Contents

Preface
1Foundations
- Introduction
- 1.1Use the Language of Algebra
- 1.2Integers
- 1.3Fractions
- 1.4Decimals
- 1.5Properties of Real Numbers
- Chapter Review
  Key Terms
  Key Concepts
- Exercises
  Review Exercises
  Practice Test
2Solving Linear Equations
3Graphs and Functions
4Systems of Linear Equations
5Polynomials and Polynomial Functions
6Factoring
7Rational Expressions and Functions
8Roots and Radicals
9Quadratic Equations and Functions
10Exponential and Logarithmic Functions
11Conics
12Sequences, Series and Binomial Theorem
- Introduction
- 12.1Sequences
- 12.2Arithmetic Sequences
- 12.3Geometric Sequences and Series
- 12.4Binomial Theorem
- Chapter Review
  Key Terms
Answer Key
Index

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

Be Prepared 8.4

Before you get started, take this readiness quiz.

Add: $3 x^{2} + 9 x - 5 - (x^{2} - 2 x + 3) . 3 x^{2} + 9 x - 5 - (x^{2} - 2 x + 3) .$
If you missed this problem, review Example 5.5.
Simplify: $(2 + a) (4 - a) . (2 + a) (4 - a) .$
If you missed this problem, review Example 5.28.
Simplify: ${(9 - 5 y)}^{2} . {(9 - 5 y)}^{2} .$
If you missed this problem, review Example 5.31.

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

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 $3 x + 8 x 3 x + 8 x$ is $11 x . 11 x .$ Similarly we add $3 \sqrt{x} + 8 \sqrt{x} 3 \sqrt{x} + 8 \sqrt{x}$ and the result is $11 \sqrt{x} . 11 \sqrt{x} .$

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.

Example 8.36

Simplify: ⓐ $2 \sqrt{2} - 7 \sqrt{2} 2$ ⓑ $5 \sqrt[3]{y} + 4 \sqrt[3]{y} 5$ ⓒ $7 \sqrt[4]{x} - 2 \sqrt[4]{y} . 7$

Show solution

Solution

ⓐ
$\begin{matrix} 2 \sqrt{2} - 7 \sqrt{2} \\ \begin{matrix} Since the radicals are like, we subtract the \\ coefficients. \end{matrix} & −5 \sqrt{2} \end{matrix} \begin{matrix} \begin{matrix} Since the radicals are like, we subtract the \\ coefficients. \end{matrix} \end{matrix}$
ⓑ
$\begin{matrix} 5 \sqrt[3]{y} + 4 \sqrt[3]{y} \\ \begin{matrix} Since the radicals are like, we add the \\ coefficients. \end{matrix} & 9 \sqrt[3]{y} \end{matrix} \begin{matrix} \begin{matrix} Since the radicals are like, we add the \\ coefficients. \end{matrix} \end{matrix}$

ⓒ
$\begin{matrix} 7 \sqrt[4]{x} - 2 \sqrt[4]{y} \end{matrix} \begin{matrix} \end{matrix}$

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.

Try It 8.71

Simplify: ⓐ $8 \sqrt{2} - 9 \sqrt{2} 8 \sqrt{2} - 9 \sqrt{2}$ ⓑ $4 \sqrt[3]{x} + 7 \sqrt[3]{x} 4 \sqrt[3]{x} + 7 \sqrt[3]{x}$ ⓒ $3 \sqrt[4]{x} - 5 \sqrt[4]{y} . 3 \sqrt[4]{x} - 5 \sqrt[4]{y} .$

Try It 8.72

Simplify: ⓐ $5 \sqrt{3} - 9 \sqrt{3} 5 \sqrt{3} - 9 \sqrt{3}$ ⓑ $5 \sqrt[3]{y} + 3 \sqrt[3]{y} 5 \sqrt[3]{y} + 3 \sqrt[3]{y}$ ⓒ $5 \sqrt[4]{m} - 2 \sqrt[3]{m} . 5 \sqrt[4]{m} - 2 \sqrt[3]{m} .$

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.

Example 8.37

Simplify: ⓐ $2 \sqrt{5 n} - 6 \sqrt{5 n} + 4 \sqrt{5 n} 2$ ⓑ $\sqrt[4]{3 x y} + 5 \sqrt[4]{3 x y} - 4 \sqrt[4]{3 x y} . \sqrt[4]{3 x y} + 5$

Show solution

Solution

ⓐ
$\begin{matrix} 2 \sqrt{5 n} - 6 \sqrt{5 n} + 4 \sqrt{5 n} \\ Since the radicals are like, we combine them. & 0 \sqrt{5 n} \\ Simplify. & 0 \end{matrix} \begin{matrix} Since the radicals are like, we combine them. \\ Simplify. \end{matrix}$

ⓑ
$\begin{matrix} \sqrt[4]{3 x y} + 5 \sqrt[4]{3 x y} - 4 \sqrt[4]{3 x y} \\ Since the radicals are like, we combine them. & 2 \sqrt[4]{3 x y} \end{matrix} \begin{matrix} Since the radicals are like, we combine them. \end{matrix}$

Try It 8.73

Simplify: ⓐ $\sqrt{7 x} - 7 \sqrt{7 x} + 4 \sqrt{7 x} \sqrt{7 x} - 7$ ⓑ $4 \sqrt[4]{5 x y} + 2 \sqrt[4]{5 x y} - 7 \sqrt[4]{5 x y} . 4$

Try It 8.74

Simplify: ⓐ $4 \sqrt{3 y} - 7 \sqrt{3 y} + 2 \sqrt{3 y} 4$ ⓑ $6 \sqrt[3]{7 m n} + \sqrt[3]{7 m n} - 4 \sqrt[3]{7 m n} . 6$

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.

Example 8.38

Simplify: ⓐ $\sqrt{20} + 3 \sqrt{5} \sqrt{20} + 3 \sqrt{5}$ ⓑ $\sqrt[3]{24} - \sqrt[3]{375} \sqrt[3]{24} - \sqrt[3]{375}$ ⓒ $\frac{1}{2} \sqrt[4]{48} - \frac{2}{3} \sqrt[4]{243} . \frac{1}{2} \sqrt[4]{48} - \frac{2}{3} \sqrt[4]{243} .$

Show solution

Solution

ⓐ
$\begin{matrix} \sqrt{20} + 3 \sqrt{5} \\ Simplify the radicals, when possible. & \sqrt{4} \cdot \sqrt{5} + 3 \sqrt{5} \\ 2 \sqrt{5} + 3 \sqrt{5} \\ Combine the like radicals. & 5 \sqrt{5} \end{matrix} \begin{matrix} Simplify the radicals, when possible. \\ Combine the like radicals. \end{matrix}$

ⓑ
$\begin{matrix} \sqrt[3]{24} - \sqrt[3]{375} \\ Simplify the radicals. & \sqrt[3]{8} \cdot \sqrt[3]{3} - \sqrt[3]{125} \cdot \sqrt[3]{3} \\ 2 \sqrt[3]{3} - 5 \sqrt[3]{3} \\ Combine the like radicals. & −3 \sqrt[3]{3} \end{matrix} \begin{matrix} Simplify the radicals. \\ Combine the like radicals. \end{matrix}$

ⓒ
$\begin{matrix} \frac{1}{2} \sqrt[4]{48} - \frac{2}{3} \sqrt[4]{243} \\ Simplify the radicals. & \frac{1}{2} \sqrt[4]{16} \cdot \sqrt[4]{3} - \frac{2}{3} \sqrt[4]{81} \cdot \sqrt[4]{3} \\ \frac{1}{2} \cdot 2 \cdot \sqrt[4]{3} - \frac{2}{3} \cdot 3 \cdot \sqrt[4]{3} \\ \sqrt[4]{3} - 2 \sqrt[4]{3} \\ Combine the like radicals. & − \sqrt[4]{3} \end{matrix} \begin{matrix} Simplify the radicals. \\ Combine the like radicals. \end{matrix}$

Try It 8.75

Simplify: ⓐ $\sqrt{18} + 6 \sqrt{2} \sqrt{18} + 6$ ⓑ $6 \sqrt[3]{16} - 2 \sqrt[3]{250} 6$ ⓒ $\frac{2}{3} \sqrt[3]{81} - \frac{1}{2} \sqrt[3]{24} . \frac{2}{3}$

Try It 8.76

Simplify: ⓐ $\sqrt{27} + 4 \sqrt{3} \sqrt{27} + 4$ ⓑ $4 \sqrt[3]{5} - 7 \sqrt[3]{40} 4$ ⓒ $\frac{1}{2} \sqrt[3]{128} - \frac{5}{3} \sqrt[3]{54} . \frac{1}{2}$

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 throughout the rest of this chapter.

Example 8.39

Simplify: ⓐ $9 \sqrt{50 m^{2}} - 6 \sqrt{48 m^{2}} 9 \sqrt{50 m^{2}} - 6 \sqrt{48 m^{2}}$ ⓑ $\sqrt[3]{54 n^{5}} - \sqrt[3]{16 n^{5}} . \sqrt[3]{54 n^{5}} - \sqrt[3]{16 n^{5}} .$

Show solution

Solution

ⓐ
$\begin{matrix} 9 \sqrt{50 m^{2}} - 6 \sqrt{48 m^{2}} \\ Simplify the radicals. & 9 \sqrt{25 m^{2}} \cdot \sqrt{2} - 6 \sqrt{16 m^{2}} \cdot \sqrt{3} \\ 9 \cdot 5 m \cdot \sqrt{2} - 6 \cdot 4 m \cdot \sqrt{3} \\ 45 m \sqrt{2} - 24 m \sqrt{3} \\ \begin{matrix} The radicals are not like and so cannot be \\ combined. \end{matrix} \end{matrix} \begin{matrix} Simplify the radicals. \\ \begin{matrix} The radicals are not like and so cannot be \\ combined. \end{matrix} \end{matrix}$

ⓑ
$\begin{matrix} \sqrt[3]{54 n^{5}} - \sqrt[3]{16 n^{5}} \\ Simplify the radicals. & \sqrt[3]{27 n^{3}} \cdot \sqrt[3]{2 n^{2}} - \sqrt[3]{8 n^{3}} \cdot \sqrt[3]{2 n^{2}} \\ 3 n \sqrt[3]{2 n^{2}} - 2 n \sqrt[3]{2 n^{2}} \\ Combine the like radicals. & n \sqrt[3]{2 n^{2}} \end{matrix} \begin{matrix} Simplify the radicals. \\ Combine the like radicals. \end{matrix}$

Try It 8.77

Simplify: ⓐ $\sqrt{32 m^{7}} - \sqrt{50 m^{7}} \sqrt{32 m^{7}} - \sqrt{50 m^{7}}$ ⓑ $\sqrt[3]{135 x^{7}} - \sqrt[3]{40 x^{7}} . \sqrt[3]{135 x^{7}} - \sqrt[3]{40 x^{7}} .$

Try It 8.78

Simplify: ⓐ $\sqrt{27 p^{3}} - \sqrt{48 p^{3}} \sqrt{27 p^{3}} - \sqrt{48 p^{3}}$ ⓑ $\sqrt[3]{256 y^{5}} - \sqrt[3]{32 n^{5}} . \sqrt[3]{256 y^{5}} - \sqrt[3]{32 n^{5}} .$

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.

Product Property of Roots

For any real numbers, $\sqrt[n]{a} \sqrt[n]{a}$ and $\sqrt[n]{b}, \sqrt[n]{b},$ and for any integer $n \geq 2 n \geq 2$

\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b} and \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b} \sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b}

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 $4 x \cdot 3 y 4 x \cdot 3 y$ we multiply the coefficients together and then the variables. The result is 12xy. Keep this in mind as you do these examples.

Example 8.40

Simplify: ⓐ $(6 \sqrt{2}) (3 \sqrt{10}) (6) (3)$ ⓑ $(−5 \sqrt[3]{4}) (−4 \sqrt[3]{6}) . (−5) (−4) .$

Show solution

Solution

ⓐ
$\begin{matrix} (6 \sqrt{2}) (3 \sqrt{10}) \\ Multiply using the Product Property. & 18 \sqrt{20} \\ Simplify the radical. & 18 \sqrt{4} \cdot \sqrt{5} \\ Simplify. & 18 \cdot 2 \cdot \sqrt{5} \\ 36 \sqrt{5} \end{matrix} \begin{matrix} Multiply using the Product Property. \\ Simplify the radical. \\ Simplify. \end{matrix}$

ⓑ
$\begin{matrix} (−5 \sqrt[3]{4}) (−4 \sqrt[3]{6}) \\ Multiply using the Product Property. & 20 \sqrt[3]{24} \\ Simplify the radical. & 20 \sqrt[3]{8} \cdot \sqrt[3]{3} \\ Simplify. & 20 \cdot 2 \cdot \sqrt[3]{3} \\ 40 \sqrt[3]{3} \end{matrix} \begin{matrix} Multiply using the Product Property. \\ Simplify the radical. \\ Simplify. \end{matrix}$

Try It 8.79

Simplify: ⓐ $(3 \sqrt{2}) (2 \sqrt{30}) (3 \sqrt{2}) (2 \sqrt{30})$ ⓑ $(2 \sqrt[3]{18}) (−3 \sqrt[3]{6}) . (2) (−3) .$

Try It 8.80

Simplify: ⓐ $(3 \sqrt{3}) (3 \sqrt{6}) (3 \sqrt{3}) (3 \sqrt{6})$ ⓑ $(−4 \sqrt[3]{9}) (3 \sqrt[3]{6}) . (−4) (3) .$

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

Example 8.41

Simplify: ⓐ $(10 \sqrt{6 p^{3}}) (4 \sqrt{3 p}) (10) (4)$ ⓑ $(2 \sqrt[4]{20 y^{2}}) (3 \sqrt[4]{28 y^{3}}) . (2) (3) .$

Show solution

Solution

ⓐ
$\begin{matrix} (10 \sqrt{6 p^{3}}) (4 \sqrt{3 p}) \\ Multiply. & 40 \sqrt{18 p^{4}} \\ Simplify the radical. & 40 \sqrt{9 p^{4}} \cdot \sqrt{2} \\ Simplify. & 40 \cdot 3 p^{2} \cdot \sqrt{3} \\ 120 p^{2} \sqrt{3} \end{matrix} \begin{matrix} Multiply. \\ Simplify the radical. \\ Simplify. \end{matrix}$

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

$\begin{matrix} (2 \sqrt[4]{20 y^{2}}) (3 \sqrt[4]{28 y^{3}}) \\ Multiply. & 6 \sqrt[4]{4 \cdot 5 \cdot 4 \cdot 7 y^{5}} \\ Simplify the radical. & 6 \sqrt[4]{16 y^{4}} \cdot \sqrt[4]{35 y^{}} \\ Simplify. & 6 \cdot 2 y \sqrt[4]{35 y^{}} \\ Multiply. & 12 y \sqrt[4]{35 y^{}} \end{matrix} \begin{matrix} Multiply. \\ Simplify the radical. \\ Simplify. \\ Multiply. \end{matrix}$

Try It 8.81

Simplify: ⓐ $(6 \sqrt{6 x^{2}}) (8 \sqrt{30 x^{4}}) (6 \sqrt{6 x^{2}}) (8 \sqrt{30 x^{4}})$ ⓑ $(−4 \sqrt[4]{12 y^{3}}) (− \sqrt[4]{8 y^{3}}) . (−4) (− \sqrt[4]{8 y^{3}}) .$

Try It 8.82

Simplify: ⓐ $(2 \sqrt{6 y^{4}}) (12 \sqrt{30 y}) (2 \sqrt{6 y^{4}}) (12 \sqrt{30 y})$ ⓑ $(−4 \sqrt[4]{9 a^{3}}) (3 \sqrt[4]{27 a^{2}}) . (−4) (3) .$

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.

Example 8.42

Simplify: ⓐ $\sqrt{6} (\sqrt{2} + \sqrt{18}) \sqrt{6} (\sqrt{2} + \sqrt{18})$ ⓑ $\sqrt[3]{9} (5 - \sqrt[3]{18}) . \sqrt[3]{9}$

Show solution

Solution

ⓐ
$\begin{matrix} \sqrt{6} (\sqrt{2} + \sqrt{18}) \\ Multiply. & \sqrt{12} + \sqrt{108} \\ Simplify. & \sqrt{4} \cdot \sqrt{3} + \sqrt{36} \cdot \sqrt{3} \\ Simplify. & 2 \sqrt{3} + 6 \sqrt{3} \\ Combine like radicals. & 8 \sqrt{3} \end{matrix} \begin{matrix} Multiply. \\ Simplify. \\ Simplify. \\ Combine like radicals. \end{matrix}$

ⓑ
$\begin{matrix} \sqrt[3]{9} (5 - \sqrt[3]{18}) \\ Distribute. & 5 \sqrt[3]{9} - \sqrt[3]{162} \\ Simplify. & 5 \sqrt[3]{9} - \sqrt[3]{27} \cdot \sqrt[3]{6} \\ Simplify. & 5 \sqrt[3]{9} - 3 \sqrt[3]{6} \end{matrix} \begin{matrix} Distribute. \\ Simplify. \\ Simplify. \end{matrix}$

Try It 8.83

Simplify: ⓐ $\sqrt{6} (1 + 3 \sqrt{6}) \sqrt{6} (1 + 3 \sqrt{6})$ ⓑ $\sqrt[3]{4} (−2 - \sqrt[3]{6}) . \sqrt[3]{4} (−2 - \sqrt[3]{6}) .$

Try It 8.84

Simplify: ⓐ $\sqrt{8} (2 - 5 \sqrt{8}) \sqrt{8} (2 - 5 \sqrt{8})$ ⓑ $\sqrt[3]{3} (− \sqrt[3]{9} - \sqrt[3]{6}) . \sqrt[3]{3} (− \sqrt[3]{9} - \sqrt[3]{6}) .$

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.

Example 8.43

Simplify: ⓐ $(3 - 2 \sqrt{7}) (4 - 2 \sqrt{7}) (3 - 2 \sqrt{7}) (4 - 2 \sqrt{7})$ ⓑ $(\sqrt[3]{x} - 2) (\sqrt[3]{x} + 4) . (\sqrt[3]{x} - 2) (\sqrt[3]{x} + 4) .$

Show solution

Solution

ⓐ
$\begin{matrix} (3 - 2 \sqrt{7}) (4 - 2 \sqrt{7}) \\ Multiply & 12 - 6 \sqrt{7} - 8 \sqrt{7} + 4 \cdot 7 \\ Simplify. & 12 - 6 \sqrt{7} - 8 \sqrt{7} + 28 \\ Combine like terms. & 40 - 14 \sqrt{7} \end{matrix} \begin{matrix} Multiply \\ Simplify. \\ Combine like terms. \end{matrix}$

ⓑ
$\begin{matrix} (\sqrt[3]{x} - 2) (\sqrt[3]{x} + 4) \\ Multiply. & \sqrt[3]{x^{2}} + 4 \sqrt[3]{x} - 2 \sqrt[3]{x} - 8 \\ Combine like terms. & \sqrt[3]{x^{2}} + 2 \sqrt[3]{x} - 8 \end{matrix} \begin{matrix} Multiply. \\ Combine like terms. \end{matrix}$

Try It 8.85

Simplify: ⓐ $(6 - 3 \sqrt{7}) (3 + 4 \sqrt{7}) (6 - 3 \sqrt{7}) (3 + 4 \sqrt{7})$ ⓑ $(\sqrt[3]{x} - 2) (\sqrt[3]{x} - 3) . (\sqrt[3]{x} - 2) (\sqrt[3]{x} - 3) .$

Try It 8.86

Simplify: ⓐ $(2 - 3 \sqrt{11}) (4 - \sqrt{11}) (2 - 3 \sqrt{11}) (4 - \sqrt{11})$ ⓑ $(\sqrt[3]{x} + 1) (\sqrt[3]{x} + 3) . (\sqrt[3]{x} + 1) (\sqrt[3]{x} + 3) .$

Example 8.44

Simplify: $(3 \sqrt{2} - \sqrt{5}) (\sqrt{2} + 4 \sqrt{5}) . (3 \sqrt{2} - \sqrt{5}) (\sqrt{2} + 4 \sqrt{5}) .$

Show solution

Solution

$\begin{matrix} (3 \sqrt{2} - \sqrt{5}) (\sqrt{2} + 4 \sqrt{5}) \\ Multiply. & 3 \cdot 2 + 12 \sqrt{10} - \sqrt{10} - 4 \cdot 5 \\ Simplify. & 6 + 12 \sqrt{10} - \sqrt{10} - 20 \\ Combine like terms. & −14 + 11 \sqrt{10} \end{matrix} \begin{matrix} Multiply. \\ Simplify. \\ Combine like terms. \end{matrix}$

Try It 8.87

Simplify: $(5 \sqrt{3} - \sqrt{7}) (\sqrt{3} + 2 \sqrt{7}) (5 \sqrt{3} - \sqrt{7}) (\sqrt{3} + 2 \sqrt{7})$

Try It 8.88

Simplify: $(\sqrt{6} - 3 \sqrt{8}) (2 \sqrt{6} + \sqrt{8}) (\sqrt{6} - 3 \sqrt{8}) (2 \sqrt{6} + \sqrt{8})$

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.

Special Products

\begin{matrix} Binomial Squares & Product of Conjugates \\ {(a + b)}^{2} = a^{2} + 2 a b + b^{2} & (a + b) (a - b) = a^{2} - b^{2} \\ {(a - b)}^{2} = a^{2} - 2 a b + b^{2} \end{matrix} \begin{matrix} Binomial Squares & {(a + b)}^{2} = a^{2} + 2 a b + b^{2} & {(a - b)}^{2} = a^{2} - 2 a b + b^{2} \end{matrix}

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

Example 8.45

Simplify: ⓐ ${(2 + \sqrt{3})}^{2} {(2 + \sqrt{3})}^{2}$ ⓑ ${(4 - 2 \sqrt{5})}^{2} . {(4 - 2 \sqrt{5})}^{2} .$

Show solution

Solution

Be sure to include the $2 a b 2 a b$ 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.

Try It 8.89

Simplify: ⓐ ${(10 + \sqrt{2})}^{2} {(10 + \sqrt{2})}^{2}$ ⓑ ${(1 + 3 \sqrt{6})}^{2} . {(1 + 3 \sqrt{6})}^{2} .$

Try It 8.90

Simplify: ⓐ ${(6 - \sqrt{5})}^{2} {(6 - \sqrt{5})}^{2}$ ⓑ ${(9 - 2 \sqrt{10})}^{2} . {(9 - 2 \sqrt{10})}^{2} .$

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

Example 8.46

Simplify: $(5 - 2 \sqrt{3}) (5 + 2 \sqrt{3}) . (5 - 2 \sqrt{3}) (5 + 2 \sqrt{3}) .$

Show solution

Solution


Multiply, using the Product of Conjugates Pattern.
Simplify.

Try It 8.91

Simplify: $(3 - 2 \sqrt{5}) (3 + 2 \sqrt{5}) (3 - 2 \sqrt{5}) (3 + 2 \sqrt{5})$

Try It 8.92

Simplify: $(4 + 5 \sqrt{7}) (4 - 5 \sqrt{7}) . (4 + 5 \sqrt{7}) (4 - 5 \sqrt{7}) .$

Media

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

Practice Makes Perfect

Add and Subtract Radical Expressions

In the following exercises, simplify.

165.

ⓐ $8 \sqrt{2} - 5 \sqrt{2} 8 \sqrt{2} - 5 \sqrt{2}$ ⓑ $5 \sqrt[3]{m} + 2 \sqrt[3]{m} 5$ ⓒ $8 \sqrt[4]{m} - 2 \sqrt[4]{m} 8$

166.

ⓐ $7 \sqrt{2} - 3 \sqrt{2} 7 \sqrt{2} - 3 \sqrt{2}$ ⓑ $7 \sqrt[3]{p} + 2 \sqrt[3]{p} 7$ ⓒ $5 \sqrt[3]{x} - 3 \sqrt[3]{x} 5$

167.

ⓐ $3 \sqrt{5} + 6 \sqrt{5} 3 \sqrt{5} + 6 \sqrt{5}$ ⓑ $9 \sqrt[3]{a} + 3 \sqrt[3]{a} 9$ ⓒ $5 \sqrt[4]{2 z} + \sqrt[4]{2 z} 5$

168.

ⓐ $4 \sqrt{5} + 8 \sqrt{5} 4 \sqrt{5} + 8 \sqrt{5}$ ⓑ $\sqrt[3]{m} - 4 \sqrt[3]{m} \sqrt[3]{m} - 4$ ⓒ $\sqrt{n} + 3 \sqrt{n} \sqrt{n} + 3 \sqrt{n}$

169.

ⓐ $3 \sqrt{2 a} - 4 \sqrt{2 a} + 5 \sqrt{2 a} 3 \sqrt{2 a} - 4 \sqrt{2 a} + 5 \sqrt{2 a}$ ⓑ $5 \sqrt[4]{3 a b} - 3 \sqrt[4]{3 a b} - 2 \sqrt[4]{3 a b} 5$

170.

ⓐ $\sqrt{11 b} - 5 \sqrt{11 b} + 3 \sqrt{11 b} \sqrt{11 b} - 5 \sqrt{11 b} + 3 \sqrt{11 b}$ ⓑ $8 \sqrt[4]{11 c d} + 5 \sqrt[4]{11 c d} - 9 \sqrt[4]{11 c d} 8$

171.

ⓐ $8 \sqrt{3 c} + 2 \sqrt{3 c} - 9 \sqrt{3 c} 8 \sqrt{3 c} + 2 \sqrt{3 c} - 9 \sqrt{3 c}$ ⓑ $2 \sqrt[3]{4 p q} - 5 \sqrt[3]{4 p q} + 4 \sqrt[3]{4 p q} 2$

172.

ⓐ $3 \sqrt{5 d} + 8 \sqrt{5 d} - 11 \sqrt{5 d} 3 \sqrt{5 d} + 8 \sqrt{5 d} - 11 \sqrt{5 d}$ ⓑ $11 \sqrt[3]{2 r s} - 9 \sqrt[3]{2 r s} + 3 \sqrt[3]{2 r s} 11$

173.

ⓐ $\sqrt{27} - \sqrt{75} \sqrt{27} - \sqrt{75}$ ⓑ $\sqrt[3]{40} - \sqrt[3]{320} \sqrt[3]{40} - \sqrt[3]{320}$ ⓒ $\frac{1}{2} \sqrt[4]{32} + \frac{2}{3} \sqrt[4]{162} \frac{1}{2}$

174.

ⓐ $\sqrt{72} - \sqrt{98} \sqrt{72} - \sqrt{98}$ ⓑ $\sqrt[3]{24} + \sqrt[3]{81} \sqrt[3]{24} + \sqrt[3]{81}$ ⓒ $\frac{1}{2} \sqrt[4]{80} - \frac{2}{3} \sqrt[4]{405} \frac{1}{2}$

175.

ⓐ $\sqrt{48} + \sqrt{27} \sqrt{48} + \sqrt{27}$ ⓑ $\sqrt[3]{54} + \sqrt[3]{128} \sqrt[3]{54} + \sqrt[3]{128}$ ⓒ $6 \sqrt[4]{5} - \frac{3}{2} \sqrt[4]{80} 6$

176.

ⓐ $\sqrt{45} + \sqrt{80} \sqrt{45} + \sqrt{80}$ ⓑ $\sqrt[3]{81} - \sqrt[3]{192} \sqrt[3]{81} - \sqrt[3]{192}$ ⓒ $\frac{5}{2} \sqrt[4]{80} + \frac{7}{3} \sqrt[4]{405} \frac{5}{2}$

177.

ⓐ $\sqrt{72 a^{5}} - \sqrt{50 a^{5}} \sqrt{72 a^{5}} - \sqrt{50 a^{5}}$ ⓑ $9 \sqrt[4]{80 p^{4}} - 6 \sqrt[4]{405 p^{4}} 9$

178.

ⓐ $\sqrt{48 b^{5}} - \sqrt{75 b^{5}} \sqrt{48 b^{5}} - \sqrt{75 b^{5}}$ ⓑ $8 \sqrt[3]{64 q^{6}} - 3 \sqrt[3]{125 q^{6}} 8$

179.

ⓐ $\sqrt{80 c^{7}} - \sqrt{20 c^{7}} \sqrt{80 c^{7}} - \sqrt{20 c^{7}}$ ⓑ $2 \sqrt[4]{162 r^{10}} + 4 \sqrt[4]{32 r^{10}} 2$

180.

ⓐ $\sqrt{96 d^{9}} - \sqrt{24 d^{9}} \sqrt{96 d^{9}} - \sqrt{24 d^{9}}$ ⓑ $5 \sqrt[4]{243 s^{6}} + 2 \sqrt[4]{3 s^{6}} 5$

181.

$3 \sqrt{128 y^{2}} + 4 y \sqrt{162} - 8 \sqrt{98 y^{2}} 3$

182.

$3 \sqrt{75 y^{2}} + 8 y \sqrt{48} - \sqrt{300 y^{2}} 3$

Multiply Radical Expressions

In the following exercises, simplify.

183.

ⓐ $(−2 \sqrt[]{3}) (3 \sqrt[]{18}) (−2 \sqrt[]{3}) (3 \sqrt[]{18})$ ⓑ $(8 \sqrt[3]{4}) (−4 \sqrt[3]{18}) (8) (−4)$

184.

ⓐ $(−4 \sqrt[]{5}) (5 \sqrt[]{10}) (−4 \sqrt[]{5}) (5 \sqrt[]{10})$ ⓑ $(−2 \sqrt[3]{9}) (7 \sqrt[3]{9}) (−2) (7)$

185.

ⓐ $(5 \sqrt[]{6}) (− \sqrt[]{12}) (5 \sqrt[]{6}) (− \sqrt[]{12})$ ⓑ $(−2 \sqrt[4]{18}) (− \sqrt[4]{9}) (−2) (−)$

186.

ⓐ $(−2 \sqrt{7}) (−2 \sqrt{14}) (−2 \sqrt{7}) (−2 \sqrt{14})$ ⓑ $(−3 \sqrt[4]{8}) (−5 \sqrt[4]{6}) (−3) (−5)$

187.

ⓐ $(4 \sqrt{12 z^{3}}) (3 \sqrt{9 z}) (4 \sqrt{12 z^{3}}) (3 \sqrt{9 z})$ ⓑ $(5 \sqrt[3]{3 x^{3}}) (3 \sqrt[3]{18 x^{3}}) (5) (3)$

188.

ⓐ $(3 \sqrt{2 x^{3}}) (7 \sqrt{18 x^{2}}) (3 \sqrt{2 x^{3}}) (7 \sqrt{18 x^{2}})$ ⓑ $(−6 \sqrt[3]{20 a^{2}}) (−2 \sqrt[3]{16 a^{3}}) (−6) (−2)$

189.

ⓐ $(−2 \sqrt{7 z^{3}}) (3 \sqrt{14 z^{8}}) (−2 \sqrt{7 z^{3}}) (3 \sqrt{14 z^{8}})$ ⓑ $(2 \sqrt[4]{8 y^{2}}) (−2 \sqrt[4]{12 y^{3}}) (2) (−2)$

190.

ⓐ $(4 \sqrt{2 k^{5}}) (−3 \sqrt{32 k^{6}}) (4 \sqrt{2 k^{5}}) (−3 \sqrt{32 k^{6}})$ ⓑ $(− \sqrt[4]{6 b^{3}}) (3 \sqrt[4]{8 b^{3}}) (−) (3)$

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

191.

ⓐ $\sqrt{7} (5 + 2 \sqrt{7}) \sqrt{7} (5 + 2 \sqrt{7})$ ⓑ $\sqrt[3]{6} (4 + \sqrt[3]{18}) \sqrt[3]{6}$

192.

ⓐ $\sqrt{11} (8 + 4 \sqrt{11}) \sqrt{11} (8 + 4 \sqrt{11})$ ⓑ $\sqrt[3]{3} (\sqrt[3]{9} + \sqrt[3]{18}) \sqrt[3]{3}$

193.

ⓐ $\sqrt{11} (−3 + 4 \sqrt{11}) \sqrt{11} (−3 + 4 \sqrt{11})$ ⓑ $\sqrt[4]{3} (\sqrt[4]{54} + \sqrt[4]{18}) \sqrt[4]{3}$

194.

ⓐ $\sqrt{2} (−5 + 9 \sqrt{2}) \sqrt{2} (−5 + 9 \sqrt{2})$ ⓑ $\sqrt[4]{2} (\sqrt[4]{12} + \sqrt[4]{24}) \sqrt[4]{2}$

195.

$(7 + \sqrt{3}) (9 - \sqrt{3}) (7 + \sqrt{3}) (9 - \sqrt{3})$

196.

$(8 - \sqrt{2}) (3 + \sqrt{2}) (8 - \sqrt{2}) (3 + \sqrt{2})$

197.

ⓐ $(9 - 3 \sqrt{2}) (6 + 4 \sqrt{2}) (9 - 3 \sqrt{2}) (6 + 4 \sqrt{2})$ ⓑ $(\sqrt[3]{x} - 3) (\sqrt[3]{x} + 1) (\sqrt[3]{x} - 3) (\sqrt[3]{x} + 1)$

198.

ⓐ $(3 - 2 \sqrt{7}) (5 - 4 \sqrt{7}) (3 - 2 \sqrt{7}) (5 - 4 \sqrt{7})$ ⓑ $(\sqrt[3]{x} - 5) (\sqrt[3]{x} - 3) (\sqrt[3]{x} - 5) (\sqrt[3]{x} - 3)$

199.

ⓐ $(1 + 3 \sqrt{10}) (5 - 2 \sqrt{10}) (1 + 3 \sqrt{10}) (5 - 2 \sqrt{10})$ ⓑ $(2 \sqrt[3]{x} + 6) (\sqrt[3]{x} + 1) (2) (\sqrt[3]{x} + 1)$

200.

ⓐ $(7 - 2 \sqrt{5}) (4 + 9 \sqrt{5}) (7 - 2 \sqrt{5}) (4 + 9 \sqrt{5})$ ⓑ $(3 \sqrt[3]{x} + 2) (\sqrt[3]{x} - 2) (3) (\sqrt[3]{x} - 2)$

201.

$(\sqrt{3} + \sqrt{10}) (\sqrt{3} + 2 \sqrt{10}) (\sqrt{3} + \sqrt{10}) (\sqrt{3} + 2 \sqrt{10})$

202.

$(\sqrt{11} + \sqrt{5}) (\sqrt{11} + 6 \sqrt{5}) (\sqrt{11} + \sqrt{5}) (\sqrt{11} + 6 \sqrt{5})$

203.

$(2 \sqrt{7} - 5 \sqrt{11}) (4 \sqrt{7} + 9 \sqrt{11}) (2 \sqrt{7} - 5 \sqrt{11}) (4 \sqrt{7} + 9 \sqrt{11})$

204.

$(4 \sqrt{6} + 7 \sqrt{13}) (8 \sqrt{6} - 3 \sqrt{13}) (4 \sqrt{6} + 7 \sqrt{13}) (8 \sqrt{6} - 3 \sqrt{13})$

205.

ⓐ ${(3 + \sqrt{5})}^{2} {(3 + \sqrt{5})}^{2}$ ⓑ ${(2 - 5 \sqrt{3})}^{2} {(2 - 5 \sqrt{3})}^{2}$

206.

ⓐ ${(4 + \sqrt{11})}^{2} {(4 + \sqrt{11})}^{2}$ ⓑ ${(3 - 2 \sqrt{5})}^{2} {(3 - 2 \sqrt{5})}^{2}$

207.

ⓐ ${(9 - \sqrt{6})}^{2} {(9 - \sqrt{6})}^{2}$ ⓑ ${(10 + 3 \sqrt{7})}^{2} {(10 + 3 \sqrt{7})}^{2}$

208.

ⓐ ${(5 - \sqrt{10})}^{2} {(5 - \sqrt{10})}^{2}$ ⓑ ${(8 + 3 \sqrt{2})}^{2} {(8 + 3 \sqrt{2})}^{2}$

209.

$(4 + \sqrt{2}) (4 - \sqrt{2}) (4 + \sqrt{2}) (4 - \sqrt{2})$

210.

$(7 + \sqrt{10}) (7 - \sqrt{10}) (7 + \sqrt{10}) (7 - \sqrt{10})$

211.

$(4 + 9 \sqrt{3}) (4 - 9 \sqrt{3}) (4 + 9 \sqrt{3}) (4 - 9 \sqrt{3})$

212.

$(1 + 8 \sqrt{2}) (1 - 8 \sqrt{2}) (1 + 8 \sqrt{2}) (1 - 8 \sqrt{2})$

213.

$(12 - 5 \sqrt{5}) (12 + 5 \sqrt{5}) (12 - 5 \sqrt{5}) (12 + 5 \sqrt{5})$

214.

$(9 - 4 \sqrt{3}) (9 + 4 \sqrt{3}) (9 - 4 \sqrt{3}) (9 + 4 \sqrt{3})$

215.

$(\sqrt[3]{3 x} + 2) (\sqrt[3]{3 x} - 2) (\sqrt[3]{3 x} + 2) (\sqrt[3]{3 x} - 2)$

216.

$(\sqrt[3]{4 x} + 3) (\sqrt[3]{4 x} - 3) (\sqrt[3]{4 x} + 3) (\sqrt[3]{4 x} - 3)$

Mixed Practice

217.

$\frac{2}{3} \sqrt{27} + \frac{3}{4} \sqrt{48} \frac{2}{3} \sqrt{27} + \frac{3}{4} \sqrt{48}$

218.

$\sqrt{175 k^{4}} - \sqrt{63 k^{4}} \sqrt{175 k^{4}} - \sqrt{63 k^{4}}$

219.

$\frac{5}{6} \sqrt{162} + \frac{3}{16} \sqrt{128} \frac{5}{6} \sqrt{162} + \frac{3}{16} \sqrt{128}$

220.

$\sqrt[3]{24} + \sqrt[3]{/ 81} \sqrt[3]{24} + \sqrt[3]{/ 81}$

221.

$\frac{1}{2} \sqrt[4]{80} - \frac{2}{3} \sqrt[4]{405} \frac{1}{2}$

222.

$8 \sqrt[4]{13} - 4 \sqrt[4]{13} - 3 \sqrt[4]{13} 8 \sqrt[4]{13} - 4 \sqrt[4]{13} - 3 \sqrt[4]{13}$

223.

$5 \sqrt{12 c^{4}} - 3 \sqrt{27 c^{6}} 5 \sqrt{12 c^{4}} - 3 \sqrt{27 c^{6}}$

224.

$\sqrt{80 a^{5}} - \sqrt{45 a^{5}} \sqrt{80 a^{5}} - \sqrt{45 a^{5}}$

225.

$\frac{3}{5} \sqrt{75} - \frac{1}{4} \sqrt{48} \frac{3}{5} \sqrt{75} - \frac{1}{4} \sqrt{48}$

226.

$21 \sqrt[3]{9} - 2 \sqrt[3]{9} 21$

227.

$8 \sqrt[3]{64 q^{6}} - 3 \sqrt[3]{125 q^{6}} 8$

228.

$11 \sqrt{11} - 10 \sqrt{11} 11 \sqrt{11} - 10 \sqrt{11}$

229.

$\sqrt{3} \cdot \sqrt{21} \sqrt{3} \cdot \sqrt{21}$

230.

$(4 \sqrt{6}) (− \sqrt{18}) (4 \sqrt{6}) (− \sqrt{18})$

231.

$(7 \sqrt[3]{4}) (−3 \sqrt[3]{18}) (7 \sqrt[3]{4}) (−3 \sqrt[3]{18})$

232.

$(4 \sqrt{12 x^{5}}) (2 \sqrt{6 x^{3}}) (4 \sqrt{12 x^{5}}) (2 \sqrt{6 x^{3}})$

233.

${(\sqrt{29})}^{2} {(\sqrt{29})}^{2}$

234.

$(−4 \sqrt{17}) (−3 \sqrt{17}) (−4 \sqrt{17}) (−3 \sqrt{17})$

235.

$(−4 + \sqrt{17}) (−3 + \sqrt{17}) (−4 + \sqrt{17}) (−3 + \sqrt{17})$

236.

$(3 \sqrt[4]{8 a^{2}}) (\sqrt[4]{12 a^{3}}) (3) (\sqrt[4]{12 a^{3}})$

237.

${(6 - 3 \sqrt{2})}^{2} {(6 - 3 \sqrt{2})}^{2}$

238.

$\sqrt{3} (4 - 3 \sqrt{3}) \sqrt{3} (4 - 3 \sqrt{3})$

239.

$\sqrt[3]{3} (2 \sqrt[3]{9} + \sqrt[3]{18}) \sqrt[3]{3}$

240.

$(\sqrt{6} + \sqrt{3}) (\sqrt{6} + 6 \sqrt{3}) (\sqrt{6} + \sqrt{3}) (\sqrt{6} + 6 \sqrt{3})$

Writing Exercises

241.

Explain the when a radical expression is in simplest form.

242.

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.

243.

ⓐ Explain why ${(− \sqrt{n})}^{2} {(− \sqrt{n})}^{2}$ is always non-negative, for $n \geq 0 . n \geq 0 .$
ⓑ Explain why $- {(\sqrt{n})}^{2} - {(\sqrt{n})}^{2}$ is always non-positive, for $n \geq 0. n \geq 0.$

244.

Use the binomial square pattern to simplify ${(3 + \sqrt{2})}^{2} . {(3 + \sqrt{2})}^{2} .$ Explain all your steps.

Self Check

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

This table has 3 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “add and subtract radical expressions.”, “ multiply radical expressions”, and “use polynomial multiplication to multiply radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

ⓑ 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?