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Factoring Radical Expressions

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  • 0:02 Factoring to Simplify…
  • 1:02 Product Rule for Radicals
  • 1:59 Example 1
  • 3:06 Example 2
  • 3:53 Lesson Summary
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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.

Factoring to Simplify a Radical

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.

Product Rule for 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.

Example 1

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.

Example 2

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.

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