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Study Guides > Intermediate Algebra

Read: Multiple Term Radicals

Learning Objectives

  • Multiply multiple term radicals
    • Use the distributive property to multiply multiple term radicals
    • Use the power of a product rule to simplify products of multiple term radicals
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 [latex] \sqrt{x}(3\sqrt{x}-5)[/latex] may look different than [latex] a(3a-5)[/latex], you can treat them the same way. Let’s have a look at how to apply the Distributive Property. First let’s do a problem with the variable a, and then solve the same problem replacing a with [latex] \sqrt{x}[/latex].

Example

Simplify. [latex] a(3a-5)[/latex]

Answer: Use the Distributive Property of Multiplication over Subtraction.

[latex]\begin{array}{c}a(3a)-a(5)\\=3a^2-5a\end{array}[/latex]

Answer

[latex-display] a(3a-5)=3{{a}^{2}}-5a[/latex-display]

Example

Simplify. [latex] \sqrt{x}(3\sqrt{x}-5)[/latex]

Answer: Use the Distributive Property of Multiplication over Subtraction.

[latex] \sqrt{x}(3\sqrt{x})-\sqrt{x}(5)[/latex]

Apply the rules of multiplying radicals: [latex] \sqrt{a}\cdot \sqrt{b}=\sqrt{ab}[/latex] to multiply [latex] \sqrt{x}(3\sqrt{x})[/latex].

[latex] 3\sqrt{{{x}^{2}}}-5\sqrt{x}[/latex]

Be sure to simplify radicals when you can: [latex] \sqrt{{{x}^{2}}}=\left| x \right|[/latex], so [latex] 3\sqrt{{{x}^{2}}}=3\left| x \right|[/latex].

Answer

[latex-display] \sqrt{x}(3\sqrt{x}-5)=3\left| x \right|-5\sqrt{x}[/latex-display]

The answers to the previous two problems should look similar to you. The only difference is that in the second problem, [latex] \sqrt{x}[/latex] has replaced the variable a (and so [latex] \left| x \right|[/latex] 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. [latex] 7\sqrt{x}\left( 2\sqrt{xy}+\sqrt{y} \right)[/latex]

Answer: Use the Distributive Property of Multiplication over Addition to multiply each term within parentheses by [latex] 7\sqrt{x}[/latex].

[latex] 7\sqrt{x}\left( 2\sqrt{xy} \right)+7\sqrt{x}\left( \sqrt{y} \right)[/latex]

Apply the rules of multiplying radicals.

[latex] 7\cdot 2\sqrt{{{x}^{2}}y}+7\sqrt{xy}[/latex]

[latex] \sqrt{{{x}^{2}}}=\left| x \right|[/latex], so [latex] \left| x \right|[/latex] can be pulled out of the radical.

[latex] 14|x|\sqrt{y}+7\sqrt{xy}[/latex]

Answer

[latex-display] 7\sqrt{x}\left( 2\sqrt{xy}+\sqrt{y} \right)=14\left| x \right|\sqrt{y}+7\sqrt{xy}[/latex-display]

Example

Simplify. [latex] \sqrt[3]{a}\left( 2\sqrt[3]{{{a}^{2}}}-4\sqrt[3]{{{a}^{5}}}+8\sqrt[3]{{{a}^{8}}} \right)[/latex]

Answer: Use the Distributive Property.

[latex] \sqrt[3]{a}\left( 2\sqrt[3]{{{a}^{2}}} \right)-\sqrt[3]{a}\left( 4\sqrt[3]{{{a}^{5}}} \right)+\sqrt[3]{a}\left( 8\sqrt[3]{{{a}^{8}}} \right)[/latex]

Apply the rules of multiplying radicals.

[latex] \begin{array}{c}2\sqrt[3]{a\cdot {{a}^{2}}}-4\sqrt[3]{a\cdot {{a}^{5}}}+8\sqrt[3]{a\cdot {{a}^{8}}}\\2\sqrt[3]{{{a}^{3}}}-4\sqrt[3]{{{a}^{6}}}+8\sqrt[3]{{{a}^{9}}}\end{array}[/latex]

Identify cubes in each of the radicals.

[latex] 2\sqrt[3]{{{a}^{3}}}-4\sqrt[3]{{{\left( {{a}^{2}} \right)}^{3}}}+8\sqrt[3]{{{\left( {{a}^{3}} \right)}^{3}}}[/latex]

Answer

[latex-display] \sqrt[3]{a}\left( 2\sqrt[3]{{{a}^{2}}}-4\sqrt[3]{{{a}^{5}}}+8\sqrt[3]{{{a}^{8}}} \right)=2a-4{{a}^{2}}+8{{a}^{3}}[/latex-display]

In the following video we show more examples of how to multiply radical expressions using distribution. https://youtu.be/hizqmgBjW0k In all of these examples, multiplication of radicals has been shown following the pattern [latex] \sqrt{a}\cdot \sqrt{b}=\sqrt{ab}[/latex]. 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 [latex] \sqrt{x}\cdot \sqrt{x}=x[/latex] without going through the intermediate step of finding that [latex] \sqrt{x}\cdot \sqrt{x}=\sqrt{{{x}^{2}}}[/latex]. In the rest of the examples that follow, though, each step is shown.

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.

Example

Multiply. [latex] \left( 2x+5 \right)\left( 3x-2 \right)[/latex]

Answer: Use the Distributive Property.

[latex]\begin{array}{l}\text{First}:\,\,\,\,\,\,\,\,\,\,\,2x\cdot 3x=6{{x}^{2}}\\\text{Outside}:\,\,\,2x\cdot \left( -2 \right)=-4x\\\text{Inside}:\,\,\,\,\,\,\,\,5\cdot 3x=15x\\\text{Last}:\,\,\,\,\,\,\,\,\,\,\,\,5\cdot \left( -2 \right)=-10\end{array}[/latex]

Record the terms, and then combine like terms.

[latex] 6{{x}^{2}}-4x+15x-10[/latex]

Answer

[latex-display] \left( 2x+5 \right)\left( 3x-2 \right)=6{{x}^{2}}+11x-10[/latex-display]

Here is the same problem, with [latex] \sqrt{b}[/latex] replacing the variable x.

Example

Multiply. [latex] \left( 2\sqrt{b}+5 \right)\left( 3\sqrt{b}-2 \right),\,\,b\ge 0[/latex]

Answer: Use the Distributive Property to multiply. Simplify using [latex] \sqrt{x}\cdot \sqrt{x}=x[/latex].

[latex]\begin{array}{l}\text{First}:\,\,\,\,\,\,\,\,\,\,\,2\sqrt{b}\cdot 3\sqrt{b}=2\cdot 3\cdot \sqrt{b}\cdot \sqrt{b}=6b\\\text{Outside}:\,\,\,2\sqrt{b}\cdot \left( -2 \right)=-4\sqrt{b}\\\text{Inside}:\,\,\,\,\,\,\,\,5\cdot 3\sqrt{b}=15\sqrt{b}\\\text{Last}:\,\,\,\,\,\,\,\,\,\,\,\,5\cdot \left( -2 \right)=-10\end{array}[/latex]

Record the terms, and then combine like terms.

[latex] 6b-4\sqrt{b}+15\sqrt{b}-10[/latex]

Answer

[latex-display] \left( 2\sqrt{b}+5 \right)\left( 3\sqrt{b}-2 \right)=6b+11\sqrt{b}-10[/latex-display]

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);
  • Remember that [latex] \sqrt{a}\cdot \sqrt{b}=\sqrt{ab}[/latex]; and
  • Combine like terms.

Example

Multiply. [latex] \left( 4{{x}^{2}}+\sqrt[3]{x} \right)\left( \sqrt[3]{{{x}^{2}}}+2 \right)[/latex]

Answer: Use FOIL to multiply.

[latex]\begin{array}{l}\text{First}:\,\,\,\,\,\,\,\,\,\,\,4x^{2}\cdot\sqrt[3]{x^{2}}=4x^{2}\sqrt[3]{x^{2}}\\\text{Outside}:\,\,\,4x^{2}\cdot 2=8x^{2}\\\text{Inside}:\,\,\,\,\,\,\,\,\sqrt[3]{x}\cdot\sqrt[3]{x^{2}}=\sqrt[3]{x^{2}\cdot x}=\sqrt[3]{x^{3}}=x\\\text{Last}:\,\,\,\,\,\,\,\,\,\,\,\,\sqrt[3]{x}\cdot 2=2\sqrt[3]{x}\end{array}[/latex]

Record the terms, and then combine like terms (if possible). Here, there are no like terms to combine.

[latex] 4{{x}^{2}}\sqrt[3]{{{x}^{2}}}+8{{x}^{2}}+x+2\sqrt[3]{x}[/latex]

Answer

[latex-display] \left( 4{{x}^{2}}+\sqrt[3]{x} \right)\left( \sqrt[3]{{{x}^{2}}}+2 \right)=4{{x}^{2}}\sqrt[3]{{{x}^{2}}}+8{{x}^{2}}+x+2\sqrt[3]{x}[/latex-display]

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 [latex] \sqrt{a}\cdot \sqrt{b}=\sqrt{ab}[/latex], and [latex] \sqrt{x}\cdot \sqrt{x}=x[/latex] to multiply and simplify. Finally, combine like terms.

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 Learning License: CC BY: Attribution.

CC licensed content, Shared previously

  • Precalculus. Provided by: OpenStax Authored by: Abramson, Jay. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Download for free at: http://cnx.org/contents/[email protected]:1/Preface.
  • Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology Located at: https://www.nroc.org/. License: CC BY: Attribution.