MX Algebra Add, Subtract, and Multiply Radical Expressions

Section 8.4Add, Subtract, and Multiply Radical Expressions

Definition

Before you get started, take this readiness quiz.

1

Add: \(3{x}^{2}+9x-5-({x}^{2}-2x+3).\)
If you missed this problem, review Example 5.

\(2{x}^{2}+11x-8\)

Definition
2

Simplify: \((2+a)(4-a).\)
If you missed this problem, review Example 4.

\(8+2a-{a}^{2}\)

Definition
3

Simplify: \({(9-5y)}^{2}.\)
If you missed this problem, review Example 7.

\(81-90y+25{y}^{2}\)

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 \(3x+8x\) 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 1

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

Check whether each pair of radicals shares the same index and radicand before combining coefficients.

ⓐ

Table 1
\(\,2\,\sqrt{2}-7\,\sqrt{2}\)
Since the radicals are like, we subtract the
coefficients.		\(\,-5\,\sqrt{2}\)

ⓑ

Table 2
\(\,5\,\sqrt[3]{y}+4\,\sqrt[3]{y}\)
Since the radicals are like, we add the
coefficients.		\(\,9\,\sqrt[3]{y}\)

ⓒ

Table 3
\(\,7\,\sqrt[4]{x}-2\,\sqrt[4]{y}\)

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.

Try It #1

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

ⓐ \(\text{-}\sqrt{2}\) ⓑ \(11\sqrt[3]{x}\)
ⓒ \(3\sqrt[4]{x}-5\sqrt[4]{y}\)

Did you get it?
Try It #2

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

ⓐ \(-4\sqrt{3}\) ⓑ \(8\sqrt[3]{y}\)
ⓒ \(5\sqrt[4]{m}-2\sqrt[3]{m}\)

Did you get it?

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 2

Simplify: ⓐ \(2\,\sqrt{5n}-6\,\sqrt{5n}+4\,\sqrt{5n}\) ⓑ \(\sqrt[4]{3xy}+5\,\sqrt[4]{3xy}-4\,\sqrt[4]{3xy}.\)

Since all the terms already share the same index and radicand, combine just the coefficients.

ⓐ

Table 4
\(\,2\,\sqrt{5n}-6\,\sqrt{5n}+4\,\sqrt{5n}\)
Since the radicals are like, we combine them.\(\,0\,\sqrt{5n}\)
Simplify.\(\,0\)

ⓑ

Table 5
\(\,\sqrt[4]{3xy}+5\,\sqrt[4]{3xy}-4\,\sqrt[4]{3xy}\)
Since the radicals are like, we combine them.\(\,2\,\sqrt[4]{3xy}\)
Try It #3

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

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

Did you get it?
Try It #4

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

ⓐ \(\text{-}\sqrt{3y}\) ⓑ \(3\sqrt[3]{7mn}\)

Did you get it?

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 3

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

Simplify each radical first by pulling out perfect-power factors, then see if the results are like radicals.

ⓐ

Table 6
\(\,\sqrt{20}+3\,\sqrt{5}\)
Simplify the radicals, when possible.\(\,\sqrt{4}·\sqrt{5}+3\,\sqrt{5}\)
\(\,2\,\sqrt{5}+3\,\sqrt{5}\)
Combine the like radicals.\(\,5\,\sqrt{5}\)

ⓑ

Table 7
\(\,\sqrt[3]{24}-\sqrt[3]{375}\)
Simplify the radicals.\(\,\sqrt[3]{8}·\sqrt[3]{3}-\sqrt[3]{125}·\sqrt[3]{3}\)
\(\,2\,\sqrt[3]{3}-5\,\sqrt[3]{3}\)
Combine the like radicals.\(\,-3\sqrt[3]{3}\)

ⓒ

Table 8
\(\,\frac{1}{2}\,\sqrt[4]{48}-\frac{2}{3}\,\sqrt[4]{243}\)
Simplify the radicals.\(\,\frac{1}{2}\,\sqrt[4]{16}·\sqrt[4]{3}-\frac{2}{3}\,\sqrt[4]{81}·\sqrt[4]{3}\)
\(\,\frac{1}{2}·2·\sqrt[4]{3}-\frac{2}{3}·3·\sqrt[4]{3}\)
\(\,\sqrt[4]{3}-2\,\sqrt[4]{3}\)
Combine the like radicals.\(\,\text{-}\sqrt[4]{3}\)
Try It #5

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

ⓐ \(9\sqrt{2}\) ⓑ \(2\sqrt[3]{2}\) ⓒ \(\sqrt[3]{3}\)

Did you get it?
Try It #6

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

ⓐ \(7\sqrt{3}\) ⓑ \(-10\sqrt[3]{5}\) ⓒ \(-3\sqrt[3]{2}\)

Did you get it?

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 4

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

Factor the largest perfect power out of each radicand before comparing the resulting radicals.

ⓐ

Table 9
\(\,9\,\sqrt{50{m}^{2}}-6\,\sqrt{48{m}^{2}}\)
Simplify the radicals.\(\,9\,\sqrt{25{m}^{2}}·\sqrt{2}-6\,\sqrt{16{m}^{2}}·\sqrt{3}\)
\(\,9·5m·\sqrt{2}-6·4m·\sqrt{3}\)
\(\,45m\,\sqrt{2}-24m\,\sqrt{3}\)
The radicals are not like and so cannot be
combined.

ⓑ

Table 10
\(\,\sqrt[3]{54{n}^{5}}-\sqrt[3]{16{n}^{5}}\)
Simplify the radicals.\(\,\sqrt[3]{27{n}^{3}}·\sqrt[3]{2{n}^{2}}-\sqrt[3]{8{n}^{3}}·\sqrt[3]{2{n}^{2}}\)
\(\,3n\,\sqrt[3]{2{n}^{2}}-2n\,\sqrt[3]{2{n}^{2}}\)
Combine the like radicals.\(\,n\,\sqrt[3]{2{n}^{2}}\)
Try It #7

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

ⓐ \(\text{-}{m}^{3}\sqrt{2m}\) ⓑ \({x}^{2}\sqrt[3]{5x}\)

Did you get it?
Try It #8

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

ⓐ \(\text{-}p\sqrt{3p}\)
ⓑ \(4y\sqrt[3]{4{y}^{2}}-2y\sqrt[3]{4{n}^{2}}\)

Did you get it?

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, \(\sqrt[n]{a}\) and \(\sqrt[n]{b},\) and for any integer \(n\ge 2\)

\[\sqrt[n]{ab}=\sqrt[n]{a}·\sqrt[n]{b}\,\text{and}\,\sqrt[n]{a}·\sqrt[n]{b}=\sqrt[n]{ab}\]

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

Example 5

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

Multiply the coefficients together and the radicands together using the Product Property of Roots.

ⓐ

Table 11
\(\,(6\,\sqrt{2})(3\,\sqrt{10})\)
Multiply using the Product Property.\(\,18\,\sqrt{20}\)
Simplify the radical.\(\,18\,\sqrt{4}·\sqrt{5}\)
Simplify.\(\,18·2·\sqrt{5}\)
\(\,36\,\sqrt{5}\)

ⓑ

Table 12
\(\,(-5\,\sqrt[3]{4})(-4\,\sqrt[3]{6})\)
Multiply using the Product Property.\(\,20\,\sqrt[3]{24}\)
Simplify the radical.\(\,20\,\sqrt[3]{8}·\sqrt[3]{3}\)
Simplify.\(\,20·2·\sqrt[3]{3}\)
\(\,40\,\sqrt[3]{3}\)
Try It #9

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

ⓐ \(12\sqrt{15}\) ⓑ \(-18\sqrt[3]{4}\)

Did you get it?
Try It #10

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

ⓐ \(27\sqrt{2}\) ⓑ \(-36\sqrt[3]{2}\)

Did you get it?

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

Example 6

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

Multiply coefficients and radicands separately, then factor the product looking for a perfect power of the index.

ⓐ

Table 13
\(\,(10\sqrt{6{p}^{3}})(4\sqrt{3p})\)
Multiply.\(\,40\sqrt{18{p}^{4}}\)
Simplify the radical.\(\,40\sqrt{9{p}^{4}}·\sqrt{2}\)
Simplify.\(\,40·3{p}^{2}·\sqrt{2}\)
\(\,120{p}^{2}\sqrt{2}\)

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

Table 14
\(\,(2\sqrt[4]{20{y}^{2}})(3\sqrt[4]{28{y}^{3}})\)
Multiply.\(\,6\,\sqrt[4]{4·5·4·7{y}^{5}}\)
Simplify the radical.\(\,6\,\sqrt[4]{16{y}^{4}}·\sqrt[4]{35y}\)
Simplify.\(\,6·2y\,\sqrt[4]{35y}\)
Multiply.\(\,12y\,\sqrt[4]{35y}\)
Try It #11

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

ⓐ \(288{x}^{3}\sqrt{5}\) ⓑ \(8y\sqrt[4]{6{y}^{2}}\)

Did you get it?
Try It #12

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

ⓐ \(144{y}^{2}\sqrt{5y}\) ⓑ \(-36a\sqrt[4]{3a}\)

Did you get it?

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 7

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

Distribute the outer radical (or number) over each term inside the parentheses first.

ⓐ

Table 15
\(\,\sqrt{6}\,(\sqrt{2}+\sqrt{18})\)
Multiply.\(\,\sqrt{12}+\sqrt{108}\)
Simplify.\(\,\sqrt{4}·\sqrt{3}+\sqrt{36}·\sqrt{3}\)
Simplify.\(\,2\sqrt{3}+6\sqrt{3}\)
Combine like radicals.\(\,8\sqrt{3}\)

ⓑ

Table 16
\(\,\sqrt[3]{9}\,(5-\sqrt[3]{18})\)
Distribute.\(\,5\sqrt[3]{9}-\sqrt[3]{162}\)
Simplify.\(\,5\,\sqrt[3]{9}-\sqrt[3]{27}·\sqrt[3]{6}\)
Simplify.\(\,5\,\sqrt[3]{9}-3\,\sqrt[3]{6}\)
Try It #13

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

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

Did you get it?
Try It #14

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

ⓐ \(-40+4\sqrt{2}\) ⓑ \(-3-\sqrt[3]{18}\)

Did you get it?

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

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

Use FOIL to multiply the two binomials, then combine any like terms.

ⓐ

Table 17
\(\,(3-2\sqrt{7})(4-2\sqrt{7})\)
Multiply.\(\,12-6\sqrt{7}-8\sqrt{7}+4·7\)
Simplify.\(\,12-6\sqrt{7}-8\sqrt{7}+28\)
Combine like terms.\(\,40-14\sqrt{7}\)

ⓑ

Table 18
\(\,(\sqrt[3]{x}-2)(\sqrt[3]{x}+4)\)
Multiply.\(\,\sqrt[3]{{x}^{2}}+4\,\sqrt[3]{x}-2\,\sqrt[3]{x}-8\)
Combine like terms.\(\,\sqrt[3]{{x}^{2}}+2\,\sqrt[3]{x}-8\)
Try It #15

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

ⓐ \(-66+15\sqrt{7}\)
ⓑ \(\sqrt[3]{{x}^{2}}-5\sqrt[3]{x}+6\)

Did you get it?
Try It #16

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

ⓐ \(41-14\sqrt{11}\)
ⓑ \(\sqrt[3]{{x}^{2}}+4\sqrt[3]{x}+3\)

Did you get it?
Example 9

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

Use FOIL to multiply the two binomials, then combine any like terms.

Table 19
\(\,(3\sqrt{2}-\sqrt{5})(\sqrt{2}+4\sqrt{5})\)
Multiply.\(\,3·2+12\sqrt{10}-\sqrt{10}-4·5\)
Simplify.\(\,6+12\sqrt{10}-\sqrt{10}-20\)
Combine like terms.\(\,-14+11\sqrt{10}\)
Try It #17

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

\(1+9\sqrt{21}\)

Did you get it?
Try It #18

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

\(-12-20\sqrt{3}\)

Did you get it?

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

\[\begin{array}{llll} \\ \\ \text{Binomial Squares} & & & \,\text{Product of Conjugates} \\ {(a+b)}^{2}={a}^{2}+2ab+{b}^{2} & & & \,(a+b)(a-b)={a}^{2}-{b}^{2} \\ {(a-b)}^{2}={a}^{2}-2ab+{b}^{2} & & & \end{array}\]

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

Example 10

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

Apply the binomial squares pattern \({(a\pm b)}^{2}={a}^{2}\pm2ab+{b}^{2},\) being careful not to drop the middle term.

Be sure to include the \(2ab\) term when squaring a binomial.

ⓐ


ⓑ

Table 20
Two mathematical expressions are shown: (a + b)' in red text, representing the square of a sum, and directly below it, (2 + sqrt(3))', a specific example of the same algebraic form.
Multiply, using the Product of Binomial Squares Pattern.Algebraic identity for a perfect square: a^2 + 2ab + b^2, with an example where a=2 and b=sqrt(3).
Simplify.A mathematical expression reads '4 + 4 square root of 3 + 3' in black text against a white background, representing an algebraic sum involving an integer and a surd.
Combine like terms.A mathematical expression '7 + 4√3' is displayed on a white background.
Table 21
The image displays two algebraic expressions related to squaring a difference. The first expression, in red, is (a - b)^2. Below it, in black, is the specific numerical instance (4 - 2√5)^2.
Multiply, using the Product of Binomial Squares Pattern.Illustration of the perfect square trinomial identity (a-b)^2, showing a^2 - 2ab + b^2 and its numerical application with a=4 and b=2sqrt(5).
Simplify.A mathematical expression reads '16 - 16√5 + 4 × 5' against a white background.
The image shows the mathematical expression 16 - 16√5 + 20.
Combine like terms.The image shows the mathematical expression 36 - 16√5 in a bold, black font against a white background.
Try It #19

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

ⓐ \(102+20\sqrt{2}\) ⓑ \(55+6\sqrt{6}\)

Did you get it?
Try It #20

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

ⓐ \(41-12\sqrt{5}\)
ⓑ \(121-36\sqrt{10}\)

Did you get it?

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

Example 11

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

Apply the Product of Conjugates pattern \((a+b)(a-b)={a}^{2}-{b}^{2}\) to eliminate the radical entirely.

Table 22
This image displays the difference of squares formula, (a-b)(a+b), and an application of it with radical numbers: (5 - 2√3)(5 + 2√3).
Multiply, using the Product of Conjugates Pattern.The image displays a mathematical problem with two lines of expressions. The first line, in red text, shows the difference of squares formula, a^2 - b^2. Below it, in black text, is the specific calculation 5^2 - (2√3)^2.
Simplify.A mathematical expression '25 - 4 ⋅ 3' is shown in black text on a white background.
The number '13' is visible on the left side of the image.
Try It #21

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

\(-11\)

Did you get it?
Try It #22

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

\(-159\)

Did you get it?
Media

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

Key Concepts

Section Exercises

Practice Makes Perfect

Add and Subtract Radical Expressions

In the following exercises, simplify.

4

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

ⓐ \(3\sqrt{2}\) ⓑ \(7\sqrt[3]{m}\) ⓒ \(6\sqrt[4]{m}\)

5

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

6

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

ⓐ \(9\sqrt{5}\) ⓑ \(12\sqrt[3]{a}\) ⓒ \(6\sqrt[4]{2z}\)

7

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

8

ⓐ \(3\sqrt{2a}-4\sqrt{2a}+5\sqrt{2a}\) ⓑ \(5\,\sqrt[4]{3ab}-3\,\sqrt[4]{3ab}-2\,\sqrt[4]{3ab}\)

ⓐ \(4\sqrt{2a}\) ⓑ 0

9

ⓐ \(\sqrt{11b}-5\sqrt{11b}+3\sqrt{11b}\) ⓑ \(8\,\sqrt[4]{11cd}+5\,\sqrt[4]{11cd}-9\,\sqrt[4]{11cd}\)

10

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

ⓐ \(\sqrt{3c}\) ⓑ \(\sqrt[3]{4pq}\)

11

ⓐ \(3\sqrt{5d}+8\sqrt{5d}-11\sqrt{5d}\) ⓑ \(11\,\sqrt[3]{2rs}-9\,\sqrt[3]{2rs}+3\,\sqrt[3]{2rs}\)

12

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

ⓐ \(-2\sqrt{3}\) ⓑ \(-2\sqrt[3]{5}\) ⓒ \(3\sqrt[4]{2}\)

13

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

14

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

ⓐ \(7\sqrt{3}\) ⓑ \(7\sqrt[3]{2}\) ⓒ \(3\sqrt[4]{5}\)

15

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

16

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

ⓐ \({a}^{2}\sqrt{2a}\) ⓑ 0

17

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

18

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

ⓐ \(2{c}^{3}\sqrt{5c}\) ⓑ \(14{r}^{2}\sqrt[4]{2{r}^{2}}\)

19

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

20

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

\(4y\sqrt{2}\)

21

\(3\,\sqrt{75{y}^{2}}+8y\,\sqrt{48}-\sqrt{300{y}^{2}}\)

Multiply Radical Expressions

In the following exercises, simplify.

22

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

ⓐ \(-18\sqrt{6}\) ⓑ \(-64\sqrt[3]{9}\)

23

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

24

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

ⓐ \(-30\sqrt{2}\) ⓑ \(6\sqrt[4]{2}\)

25

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

26

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

ⓐ \(72{z}^{2}\sqrt{3}\) ⓑ \(45{x}^{2}\sqrt[3]{2}\)

27

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

28

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

ⓐ \(-42{z}^{5}\sqrt{2z}\) ⓑ \(-8y\sqrt[4]{6y}\)

29

ⓐ \((4\sqrt{2{k}^{5}})(-3\sqrt{32{k}^{6}})\) ⓑ \((\text{-}\,\sqrt[4]{6{b}^{3}})(3\,\sqrt[4]{8{b}^{3}})\)

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

30

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

ⓐ \(14+5\sqrt{7}\) ⓑ \(4\sqrt[3]{6}+3\sqrt[3]{4}\)

31

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

32

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

ⓐ \(44-3\sqrt{11}\) ⓑ \(3\sqrt[4]{2}+\sqrt[4]{54}\)

33

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

34

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

\(60+2\sqrt{3}\)

35

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

36

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

ⓐ \(30+18\sqrt{2}\) ⓑ \(\sqrt[3]{{x}^{2}}-2\sqrt[3]{x}-3\)

37

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

38

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

ⓐ \(-55+13\sqrt{10}\)
ⓑ \(2\sqrt[3]{{x}^{2}}+8\sqrt[3]{x}+6\)

39

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

40

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

\(23+3\sqrt{30}\)

41

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

42

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

\(-439-2\sqrt{77}\)

43

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

44

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

ⓐ \(14+6\sqrt{5}\) ⓑ \(79-20\sqrt{3}\)

45

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

46

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

ⓐ \(87-18\sqrt{6}\)
ⓑ \(163+60\sqrt{7}\)

47

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

48

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

14

49

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

50

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

\(-227\)

51

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

52

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

\(19\)

53

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

54

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

\(\sqrt[3]{9{x}^{2}}-4\)

55

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

Mixed Practice

56

\(\frac{2}{3}\sqrt{27}+\frac{3}{4}\sqrt{48}\)

\(5\sqrt{3}\)

57

\(\sqrt{175{k}^{4}}-\sqrt{63{k}^{4}}\)

58

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

\(9\sqrt{2}\)

59

\(\sqrt[3]{24}+\sqrt[3]{\text{/}81}\)

60

\(\frac{1}{2}\,\sqrt[4]{80}-\frac{2}{3}\,\sqrt[4]{405}\)

\(\text{-}\sqrt[4]{5}\)

61

\(8\sqrt[4]{13}-4\sqrt[4]{13}-3\sqrt[4]{13}\)

62

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

\(10{c}^{2}\sqrt{3}-9{c}^{3}\sqrt{3}\)

63

\(\sqrt{80{a}^{5}}-\sqrt{45{a}^{5}}\)

64

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

\(2\sqrt{3}\)

65

\(21\,\sqrt[3]{9}-2\,\sqrt[3]{9}\)

66

\(8\,\sqrt[3]{64{q}^{6}}-3\,\sqrt[3]{125{q}^{6}}\)

\(17{q}^{2}\)

67

\(11\sqrt{11}-10\sqrt{11}\)

68

\(\sqrt{3}·\sqrt{21}\)

\(3\sqrt{7}\)

69

\((4\sqrt{6})(\text{-}\sqrt{18})\)

70

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

\(-42\sqrt[3]{9}\)

71

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

72

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

29

73

\((-4\sqrt{17})(-3\sqrt{17})\)

74

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

\(29-7\sqrt{17}\)

75

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

76

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

\(54-36\sqrt{2}\)

77

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

78

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

\(6+3\sqrt[3]{2}\)

79

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

Writing Exercises

80

Explain when a radical expression is in simplest form.

Answers will vary.

81

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.

82


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

Answers will vary.

83

Use the binomial square pattern to simplify \({(3+\sqrt{2})}^{2}.\) 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?

Glossary

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