Use the distributive property to multiply multiple term radicals and then simplify
When multiplying multiple term radical expressions, it is important to follow the Distributive Property of Multiplication, as when you are multiplying regular, non-radical expressions.
Radicals follow the same mathematical rules that other real numbers do. So, although the expression x(3x−5) may look different than a(3a−5), you can treat them the same way.
Let us have a look at how to apply the Distributive Property. First let us do a problem with the variable a, and then solve the same problem replacing a with x.
Example
Simplify. a(3a−5)
Answer:
Use the Distributive Property of Multiplication over Subtraction.
a(3a)−a(5)=3a2−5a
Example
Simplify. x(3x−5)
Answer:
Use the Distributive Property of Multiplication over Subtraction.
x(3x)−x(5)
Apply the rules of multiplying radicals: a⋅b=ab to multiply x(3x).
3x2−5x
Be sure to simplify radicals when you can: x2=∣x∣, so 3x2=3∣x∣.
The answer is 3∣x∣−5x
The answers to the previous two problems should look similar to you. The only difference is that in the second problem, x has replaced the variable a (and so ∣x∣ has replaced a2). The process of multiplying is very much the same in both problems.
In these next two problems, each term contains a radical.
Example
Simplify. 7x(2xy+y)
Answer:
Use the Distributive Property of Multiplication over Addition to multiply each term within parentheses by 7x.
7x(2xy)+7x(y)
Apply the rules of multiplying radicals.
7⋅2x2y+7xy
x2=∣x∣, so ∣x∣ can be pulled out of the radical.
14∣x∣y+7xy
Example
Simplify. 3a(23a2−43a5+83a8)
Answer:
Use the Distributive Property.
3a(23a2)−3a(43a5)+3a(83a8)
Apply the rules of multiplying radicals.
23a⋅a2−43a⋅a5+83a⋅a823a3−43a6+83a9
Identify cubes in each of the radicals.
23a3−43(a2)3+83(a3)3
The solution is 2a−4a2+8a3.
In the following video, we show more examples of how to multiply radical expressions using the distributive property.
https://youtu.be/hizqmgBjW0k
In all of these examples, multiplication of radicals has been shown following the pattern a⋅b=ab. Then, only after multiplying, some radicals have been simplified—like in the last problem. After you have worked with radical expressions a bit more, you may feel more comfortable identifying quantities such as x⋅x=x without going through the intermediate step of finding that x⋅x=x2. In the rest of the examples that follow, though, each step is shown.
Try It
[ohm_question]196059[/ohm_question]
Multiply Binomial Expressions That Contain Radicals
You can use the same technique for multiplying binomials to multiply binomial expressions with radicals.
As a refresher, here is the process for multiplying two binomials. If you like using the expression “FOIL” (First, Outside, Inside, Last) to help you figure out the order in which the terms should be multiplied, you can use it here, too.
The multiplication works the same way in both problems; you just have to pay attention to the index of the radical (that is, whether the roots are square roots, cube roots, etc.) when multiplying radical expressions.
Multiplying Two-Term Radical Expressions
To multiply radical expressions, use the same method as used to multiply polynomials.
Use the Distributive Property (or, if you prefer, the shortcut FOIL method)
Record the terms, and then combine like terms (if possible). Here, there are no like terms to combine.
4x23x2+8x2+x+23x
In the following video, we show more examples of how to multiply two binomials that contain radicals.
https://youtu.be/VUWIBk3ga5I
Summary
To multiply radical expressions that contain more than one term, use the same method that you use to multiply polynomials. First, use the Distributive Property (or, if you prefer, the shortcut FOIL method) to multiply the terms. Then, apply the rules a⋅b=ab, and x⋅x=x to multiply and simplify. Finally, combine like terms.
Contribute!
Did you have an idea for improving this content? We’d love your input.
Licenses & Attributions
CC licensed content, Original
Multiplying Radical Expressions with Variables Using Distribution.Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.
Multiplying Binomial Radical Expressions with Variables.Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.
Revision and Adaptation.Provided by: Lumen LearningLicense: CC BY: Attribution.
CC licensed content, Shared previously
Precalculus.Provided by: OpenStaxAuthored by: Abramson, Jay.Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface.License: CC BY: Attribution. License terms: Download for free at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface.
Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program.Provided by: Monterey Institute of TechnologyLocated at: https://www.nroc.org/.License: CC BY: Attribution.