8.3: Simplify Radical Expressions

Learning Objectives

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

• Use the Product Property to simplify radical expressions
• Use the Quotient Property to simplify radical expressions

BE PREPARED 8.4

Before you get started, take this readiness quiz.

Simplify: x9x4.x9x4.
If you missed this problem, review Example 5.13.

BE PREPARED 8.5

Simplify: y3y11.y3y11.
If you missed this problem, review Example 5.13.

BE PREPARED 8.6

Simplify: (n2)6.(n2)6.
If you missed this problem, review Example 5.17.

Use the Product Property to Simplify Radical Expressions

We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator.

A radical expression, a−−√n,an, is considered simplified if it has no factors of mn.mn. So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.

SIMPLIFIED RADICAL EXPRESSION

For real numbers a and m, and n≥2,n≥2,

a−−√nis considered simplified ifahas no factors ofmnanis considered simplified ifahas no factors ofmn

For example, 5–√5 is considered simplified because there are no perfect square factors in 5. But 12−−√12 is not simplified because 12 has a perfect square factor of 4.

Similarly, 4–√343 is simplified because there are no perfect cube factors in 4. But 24−−√3243 is not simplified because 24 has a perfect cube factor of 8.

To simplify radical expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know that (ab)n=anbn.(ab)n=anbn. The corresponding of Product Property of Roots says that ab−−√n=a−−√n⋅b√n.abn=an·bn.

PRODUCT PROPERTY OF N^TH ROOTS

If a−−√nan and b√nbn are real numbers, and n≥2n≥2 is an integer, then

ab−−√n=a−−√n⋅b√nanda−−√n⋅b√n=ab−−√nabn=an·bnandan·bn=abn

We use the Product Property of Roots to remove all perfect square factors from a square root.

EXAMPLE 8.13

Simplify Square Roots Using the Product Property of Roots

Simplify: 98−−√.98.

Answer

TRY IT 8.25

Simplify: 48−−√.48.

TRY IT 8.26

Simplify: 45−−√.45.

Notice in the previous example that the simplified form of 98−−√98 is 72–√,72, which is the product of an integer and a square root. We always write the integer in front of the square root.

Be careful to write your integer so that it is not confused with the index. The expression 72–√72 is very different from 2–√7.27.

HOW TO

Simplify a radical expression using the Product Property.

1. Step 1. Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
2. Step 2. Use the product rule to rewrite the radical as the product of two radicals.
3. Step 3. Simplify the root of the perfect power.

We will apply this method in the next example. It may be helpful to have a table of perfect squares, cubes, and fourth powers.

EXAMPLE 8.14

Simplify: ⓐ 500−−−√500 ⓑ 16−−√3163 ⓒ 243−−−√4.2434.

Answer

TRY IT 8.27

Simplify: ⓐ 288−−−√288 ⓑ 81−−√3813 ⓒ 64−−√4.644.

TRY IT 8.28

Simplify: ⓐ 432−−−√432 ⓑ 625−−−√36253 ⓒ 729−−−√4.7294.

The next example is much like the previous examples, but with variables. Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

EXAMPLE 8.15

Simplify: ⓐ x3−−√x3 ⓑ x4−−√3x43 ⓒ x7−−√4.x74.

Answer

TRY IT 8.29

Simplify: ⓐ b5−−√b5 ⓑ y6−−√4y64 ⓒ z5−−√3z53

TRY IT 8.30

Simplify: ⓐ p9−−√p9 ⓑ y8−−√5y85 ⓒ q13−−−√6q136

We follow the same procedure when there is a coefficient in the radicand. In the next example, both the constant and the variable have perfect square factors.

EXAMPLE 8.16

Simplify: ⓐ 72n7−−−−√72n7 ⓑ 24x7−−−−√324x73 ⓒ 80y14−−−−−√4.80y144.

Answer

TRY IT 8.31

Simplify: ⓐ 32y5−−−−√32y5 ⓑ 54p10−−−−√354p103 ⓒ 64q10−−−−−√4.64q104.

TRY IT 8.32

Simplify: ⓐ 75a9−−−−√75a9 ⓑ 128m11−−−−−−√3128m113 ⓒ 162n7−−−−−√4.162n74.

In the next example, we continue to use the same methods even though there are more than one variable under the radical.

EXAMPLE 8.17

Simplify: ⓐ 63u3v5−−−−−−√63u3v5 ⓑ 40x4y5−−−−−−√340x4y53 ⓒ 48x4y7−−−−−−√4.48x4y74.

Answer

TRY IT 8.33

Simplify: ⓐ 98a7b5−−−−−√98a7b5 ⓑ 56x5y4−−−−−−√356x5y43 ⓒ 32x5y8−−−−−−√4.32x5y84.

TRY IT 8.34

Simplify: ⓐ 180m9n11−−−−−−−−√180m9n11 ⓑ 72x6y5−−−−−−√372x6y53 ⓒ 80x7y4−−−−−−√4.80x7y44.

EXAMPLE 8.18

Simplify: ⓐ −27−−−−√3−273 ⓑ −16−−−−√4.−164.

Answer

TRY IT 8.35

Simplify: ⓐ −64−−−−√3−643 ⓑ −81−−−−√4.−814.

TRY IT 8.36

Simplify: ⓐ −625−−−−√3−6253 ⓑ −324−−−−√4.−3244.

We have seen how to use the order of operations to simplify some expressions with radicals. In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer since one term contains a radical and the other does not. The next example also includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

EXAMPLE 8.19

Simplify: ⓐ 3+32−−√3+32 ⓑ 4−48√2.4−482.

Answer

TRY IT 8.37

Simplify: ⓐ 5+75−−√5+75 ⓑ 10−75√510−755

TRY IT 8.38

Simplify: ⓐ 2+98−−√2+98 ⓑ 6−45√36−453

Use the Quotient Property to Simplify Radical Expressions

Whenever you have to simplify a radical expression, the first step you should take is to determine whether the radicand is a perfect power of the index. If not, check the numerator and denominator for any common factors, and remove them. You may find a fraction in which both the numerator and the denominator are perfect powers of the index.

EXAMPLE 8.20

Simplify: ⓐ 4580−−√4580 ⓑ 1654−−√316543 ⓒ 580−−√4.5804.

Answer

TRY IT 8.39

Simplify: ⓐ 7548−−√7548 ⓑ 54250−−−√3542503 ⓒ 32162−−−√4.321624.

TRY IT 8.40

Simplify: ⓐ 98162−−−√98162 ⓑ 24375−−−√3243753 ⓒ 4324−−−√4.43244.

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents,

aman=am−n,a≠0aman=am−n,a≠0

EXAMPLE 8.21

Simplify: ⓐ m6m4−−−√m6m4 ⓑ a8a5−−√3a8a53 ⓒ a10a2−−−√4.a10a24.

Answer

TRY IT 8.41

Simplify: ⓐ a8a6−−√a8a6 ⓑ x7x3−−√4x7x34 ⓒ y17y5−−−√4.y17y54.

TRY IT 8.42

Simplify: ⓐ x14x10−−−√x14x10 ⓑ m13m7−−−√3m13m73 ⓒ n12n2−−−√5.n12n25.

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

(ab)m=ambm,b≠0(ab)m=ambm,b≠0

We can use a similar property to simplify a root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect power of the index, we simplify the numerator and denominator separately.

QUOTIENT PROPERTY OF RADICAL EXPRESSIONS

If a−−√nan and b√nbn are real numbers,b≠0,b≠0, and for any integer n≥2n≥2 then,

ab−−√n=a−−√nb√nanda−−√nb√n=ab−−√nabn=anbnandanbn=abn

EXAMPLE 8.22

How to Simplify the Quotient of Radical Expressions

Simplify: 27m3196−−−−√.27m3196.

Answer

TRY IT 8.43

Simplify: 24p349−−−√.24p349.

TRY IT 8.44

Simplify: 48x5100−−−−√.48x5100.

HOW TO

Simplify a square root using the Quotient Property.

1. Step 1. Simplify the fraction in the radicand, if possible.
2. Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
3. Step 3. Simplify the radicals in the numerator and the denominator.

EXAMPLE 8.23

Simplify: ⓐ 45x5y4−−−−√45x5y4 ⓑ 24x7y3−−−−√324x7y33 ⓒ 48x10y8−−−−√4.48x10y84.

Answer

TRY IT 8.45

Simplify: ⓐ 80m3n6−−−−√80m3n6 ⓑ 108c10d6−−−−√3108c10d63 ⓒ 80x10y4−−−−√4.80x10y44.

TRY IT 8.46

Simplify: ⓐ 54u7v8−−−−√54u7v8 ⓑ 40r3s6−−−√340r3s63 ⓒ 162m14n12−−−−−√4.162m14n124.

Be sure to simplify the fraction in the radicand first, if possible.

EXAMPLE 8.24

Simplify: ⓐ 18p5q732pq2−−−−−√18p5q732pq2 ⓑ 16x5y754x2y2−−−−−√316x5y754x2y23 ⓒ 5a8b680a3b2−−−−−√4.5a8b680a3b24.

Answer

TRY IT 8.47

Simplify: ⓐ 50x5y372x4y−−−−−√50x5y372x4y ⓑ 16x5y754x2y2−−−−−√316x5y754x2y23 ⓒ 5a8b680a3b2−−−−−√4.5a8b680a3b24.

TRY IT 8.48

Simplify: ⓐ 48m7n2100m5n8−−−−−−√48m7n2100m5n8 ⓑ 54x7y5250x2y2−−−−−√354x7y5250x2y23 ⓒ 32a9b7162a3b3−−−−−√4.32a9b7162a3b34.

In the next example, there is nothing to simplify in the denominators. Since the index on the radicals is the same, we can use the Quotient Property again, to combine them into one radical. We will then look to see if we can simplify the expression.

EXAMPLE 8.25

Simplify: ⓐ 48a7√3a√48a73a ⓑ −108√32√3−108323 ⓒ 96x7√43x2√4.96x743x24.

Answer

TRY IT 8.49

Simplify: ⓐ 98z5√2z√98z52z ⓑ −500√32√3−500323 ⓒ 486m11√43m5√4.486m1143m54.

TRY IT 8.50

Simplify: ⓐ 128m9√2m√128m92m ⓑ −192√33√3−192333 ⓒ 324n7√42n3√4.324n742n34.

MEDIA

Access these online resources for additional instruction and practice with simplifying radical expressions.

• Simplifying Square Root and Cube Root with Variables
• Express a Radical in Simplified Form-Square and Cube Roots with Variables and Exponents
• Simplifying Cube Roots

Section 8.2 Exercises

Practice Makes Perfect

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

55.

27−−√27

56.

80−−√80

57.

125−−−√125

58.

96−−√96

59.

147−−−√147

60.

450−−−√450

61.

800−−−√800

62.

675−−−√675

63.

ⓐ 32−−√4324 ⓑ 64−−√5645

64.

ⓐ 625−−−√36253 ⓑ 128−−−√61286

65.

ⓐ 64−−√4644 ⓑ 256−−−√32563

66.

ⓐ 3125−−−−√431254 ⓑ 81−−√3813

In the following exercises, simplify using absolute value signs as needed.

67.

ⓐ y11−−−√y11 ⓑ r5−−√3r53 ⓒ s10−−−√4s104

68.

ⓐ m13−−−√m13 ⓑ u7−−√5u75 ⓒ v11−−−√6v116

69.

ⓐ n21−−−√n21 ⓑ q8−−√3q83 ⓒ n10−−−√8n108

70.

ⓐ r25−−−√r25 ⓑ p8−−√5p85 ⓒ m5−−−√4m54

71.

ⓐ 125r13−−−−−√125r13 ⓑ 108x5−−−−−√3108x53 ⓒ 48y6−−−−√448y64

72.

ⓐ 80s15−−−−√80s15 ⓑ 96a7−−−−√596a75 ⓒ 128b7−−−−−√6128b76

73.

ⓐ 242m23−−−−−−√242m23 ⓑ 405m10−−−−−−√4405m104 ⓒ 160n8−−−−−√5160n85

74.

ⓐ 175n13−−−−−−√175n13 ⓑ 512p5−−−−−√5512p55 ⓒ 324q7−−−−−√4324q74

75.

ⓐ 147m7n11−−−−−−−−√147m7n11 ⓑ 48x6y7−−−−−−√348x6y73 ⓒ 32x5y4−−−−−−√432x5y44

76.

ⓐ 96r3s3−−−−−√96r3s3 ⓑ 80x7y6−−−−−−√380x7y63 ⓒ 80x8y9−−−−−−√480x8y94

77.

ⓐ 192q3r7−−−−−−√192q3r7 ⓑ 54m9n10−−−−−−−√354m9n103 ⓒ 81a9b8−−−−−√481a9b84

78.

ⓐ 150m9n3−−−−−−−√150m9n3 ⓑ 81p7q8−−−−−−√381p7q83 ⓒ 162c11d12−−−−−−−−√4162c11d124

79.

ⓐ −864−−−−√3−8643 ⓑ −256−−−−√4−2564

80.

ⓐ −486−−−−√5−4865 ⓑ −64−−−−√6−646

81.

ⓐ −32−−−−√5−325 ⓑ −1−−−√8−18

82.

ⓐ −8−−−√3−83 ⓑ −16−−−−√4−164

83.

ⓐ 5+12−−√5+12 ⓑ 10−24√210−242

84.

ⓐ 8+96−−√8+96 ⓑ 8−80√48−804

85.

ⓐ 1+45−−√1+45 ⓑ 3+90√33+903

86.

ⓐ 3+125−−−√3+125 ⓑ 15+75√515+755

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

87.

ⓐ 4580−−√4580 ⓑ 827−−√38273 ⓒ 181−−√41814

88.

ⓐ 7298−−√7298 ⓑ 2481−−√324813 ⓒ 696−−√46964

89.

ⓐ 10036−−−√10036 ⓑ 81375−−−√3813753 ⓒ 1256−−−√412564

90.

ⓐ 12116−−−√12116 ⓑ 16250−−−√3162503 ⓒ 32162−−−√4321624

91.

ⓐ x10x6−−−√x10x6 ⓑ p11p2−−−√3p11p23 ⓒ q17q13−−−√4q17q134

92.

ⓐ p20p10−−−√p20p10 ⓑ d12d7−−−√5d12d75 ⓒ m12m4−−−√8m12m48

93.

ⓐ y4y8−−√y4y8 ⓑ u21u11−−−√5u21u115 ⓒ v30v12−−−√6v30v126

94.

ⓐ q8q14−−−√q8q14 ⓑ r14r5−−−√3r14r53 ⓒ c21c9−−−√4c21c94

95.

96x7121−−−−√96x7121

96.

108y449−−−−√108y449

97.

300m564−−−−−√300m564

98.

125n7169−−−−√125n7169

99.

98r5100−−−√98r5100

100.

180s10144−−−−−√180s10144

101.

28q6225−−−−√28q6225

102.

150r3256−−−−√150r3256

103.

ⓐ 75r9s8−−−√75r9s8 ⓑ 54a8b3−−−−√354a8b33 ⓒ 64c5d4−−−√464c5d44

104.

ⓐ 72x5y6−−−−√72x5y6 ⓑ 96r11s5−−−−√596r11s55 ⓒ 128u7v12−−−−√6128u7v126

105.

ⓐ 28p7q2−−−√28p7q2 ⓑ 81s8t3−−−√381s8t33 ⓒ 64p15q12−−−−√464p15q124

106.

ⓐ 45r3s10−−−√45r3s10 ⓑ 625u10v3−−−−−√3625u10v33 ⓒ 729c21d8−−−−√4729c21d84

107.

ⓐ 32x5y318x3y−−−−−√32x5y318x3y ⓑ 5x6y940x5y3−−−−−√35x6y940x5y33 ⓒ 5a8b680a3b2−−−−−√45a8b680a3b24

108.

ⓐ 75r6s848rs4−−−−−√75r6s848rs4 ⓑ 24x8y481x2y−−−−−√324x8y481x2y3 ⓒ 32m9n2162mn2−−−−−√432m9n2162mn24

109.

ⓐ 27p2q108p4q3−−−−−√27p2q108p4q3 ⓑ 16c5d7250c2d2−−−−−√316c5d7250c2d23 ⓒ 2m9n7128m3n−−−−−√62m9n7128m3n6

110.

ⓐ 50r5s2128r2s6−−−−−√50r5s2128r2s6 ⓑ 24m9n7375m4n−−−−−√324m9n7375m4n3 ⓒ 81m2n8256m1n2−−−−−−√481m2n8256m1n24

111.

ⓐ 45p9√5q2√45p95q2 ⓑ 64√42√464424 ⓒ 128x8√52x2√5128x852x25

112.

ⓐ 80q5√5q√80q55q ⓑ −625√35√3−625353 ⓒ 80m7√45m√480m745m4

113.

ⓐ 50m7√2m√50m72m ⓑ 12502−−−−√3125023 ⓒ 486y92y3−−−−√4486y92y34

114.

ⓐ 72n11√2n√72n112n ⓑ 1626−−−√316263 ⓒ 160r105r3−−−−−√4160r105r34

Writing Exercises

115.

Explain why x4−−√=x2.x4=x2. Then explain why x16−−−√=x8.x16=x8.

116.

Explain why 7+9–√7+9 is not equal to 7+9−−−−√.7+9.

117.

Explain how you know that x10−−−√5=x2.x105=x2.

118.

Explain why −64−−−−√4−644 is not a real number but −64−−−−√3−643 is.

Self Check

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

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

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