside the two radicals. To simplify our radical, we first find the different ways that we can factor our number inside the radical.

Next, we look at our factors and see if we know one of the radicals. If we do, then we will use that group of factors to simplify our radical. We then evaluate the radical that we know and rewrite our original radical in its simplified form with the radical that we just evaluated times the left-over radical that we can't evaluate.

Gain the ability to do the following through this video lesson:

- Recall what it means to factor numbers
- Identify a radical expression
- Simplify a radical expression using the product rule for radicals

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- How to Find the Square Root of a Number 5:42
- Estimating Square Roots 5:10
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Expressions Containing Square Roots 7:03
- Division and Reciprocals of Radical Expressions 5:53
- Radicands and Radical Expressions 4:29
- Evaluating Square Roots of Perfect Squares 5:12
- Factoring Radical Expressions 4:45
- Multiplying then Simplifying Radical Expressions 3:57
- Dividing Radical Expressions 7:07
- Simplify Square Roots of Quotients 4:49
- Rationalizing Denominators in Radical Expressions 7:01
- Addition and Subtraction Using Radical Notation 3:08
- Multiplying Radical Expressions with Two or More Terms 6:35
- Solving Radical Equations: Steps and Examples 6:48
- Solving Radical Equations with Two Radical Terms 6:00
- Go to High School Algebra: Radical Expressions

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