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

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

Multiplying radical expressions with more than two terms can be confusing. This lesson will take some of the confusion away by giving clear steps for multiplying these expressions. It will also provide some examples to help solidify the steps.

Multiplying radical expressions is not much different from multiplying any other expression except the terms are under a square root symbol, which adds a few extra steps. To multiply two or more of the same variable, you need to add the exponents of the variables.

For example, 5*bc*^2 * 3*(b*^3)*c* = 15(*b*^4)(*c*^3).

Variables that are different, like *b* and *c*, cannot be combined.

When multiplying a single-term radical expression by a multiple-term expression, you need to use the **distributive property**. The distributive property means that you multiply the single term to both of the terms in the multiple-term expression. It might be easier to see with an example.

Simplify sqrt 2*x*(sqrt 3 + *y*).

First, multiply the number outside the parenthesis to the first term inside the parentheses. sqrt 2*x* * sqrt 3 = sqrt 6*x*.

Then, multiply the number outside the parenthesis to the second term inside the parentheses. sqrt 2*x* * *y* = *y*sqrt 2*x*.

Then, write out both terms. Remember, they cannot be combined unless their variables are exactly the same. (sqrt 6*x*) + y sqrt 2*x*.

Let's try another one. Simplify 2 sqrt *x* * (sqrt 7*xy* - sqrt *y*).

Multiply the outside term by the first inside term. 2 sqrt *x* * sqrt 7*xy* = 2 sqrt 7(*x*^2)*y*.

Then, multiply the outside term by the second inside term. 2 sqrt *x* * sqrt *y* = 2 sqrt *xy*.

Next, write down the two terms, making sure to keep the subtraction sign in the middle. 2 sqrt 7(*x*^2)*y* - 2 sqrt *xy*.

Before we can finish this one, we notice that there is an *x*^2 inside the radical of the first term. That can be simplified further because the square root of *x*^2 is *x*, so we can move an *x* outside the radical, and this problem is complete.

2*x* sqrt 7*y* - 2 sqrt *xy*.

When you are multiplying two **binomials**, expressions with two terms, you also use the distributive property, but it becomes more complex because you have to make sure that each term in the first expression is multiplied by every term in the second expression.

Mathematicians have come up with a mnemonic to help us remember what terms to multiply together. The mnemonic is FOIL, and it stands for:

- F = the
**f**irst term in each binomial - O = the
**o**utside term in each binomial - I = the
**i**nside term in each binomial - L = the
**l**ast term in each binomial

Here is an example using the FOIL method of multiplying.

(2 + sqrt *x*)(5 - sqrt 3).

Start by multiplying the first term in each set of parentheses. 2 * 5 = 10.

Then, the outside term in each set of parentheses: 2 * -sqrt 3 = -2 sqrt 3.

Next, the inside term of each set of parentheses: sqrt *x* * 5 = 5 sqrt *x*.

And finally, the last term in each set of parentheses: sqrt *x* * -sqrt 3 = -sqrt 3*x*.

Now we can write out each term, and if there are any that can be combined, we can combine them. Don't forget, they can only be combined if the variable terms are exactly the same. For this example, there are no like terms that can be combined, and this is our answer: 10 - 2 sqrt 3 + 5 sqrt *x* - sqrt 3*x*.

Let's do one more: (4 sqrt *x* + 3)(2 sqrt *x* - 1).

Start with **F** - the first terms: 4 sqrt *x* * 2 sqrt *x* = 8 sqrt *x*^2.

Then, **O** - the outside terms: 4 sqrt *x* * -1 = -4 sqrt *x*.

**I** - the inside terms: 3 * 2 sqrt *x* = 6 sqrt *x*.

And **L** - the last terms: 3 * -1 = -3.

Now we can write out the terms and combine where possible. 8 sqrt *x*^2 - 4 sqrt *x* + 6 sqrt *x* - 3.

We can simplify this problem in two different places. First, the square root of *x*^2 is *x*, so that term becomes 8*x*. Then the -4 sqrt *x* and the 6 sqrt *x* can be combined to equal 2 sqrt *x*. So our final answer is 8*x* + 2 sqrt *x* - 3.

When multiplying radical expressions containing more than one term, you need to use the distributive property. This will allow you to make sure every term in one expression is multiplied to every term in the second expression. If both expressions are binomials, you can use the FOIL method to remember what terms to multiply together. After multiplying, you need to make sure that everything under the radical is simplified as much as possible.

After reviewing this lesson, you'll have the ability to:

- Describe how to multiply radical expressions containing more than one term using the distributive property
- Explain how to use the FOIL method to multiply when both expressions are binomials

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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
- Simplifying Square Roots of Powers in Radical Expressions 3:51
- 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 with Two Radical Terms 6:00
- Go to High School Algebra: Radical Expressions

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