Learning Outcome

• 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 $\sqrt{x}(3\sqrt{x}-5)$ 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 $\sqrt{x}$.

Example

Simplify. $a(3a-5)$

Example

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

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 $\left| x \right|$ 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. $7\sqrt{x}\left( 2\sqrt{xy}+\sqrt{y} \right)$

Example

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

In the following video, we show more examples of how to multiply radical expressions using the distributive property.

In all of these examples, multiplication of radicals has been shown following the pattern $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$. 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 $\sqrt{x}\cdot \sqrt{x}=x$ 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.

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. $\left( 2x+5 \right)\left( 3x-2 \right)$

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

Example

Multiply. $\left( 2\sqrt{b}+5 \right)\left( 3\sqrt{b}-2 \right),\,\,b\ge 0$

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.

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 $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$
• Combine like terms

Example

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

In the following video, we show more examples of how to multiply two binomials that contain radicals.

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