8.4 Add, Subtract, and Multiply Radical Expressions

Learning objectives

Learning objectives

• Add and subtract radical expressions

• Multiply radical expressions

• Use polynomial multiplication to multiply radical expressions

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.

We add and subtract like radicals in the same way we add and subtract like terms. We know that $3x+8x$ is $11x.$ Similarly we add $3\sqrt{x}+8\sqrt{x}$ and the result is $11\sqrt{x}.$

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

Solution

ⓐ

$\,2\,\sqrt{2}-7\,\sqrt{2}$

Since the radicals are like, we subtract thecoefficients. · $\,-5\,\sqrt{2}$

ⓑ

$\,5\,\sqrt[3]{y}+4\,\sqrt[3]{y}$

Since the radicals are like, we add thecoefficients. · $\,9\,\sqrt[3]{y}$

ⓒ

$\,7\,\sqrt[4]{x}-2\,\sqrt[4]{y}$

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.

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

Solution

ⓐ

$\,2\,\sqrt{5n}-6\,\sqrt{5n}+4\,\sqrt{5n}$

Since the radicals are like, we combine them. · $\,0\,\sqrt{5n}$ Simplify. · $\,0$

ⓑ

$\,\sqrt[4]{3xy}+5\,\sqrt[4]{3xy}-4\,\sqrt[4]{3xy}$

Since the radicals are like, we combine them. · $\,2\,\sqrt[4]{3xy}$

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

Solution

ⓐ

$\,\sqrt{20}+3\,\sqrt{5}$

Simplify the radicals, when possible. · $\,\sqrt{4}\cdot\sqrt{5}+3\,\sqrt{5}$

$\,2\,\sqrt{5}+3\,\sqrt{5}$

Combine the like radicals. · $\,5\,\sqrt{5}$

ⓑ

$\,\sqrt[3]{24}-\sqrt[3]{375}$

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. · $\,-3\sqrt[3]{3}$

ⓒ

$\,\frac{1}{2}\,\sqrt[4]{48}-\frac{2}{3}\,\sqrt[4]{243}$

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. · $\,-\sqrt[4]{3}$

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

Solution

ⓐ

$\,9\,\sqrt{50{m}^{2}}-6\,\sqrt{48{m}^{2}}$

Simplify the radicals. · $\,9\,\sqrt{25{m}^{2}}\cdot\sqrt{2}-6\,\sqrt{16{m}^{2}}\cdot\sqrt{3}$

$\,9\cdot{}5m\cdot\sqrt{2}-6\cdot{}4m\cdot\sqrt{3}$

$\,45m\,\sqrt{2}-24m\,\sqrt{3}$

The radicals are not like and so cannot becombined.

ⓑ

$\,\sqrt[3]{54{n}^{5}}-\sqrt[3]{16{n}^{5}}$

Simplify the radicals. · $\,\sqrt[3]{27{n}^{3}}\cdot\sqrt[3]{2{n}^{2}}-\sqrt[3]{8{n}^{3}}\cdot\sqrt[3]{2{n}^{2}}$

$\,3n\,\sqrt[3]{2{n}^{2}}-2n\,\sqrt[3]{2{n}^{2}}$

Combine the like radicals. · $\,n\,\sqrt[3]{2{n}^{2}}$

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.

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

Example 8.40

Solution

ⓐ

$\,(6\,\sqrt{2})(3\,\sqrt{10})$

Multiply using the Product Property. · $\,18\,\sqrt{20}$ Simplify the radical. · $\,18\,\sqrt{4}\cdot\sqrt{5}$ Simplify. · $\,18\cdot{}2\cdot\sqrt{5}$

$\,36\,\sqrt{5}$

ⓑ

$\,(-5\,\sqrt[3]{4})(-4\,\sqrt[3]{6})$

Multiply using the Product Property. · $\,20\,\sqrt[3]{24}$ Simplify the radical. · $\,20\,\sqrt[3]{8}\cdot\sqrt[3]{3}$ Simplify. · $\,20\cdot{}2\cdot\sqrt[3]{3}$

$\,40\,\sqrt[3]{3}$

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

Example 8.41

Solution

ⓐ

$\,(10\sqrt{6{p}^{3}})(4\sqrt{3p})$

Multiply. · $\,40\sqrt{18{p}^{4}}$ Simplify the radical. · $\,40\sqrt{9{p}^{4}}\cdot\sqrt{2}$ Simplify. · $\,40\cdot{}3{p}^{2}\cdot\sqrt{2}$

$\,120{p}^{2}\sqrt{2}$

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

$\,(2\sqrt[4]{20{y}^{2}})(3\sqrt[4]{28{y}^{3}})$

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. · $\,6\cdot{}2y\,\sqrt[4]{35{y}^{}}$ Multiply. · $\,12y\,\sqrt[4]{35{y}^{}}$

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

Solution

ⓐ

$\,\sqrt{6}\,(\sqrt{2}+\sqrt{18})$

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. · $\,8\sqrt{3}$

ⓑ

$\,\sqrt[3]{9}\,(5-\sqrt[3]{18})$

Distribute. · $\,5\sqrt[3]{9}-\sqrt[3]{162}$ Simplify. · $\,5\,\sqrt[3]{9}-\sqrt[3]{27}\cdot\sqrt[3]{6}$ Simplify. · $\,5\,\sqrt[3]{9}-3\,\sqrt[3]{6}$

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

Solution

ⓐ

$\,(3-2\sqrt{7})(4-2\sqrt{7})$

Multiply. · $\,12-6\sqrt{7}-8\sqrt{7}+4\cdot{}7$ Simplify. · $\,12-6\sqrt{7}-8\sqrt{7}+28$ Combine like terms. · $\,40-14\sqrt{7}$

ⓑ

$\,(\sqrt[3]{x}-2)(\sqrt[3]{x}+4)$

Multiply. · $\,\sqrt[3]{{x}^{2}}+4\,\sqrt[3]{x}-2\,\sqrt[3]{x}-8$ Combine like terms. · $\,\sqrt[3]{{x}^{2}}+2\,\sqrt[3]{x}-8$

Example 8.44

Solution

$\,(3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})$

Multiply. · $\,3\cdot{}2+12\sqrt{10}-\sqrt{10}-4\cdot{}5$ Simplify. · $\,6+12\sqrt{10}-\sqrt{10}-20$ Combine like terms. · $\,-14+11\sqrt{10}$

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.

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

Solution

Be sure to include the $2ab$ term when squaring a binomial.

ⓐ

ⓑ

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

Example 8.46

Solution

Section 8.4 Exercises

Section 8.4 Exercises

Add and Subtract Radical Expressions

In the following exercises, simplify.

Multiply Radical Expressions

In the following exercises, simplify.

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

Mixed Practice

Writing Exercises

ⓐ 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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1. ⓐ $3\sqrt{2a}-4\sqrt{2a}+5\sqrt{2a}$ ⓑ $5\,\sqrt[4]{3ab}-3\,\sqrt[4]{3ab}-2\,\sqrt[4]{3ab}$

2. ⓐ $8\sqrt{3c}+2\sqrt{3c}-9\sqrt{3c}$ ⓑ $2\,\sqrt[3]{4pq}-5\,\sqrt[3]{4pq}+4\,\sqrt[3]{4pq}$

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

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

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

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

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

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

1. ⓐ $(5\sqrt[]{6})(-\sqrt[]{12})$ ⓑ $(-2\,\sqrt[4]{18})(-\,\sqrt[4]{9})$

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

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

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

1. ⓐ $\sqrt{11}(-3+4\sqrt{11})$ ⓑ $\sqrt[4]{3}\,(\sqrt[4]{54}+\sqrt[4]{18})$

2. $(7+\sqrt{3})(9-\sqrt{3})$

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

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

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

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

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

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

1. $(4+9\sqrt{3})(4-9\sqrt{3})$

2. $(12-5\sqrt{5})(12+5\sqrt{5})$

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

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

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

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

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

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

1. $8\,\sqrt[3]{64{q}^{6}}-3\,\sqrt[3]{125{q}^{6}}$

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

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

4. $(-4+\sqrt{17})(-3+\sqrt{17})$

1. ${(6-3\sqrt{2})}^{2}$

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

3. Explain when a radical expression is in simplest form.

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