16.2.3: Multiplication of Multiple Term Radicals

Learning Objectives

• Multiply and simplify radical expressions that contain more than one term.

Introduction

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 \(\ \sqrt{x}(3 \sqrt{x}-5)\) may look different than \(\ a(3 a-5)\), you can treat them the same way.

Using the Distributive Property

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 \(\ \sqrt{x}\).

Example

Simplify. \(\ a(3 a-5)\)

Solution

\(\ a(3 a)-a(5)\)

Use the Distributive Property of Multiplication over Subtraction.

\(\ a(3 a-5)=3 a^{2}-5 a\)

Example

Simplify. \(\ \sqrt{x}(3 \sqrt{x}-5)\)

Solution

\(\ \sqrt{x}(3 \sqrt{x})-\sqrt{x}(5)\)	Use the Distributive Property of Multiplication over Subtraction.
\(\ 3 \sqrt{x^{2}}-5 \sqrt{x}\)	Apply the rules of multiplying radicals: \(\ \sqrt{a} \cdot \sqrt{b}=\sqrt{a b}\) to multiply \(\ \sqrt{x}(3 \sqrt{x})\).
	Be sure to simplify radicals when you can: \(\ \sqrt{x^{2}}=|x|\), so \(\ 3 \sqrt{x^{2}}=3|x|\).

\(\ \sqrt{x}(3 \sqrt{x}-5)=3|x|-5 \sqrt{x}\)

The answers to the previous two problems should look similar to you. The only difference is that in the second problem, \(\ \sqrt{x}\) has replaced the variable \(\ a\) (and so \(\ |x|\) has replaced \(\ a^{2}\)). The process of multiplying is very much the same in both problems.

In these next two problems, each term contains a radical.

Example

Simplify. \(\ 7 \sqrt{x}(2 \sqrt{x y}+\sqrt{y})\)

Solution

\(\ 7 \sqrt{x}(2 \sqrt{x y})+7 \sqrt{x}(\sqrt{y})\)	Use the Distributive Property of Multiplication over Addition to multiply each term within parentheses by \(\ 7 \sqrt{x}\).
\(\ 7 \cdot 2 \sqrt{x^{2} y}+7 \sqrt{x y}\)	Apply the rules of multiplying radicals.
\(\ 14 \sqrt{x^{2} \cdot y}+7 \sqrt{x y}\)	\(\ \sqrt{x^{2}}=|x|\), so \(\ |x|\) can be pulled out of the radical.

\(\ 7 \sqrt{x}(2 \sqrt{x y}+\sqrt{y})=14|x| \sqrt{y}+7 \sqrt{x y}\)

Example

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

Solution

\(\ \sqrt[3]{a}\left(2 \sqrt[3]{a^{2}}\right)-\sqrt[3]{a}\left(4 \sqrt[3]{a^{5}}\right)+\sqrt[3]{a}\left(8 \sqrt[3]{a^{8}}\right)\)	Use the Distributive Property.
\(\ 2 \sqrt[3]{a \cdot a^{2}}-4 \sqrt[3]{a \cdot a^{5}}+8 \sqrt[3]{a \cdot a^{8}}\)	Apply the rules of multiplying radicals.
\(\ 2 \sqrt[3]{a^{3}}-4 \sqrt[3]{a^{6}}+8 \sqrt[3]{a^{9}}\)
\(\ 2 \sqrt[3]{a^{3}}-4 \sqrt[3]{\left(a^{2}\right)^{3}}+8 \sqrt[3]{\left(a^{3}\right)^{3}}\)	Identify cubes in each of the radicals.

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

In all of these examples, multiplication of radicals has been shown following the pattern \(\ \sqrt{a} \cdot \sqrt{b} \cdot \sqrt{a b}\). 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 \(\ \sqrt{x} \cdot \sqrt{x}=x\) without going through the intermediate step of finding that \(\ \sqrt{x} \cdot \sqrt{x}=\sqrt{x^{2}}\). In the rest of the examples that follow, though, each step is shown.

Exercise

Multiply and simplify. \(\ \sqrt{10}(\sqrt{10}-\sqrt{5})\)

1. \(\ 10-5 \sqrt{2}\)
2. \(\ 10\)
3. \(\ 5 \sqrt{2}\)
4. \(\ \sqrt{100}-\sqrt{50}\)

Answer

1. Correct. Multiplying \(\ \sqrt{10}\) by \(\ \sqrt{10}\) and \(\ -\sqrt{5}\), you find it is equal to \(\ \sqrt{100}-\sqrt{50}\), or \(\ 10-5 \sqrt{2}\).
2. Incorrect. \(\ \sqrt{10} \cdot \sqrt{10}=10\), but what does \(\ \sqrt{10} \cdot(-\sqrt{5})\) simplify to? The correct answer is \(\ 10-5 \sqrt{2}\).
3. Incorrect. You subtracted \(\ \sqrt{10}-\sqrt{5}=\sqrt{5}\), and then multiplied by \(\ \sqrt{10}\). Remember that you cannot subtract radicals unless the indices and the radicands are the same. The correct answer is \(\ 10-5 \sqrt{2}\).
4. Incorrect. This is the correct product, but it is not in simplest form. Look for squares that exist within \(\ \sqrt{100}\) and \(\ \sqrt{50}\). The correct answer is \(\ 10-5 \sqrt{2}\).

Multiplying Radical Expressions as Binomials

Sometimes, radical expressions appear in binomials as well. In these cases, you still follow the rules of binomial multiplication, but it is very important that you be precise and structured when you are multiplying the different terms.

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. \(\ (2 x+5)(3 x-2)\)

Solution

First: \(\ 2 x \cdot 3 x=6 x^{2}\) Outside: \(\ 2 x \cdot(-2)=-4 x\) Inside: \(\ 5 \cdot 3 x=15 x\) Last: \(\ 5 \cdot(-2)=-10\)	Use the Distributive Property.
\(\ 6 x^{2}-4 x+15 x-10\)	Record the terms, and then combine like terms.

\(\ (2 x+5)(3 x-2)=6 x^{2}+11 x-10\)

Here is the same problem, with \(\ \sqrt{b}\) replacing the variable \(\ x\).

Example

Multiply. \(\ (2 \sqrt{b}+5)(3 \sqrt{b}-2), b \geq 0\)

Solution

First: \(\ 2 \sqrt{b} \cdot 3 \sqrt{b}=2 \cdot 3 \cdot \sqrt{b} \cdot \sqrt{b}=6 b\) Outside: \(\ 2 \sqrt{b} \cdot(-2)=-4 \sqrt{b}\) Inside: \(\ 5 \cdot 3 \sqrt{b}=15 \sqrt{b}\) Last: \(\ 5 \cdot(-2)=-10\)	Use the Distributive Property to multiply. Simplify using \(\ \sqrt{x} \cdot \sqrt{x}=x\).
\(\ 6 b-4 \sqrt{b}+15 \sqrt{b}-10\)	Record the terms, and then combine like terms.

\(\ (2 \sqrt{b}+5)(3 \sqrt{b}-2)=6 b+11 \sqrt{b}-10\)

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 Binomial 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 \(\ \sqrt{a} \cdot \sqrt{b}=\sqrt{a b}\); and
• Combine like terms.

Example

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

Solution

First: \(\ 4 x^{2} \cdot \sqrt[3]{x^{2}}=4 x^{2} \sqrt[3]{x^{2}}\) Outside: \(\ 4 x^{2} \cdot 2=8 x^{2}\) Inside: \(\ \sqrt[3]{x} \cdot \sqrt[3]{x^{2}}=\sqrt[3]{x^{2} \cdot x}=\sqrt[3]{x^{3}}=x\) Last: \(\ \sqrt[3]{x} \cdot 2=2 \sqrt[3]{x}\)	Use FOIL to multiply.
\(\ 4 x^{2} \sqrt[3]{x^{2}}+8 x^{2}+x+2 \sqrt[3]{x}\)	Record the terms, and then combine like terms (if possible). Here, there are no like terms to combine.

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

Exercise

Multiply and simplify. \(\ (4 \sqrt{x}+3)(2 \sqrt{x}-1), x \geq 0\)

1. \(\ 7\)
2. \(\ 8 x+2 \sqrt{x}-3\)
3. \(\ 8 x^{2}+2 x+3\)
4. \(\ 4 x+8 \sqrt{x}-3\)

Answer

1. Incorrect. Using the FOIL method, the product is \(\ 8 x-4 \sqrt{x}+6 \sqrt{x}-3\), which simplifies to \(\ 8 x+2 \sqrt{x}-3\). It looks like you forgot the variables and added the coefficients \(\ 8+2-3\) in order to arrive at \(\ 7\). The correct answer is \(\ 8 x+2 \sqrt{x}-3\).
2. Correct. Using the FOIL method, you find that the product of the binomials is \(\ 8 \cdot \sqrt{x} \cdot \sqrt{x}-4 \sqrt{x}+6 \sqrt{x}-3\), which simplifies to \(\ 8 x+2 \sqrt{x}-3\).
3. Incorrect. Remember that \(\ 4 \sqrt{x} \cdot 2 \sqrt{x}=4 \cdot 2 \cdot \sqrt{x} \cdot \sqrt{x}=8 x\), not \(\ 8 x^{2}\). The correct answer is \(\ 8 x+2 \sqrt{x}-3\).
4. Incorrect. You have multiplied the variable terms correctly, but you have the incorrect coefficients. Using the FOIL method, the product is \(\ 8 x-4 \sqrt{x}+6 \sqrt{x}-3\), which simplifies to \(\ 8 x+2 \sqrt{x}-3\). The correct answer is \(\ 8 x+2 \sqrt{x}-3\).

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 \(\ \sqrt{a} \cdot \sqrt{b}=\sqrt{a b}\), and \(\ \sqrt{x} \cdot \sqrt{x}=x\) to multiply and simplify. Finally, combine like terms.