Roots and Radicals

# Add, Subtract, and Multiply Radical Expressions

### Learning Objectives

By the end of this section, you will be able to:

- Add and subtract radical expressions
- Multiply radical expressions
- Use polynomial multiplication to multiply radical expressions

Before you get started, take this readiness quiz.

### Add and Subtract Radical Expressions

Adding radical expressions with the same index and the same radicand is just like adding like terms. We call radicals with the same index and the same radicand like radicals to remind us they work the same as like terms.

**Like radicals** are radical expressions with the same index and the same radicand.

We add and subtract like radicals in the same way we add and subtract like terms. We know that is Similarly we add and the result is

Think about adding like terms with variables as you do the next few examples. When you have like radicals, you just add or subtract the coefficients. When the radicals are not like, you cannot combine the terms.

Simplify: ⓐ ⓑ ⓒ

ⓐ

ⓑ

ⓒ

The indices are the same but the radicals are different. These are not like radicals. Since the radicals are not like, we cannot subtract them.

Simplify: ⓐ ⓑ ⓒ

ⓐⓑ

ⓒ

Simplify: ⓐ ⓑ ⓒ

ⓐⓑ

ⓒ

For radicals to be like, they must have the same index and radicand. When the radicands contain more than one variable, as long as all the variables and their exponents are identical, the radicands are the same.

Simplify: ⓐ ⓑ

ⓐ

ⓑ

Simplify: ⓐ ⓑ

ⓐⓑ

Simplify: ⓐ ⓑ

ⓐⓑ

Remember that we always simplify radicals by removing the largest factor from the radicand that is a power of the index. Once each radical is simplified, we can then decide if they are like radicals.

Simplify: ⓐ ⓑ ⓒ

ⓐ

ⓑ

ⓒ

Simplify: ⓐ ⓑ ⓒ

ⓐⓑⓒ

Simplify: ⓐ ⓑ ⓒ

ⓐⓑⓒ

In the next example, we will remove both constant and variable factors from the radicals. Now that we have practiced taking both the even and odd roots of variables, it is common practice at this point for us to assume all variables are greater than or equal to zero so that absolute values are not needed. We will use this assumption thoughout the rest of this chapter.

Simplify: ⓐ ⓑ

ⓐ

ⓑ

Simplify: ⓐ ⓑ

ⓐⓑ

Simplify: ⓐ ⓑ

ⓐ

ⓑ

### Multiply Radical Expressions

We have used the Product Property of Roots to simplify square roots by removing the perfect square factors. We can use the Product Property of Roots ‘in reverse’ to multiply square roots. Remember, we assume all variables are greater than or equal to zero.

We will rewrite the Product Property of Roots so we see both ways together.

For any real numbers, and and for any integer

When we multiply two radicals they must have the same index. Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible.

Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply we multiply the coefficients together and then the variables. The result is 12*xy*. Keep this in mind as you do these examples.

Simplify: ⓐ ⓑ

ⓐ

ⓑ

Simplify: ⓐ ⓑ

ⓐⓑ

Simplify: ⓐ ⓑ

ⓐⓑ

We follow the same procedures when there are variables in the radicands.

Simplify: ⓐ ⓑ

ⓐ

ⓑ When the radicands involve large numbers, it is often advantageous to factor them in order to find the perfect powers.

Simplify: ⓐ ⓑ

ⓐⓑ

Simplify: ⓐ ⓑ

ⓐⓑ

### Use Polynomial Multiplication to Multiply Radical Expressions

In the next a few examples, we will use the Distributive Property to multiply expressions with radicals. First we will distribute and then simplify the radicals when possible.

Simplify: ⓐ ⓑ

ⓐ

ⓑ

Simplify: ⓐ ⓑ

ⓐⓑ

Simplify: ⓐ ⓑ

ⓐⓑ

When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.

Simplify: ⓐ ⓑ

ⓐ

ⓑ

Simplify: ⓐ ⓑ

ⓐ

ⓑ

Simplify: ⓐ ⓑ

ⓐ

ⓑ

Simplify:

Simplify:

Simplify:

Recognizing some special products made our work easier when we multiplied binomials earlier. This is true when we multiply radicals, too. The special product formulas we used are shown here.

*** QuickLaTeX cannot compile formula: \begin{array}{}\\ \\ \hfill \mathbf{\text{Binomial Squares}}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\mathbf{\text{Product of Conjugates}}\hfill \\ \hfill {\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}\hfill \\ \hfill {\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}\hfill & & & \end{array} *** Error message: Missing # inserted in alignment preamble. leading text: $\begin{array}{} \mathbf allowed only in math mode. leading text: ... \\ \hfill \mathbf{\text{Binomial Squares}} Extra alignment tab has been changed to \cr. leading text: ...ll \mathbf{\text{Binomial Squares}}\hfill & Extra alignment tab has been changed to \cr. leading text: ... \mathbf{\text{Binomial Squares}}\hfill & & Extra alignment tab has been changed to \cr. leading text: ...mathbf{\text{Binomial Squares}}\hfill & & & \mathbf allowed only in math mode. leading text: ...{0ex}}\mathbf{\text{Product of Conjugates}} Missing $ inserted. leading text: ...duct of Conjugates}}\hfill \\ \hfill {\left Extra }, or forgotten $. leading text: ...ugates}}\hfill \\ \hfill {\left(a+b\right)} Missing } inserted.

We will use the special product formulas in the next few examples. We will start with the Product of Binomial Squares Pattern.

Simplify: ⓐ ⓑ

Be sure to include the term when squaring a binomial.

ⓐ

Multiply, using the Product of Binomial Squares Pattern. | |

Simplify. | |

Combine like terms. |

ⓑ

Multiply, using the Product of Binomial Squares Pattern. | |

Simplify. | |

Combine like terms. |

Simplify: ⓐ ⓑ

ⓐⓑ

Simplify: ⓐ ⓑ

ⓐ

ⓑ

In the next example, we will use the Product of Conjugates Pattern. Notice that the final product has no radical.

Simplify:

Multiply, using the Product of Conjugates Pattern. | |

Simplify. | |

Simplify:

Simplify:

Access these online resources for additional instruction and practice with adding, subtracting, and multiplying radical expressions.

### Key Concepts

**Product Property of Roots**- For any real numbers, and and for any integer
and

- For any real numbers, and and for any integer
**Special Products**

#### Practice Makes Perfect

**Add and Subtract Radical Expressions**

In the following exercises, simplify.

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓐⓑ 0

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓐⓑ 0

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

**Multiply Radical Expressions**

In the following exercises, simplify.

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ) ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

**Use Polynomial Multiplication to Multiply Radical Expressions**

In the following exercises, multiply.

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐⓑ

ⓐⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐⓑ

14

**Mixed Practice**

29

#### Writing Exercises

Explain the when a radical expression is in simplest form.

Answers will vary.

Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.

ⓐ Explain why is always non-negative, for

ⓑ Explain why is always non-positive, for

Answers will vary.

Use the binomial square pattern to simplify Explain all your steps.

#### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

### Glossary

- like radicals
- Like radicals are radical expressions with the same index and the same radicand.