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Intermediate Algebra 2e

8.2 Simplify Radical Expressions

Intermediate Algebra 2e8.2 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, an,an, is considered simplified if it has no factors of 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 n2,n2,

anis considered simplified ifahas no factors ofmnanis considered simplified ifahas no factors ofmn

For example, 55 is considered simplified because there are no perfect square factors in 5. But 1212 is not simplified because 12 has a perfect square factor of 4.

Similarly, 4343 is simplified because there are no perfect cube factors in 4. But 243243 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 abn=an·bn.abn=an·bn.

Product Property of nth Roots

If anan and bnbn are real numbers, and n2n2 is an integer, then


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.

Try It 8.25

Simplify: 48.48.

Try It 8.26

Simplify: 45.45.

Notice in the previous example that the simplified form of 9898 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 7272 is very different from 27.27.

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: 500500 163163 2434.2434.

Try It 8.27

Simplify: 288288 813813 644.644.

Try It 8.28

Simplify: 432432 62536253 7294.7294.

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: x3x3 x43x43 x74.x74.

Try It 8.29

Simplify: b5b5 y64y64 z53z53

Try It 8.30

Simplify: p9p9 y85y85 q136q136

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: 72n772n7 24x7324x73 80y144.80y144.

Try It 8.31

Simplify: 32y532y5 54p10354p103 64q104.64q104.

Try It 8.32

Simplify: 75a975a9 128m113128m113 162n74.162n74.

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: 63u3v563u3v5 40x4y5340x4y53 48x4y74.48x4y74.

Try It 8.33

Simplify: 98a7b598a7b5 56x5y4356x5y43 32x5y84.32x5y84.

Try It 8.34

Simplify: 180m9n11180m9n11 72x6y5372x6y53 80x7y44.80x7y44.

Example 8.18

Simplify: −273−273 −164.−164.

Try It 8.35

Simplify: −643−643 −814.−814.

Try It 8.36

Simplify: −6253−6253 −3244.−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+323+32 4482.4482.

Try It 8.37

Simplify: 5+755+75 1075510755

Try It 8.38

Simplify: 2+982+98 64536453

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: 45804580 1654316543 5804.5804.

Try It 8.39

Simplify: 75487548 542503542503 321624.321624.

Try It 8.40

Simplify: 9816298162 243753243753 43244.43244.

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,


Example 8.21

Simplify: m6m4m6m4 a8a53a8a53 a10a24.a10a24.

Try It 8.41

Simplify: a8a6a8a6 x7x34x7x34 y17y54.y17y54.

Try It 8.42

Simplify: x14x10x14x10 m13m73m13m73 n12n25.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.


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 anan and bnbn are real numbers,b0,b0, and for any integer n2n2 then,


Example 8.22

How to Simplify the Quotient of Radical Expressions

Simplify: 27m3196.27m3196.

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: 45x5y445x5y4 24x7y3324x7y33 48x10y84.48x10y84.

Try It 8.45

Simplify: 80m3n680m3n6 108c10d63108c10d63 80x10y44.80x10y44.

Try It 8.46

Simplify: 54u7v854u7v8 40r3s6340r3s63 162m14n124.162m14n124.

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

Example 8.24

Simplify: 18p5q732pq218p5q732pq2 16x5y754x2y2316x5y754x2y23 5a8b680a3b24.5a8b680a3b24.

Try It 8.47

Simplify: 50x5y372x4y50x5y372x4y 16x5y754x2y2316x5y754x2y23 5a8b680a3b24.5a8b680a3b24.

Try It 8.48

Simplify: 48m7n2100m5n848m7n2100m5n8 54x7y5250x2y2354x7y5250x2y23 32a9b7162a3b34.32a9b7162a3b34.

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: 48a73a48a73a −108323−108323 96x743x24.96x743x24.

Try It 8.49

Simplify: 98z52z98z52z −500323−500323 486m1143m54.486m1143m54.

Try It 8.50

Simplify: 128m92m128m92m −192333−192333 324n742n34.324n742n34.


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

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.


27 27


80 80


125 125


96 96


147 147


450 450


800 800


675 675


324324 645645


62536253 12861286


644644 25632563


3125431254 813813

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


y11y11 r53r53 s104s104


m13m13 u75u75 v116v116


n21n21 q83q83 n108n108


r25r25 p85p85 m54m54


125r13125r13 108x53108x53 48y6448y64


80s1580s15 96a7596a75 128b76128b76


242m23242m23 405m104405m104 160n85160n85


175n13175n13 512p55512p55 324q74324q74


147m7n11147m7n11 48x6y7348x6y73 32x5y4432x5y44


96r3s396r3s3 80x7y6380x7y63 80x8y9480x8y94


192q3r7192q3r7 54m9n10354m9n103 81a9b8481a9b84


150m9n3150m9n3 81p7q8381p7q83 162c11d124162c11d124


−8643−8643 −2564−2564


−4865−4865 −646−646


−325−325 −18−18


−83−83 −164−164


5+125+12 1024210242


8+968+96 88048804


1+451+45 3+9033+903


3+1253+125 15+75515+755

Use the Quotient Property to Simplify Radical Expressions

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


45804580 82738273 18141814


72987298 2481324813 69646964


1003610036 813753813753 1256412564


1211612116 162503162503 321624321624


x10x6x10x6 p11p23p11p23 q17q134q17q134


p20p10p20p10 d12d75d12d75 m12m48m12m48


y4y8y4y8 u21u115u21u115 v30v126v30v126


q8q14q8q14 r14r53r14r53 c21c94c21c94


96 x 7 121 96 x 7 121


108 y 4 49 108 y 4 49


300 m 5 64 300 m 5 64


125 n 7 169 125 n 7 169


98 r 5 100 98 r 5 100


180 s 10 144 180 s 10 144


28 q 6 225 28 q 6 225


150 r 3 256 150 r 3 256


75r9s875r9s8 54a8b3354a8b33 64c5d4464c5d44


72x5y672x5y6 96r11s5596r11s55 128u7v126128u7v126


28p7q228p7q2 81s8t3381s8t33 64p15q12464p15q124


45r3s1045r3s10 625u10v33625u10v33 729c21d84729c21d84


32x5y318x3y32x5y318x3y 5x6y940x5y335x6y940x5y33 5a8b680a3b245a8b680a3b24


75r6s848rs475r6s848rs4 24x8y481x2y324x8y481x2y3 32m9n2162mn2432m9n2162mn24


27p2q108p4q327p2q108p4q3 16c5d7250c2d2316c5d7250c2d23 2m9n7128m3n62m9n7128m3n6


50r5s2128r2s650r5s2128r2s6 24m9n7375m4n324m9n7375m4n3 81m2n8256m1n2481m2n8256m1n24


45p95q245p95q2 6442464424 128x852x25128x852x25


80q55q80q55q −625353−625353 80m745m480m745m4


50m72m50m72m 125023125023 486y92y34486y92y34


72n112n72n112n 1626316263 160r105r34160r105r34

Writing Exercises


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


Explain why 7+97+9 is not equal to 7+9.7+9.


Explain how you know that x105=x2.x105=x2.


Explain why −644−644 is not a real number but −643−643 is.

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 “use the product property to simplify radical expressions” and “use the quotient property to simplify radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

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

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