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Prealgebra 2e

10.4 Divide Monomials

Prealgebra 2e10.4 Divide Monomials
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  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Simplify expressions using the Quotient Property of Exponents
  • Simplify expressions with zero exponents
  • Simplify expressions using the Quotient to a Power Property
  • Simplify expressions by applying several properties
  • Divide monomials
Be Prepared 10.9

Before you get started, take this readiness quiz.

Simplify: 824.824.
If you missed the problem, review Example 4.19.

Be Prepared 10.10

Simplify: (2m3)5.(2m3)5.
If you missed the problem, review Example 10.23.

Be Prepared 10.11

Simplify: 12x12y.12x12y.
If you missed the problem, review Example 4.23.

Simplify Expressions Using the Quotient Property of Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.

Summary of Exponent Properties for Multiplication

If a,ba,b are real numbers and m,nm,n are whole numbers, then

Product Propertyaman=am+nPower Property(am)n=amnProduct to a Power(ab)m=ambmProduct Propertyaman=am+nPower Property(am)n=amnProduct to a Power(ab)m=ambm

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions—which are also quotients.

Equivalent Fractions Property

If a,b,ca,b,c are whole numbers where b0,c0,b0,c0, then

ab=a·cb·canda·cb·c=abab=a·cb·canda·cb·c=ab

As before, we'll try to discover a property by looking at some examples.

Considerx5x2andx2x3What do they mean?xxxxxxxxxxxxUse the Equivalent Fractions Property.xxxxxxx1xx1xxxSimplify.x31xConsiderx5x2andx2x3What do they mean?xxxxxxxxxxxxUse the Equivalent Fractions Property.xxxxxxx1xx1xxxSimplify.x31x

Notice that in each case the bases were the same and we subtracted the exponents.

  • When the larger exponent was in the numerator, we were left with factors in the numerator and 11 in the denominator, which we simplified.
  • When the larger exponent was in the denominator, we were left with factors in the denominator, and 11 in the numerator, which could not be simplified.

We write:

x5x2x2x3x521x32x31xx5x2x2x3x521x32x31x

Quotient Property of Exponents

If aa is a real number, a0,a0, and m,nm,n are whole numbers, then

aman=amn,m>nandaman=1anm,n>maman=amn,m>nandaman=1anm,n>m

A couple of examples with numbers may help to verify this property.

3432=?3425253=?1532819=?3225125=?1519=915=153432=?3425253=?1532819=?3225125=?1519=915=15

When we work with numbers and the exponent is less than or equal to 3,3, we will apply the exponent. When the exponent is greater than 33, we leave the answer in exponential form.

Example 10.45

Simplify:

  1. x10x8x10x8
  2. 29222922
Try It 10.89

Simplify:

  1. x12x9x12x9
  2. 7147571475
Try It 10.90

Simplify:

  1. y23y17y23y17
  2. 8158781587

Example 10.46

Simplify:

  1. b10b15b10b15
  2. 33353335
Try It 10.91

Simplify:

  1. x8x15x8x15
  2. 1211122112111221
Try It 10.92

Simplify:

  1. m17m26m17m26
  2. 7871478714

Example 10.47

Simplify:

  1. a5a9a5a9
  2. x11x7x11x7
Try It 10.93

Simplify:

  1. b19b11b19b11
  2. z5z11z5z11
Try It 10.94

Simplify:

  1. p9p17p9p17
  2. w13w9w13w9

Simplify Expressions with Zero Exponents

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like amam.amam. From earlier work with fractions, we know that

22=11717=1−43−43=122=11717=1−43−43=1

In words, a number divided by itself is 1.1. So xx=1,xx=1, for any xx (x0x0), since any number divided by itself is 1.1.

The Quotient Property of Exponents shows us how to simplify amanaman when m>nm>n and when n<mn<m by subtracting exponents. What if m=nm=n?

Now we will simplify amamamam in two ways to lead us to the definition of the zero exponent.

Consider first 88,88, which we know is 1.1.

88=188=1
Write 8 as 2323. 2323=12323=1
Subtract exponents. 233=1233=1
Simplify. 20=120=1
.

We see amanaman simplifies to a a0a0 and to 11. So a0=1a0=1.

Zero Exponent

If aa is a non-zero number, then a0=1.a0=1.

Any nonzero number raised to the zero power is 1.1.

In this text, we assume any variable that we raise to the zero power is not zero.

Example 10.48

Simplify:

  1. 120120
  2. y0y0
Try It 10.95

Simplify:

  1. 170170
  2. m0m0
Try It 10.96

Simplify:

  1. k0k0
  2. 290290

Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.

What about raising an expression to the zero power? Let's look at (2x)0.(2x)0. We can use the product to a power rule to rewrite this expression.

(2x)0(2x)0
Use the Product to a Power Rule. 20x020x0
Use the Zero Exponent Property. 1111
Simplify. 1

This tells us that any non-zero expression raised to the zero power is one.

Example 10.49

Simplify: (7z)0.(7z)0.

Try It 10.97

Simplify: (−4y)0.(−4y)0.

Try It 10.98

Simplify: (23x)0.(23x)0.

Example 10.50

Simplify:

  1. (−3x2y)0(−3x2y)0
  2. −3x2y0−3x2y0
Try It 10.99

Simplify:

  1. (7x2y)0(7x2y)0
  2. 7x2y07x2y0
Try It 10.100

Simplify:

  1. −23x2y0−23x2y0
  2. (−23x2y)0(−23x2y)0

Simplify Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.

(xy)3(xy)3
This means xyxyxyxyxyxy
Multiply the fractions. xxxyyyxxxyyy
Write with exponents. x3y3x3y3

Notice that the exponent applies to both the numerator and the denominator.

We see that (xy)3(xy)3 is x3y3.x3y3.

We write:(xy)3x3y3We write:(xy)3x3y3

This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property of Exponents

If aa and bb are real numbers, b0,b0, and mm is a counting number, then

(ab)m=ambm(ab)m=ambm

To raise a fraction to a power, raise the numerator and denominator to that power.

An example with numbers may help you understand this property:

(23)3=?2333232323=?827827=827(23)3=?2333232323=?827827=827

Example 10.51

Simplify:

  1. (58)2(58)2
  2. (x3)4(x3)4
  3. (ym)3(ym)3
Try It 10.101

Simplify:

  1. (79)2(79)2
  2. (y8)3(y8)3
  3. (pq)6(pq)6
Try It 10.102

Simplify:

  1. (18)2(18)2
  2. (−5m)3(−5m)3
  3. (rs)4(rs)4

Simplify Expressions by Applying Several Properties

We'll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

Summary of Exponent Properties

If a,ba,b are real numbers and m,nm,n are whole numbers, then

Product Propertyaman=am+nPower Property(am)n=amnProduct to a Power Property(ab)m=ambmQuotient Propertyaman=amn,a0,m>naman=1anm,a0,n>mZero Exponent Definitiona0=1,a0Quotient to a Power Property(ab)m=ambm,b0Product Propertyaman=am+nPower Property(am)n=amnProduct to a Power Property(ab)m=ambmQuotient Propertyaman=amn,a0,m>naman=1anm,a0,n>mZero Exponent Definitiona0=1,a0Quotient to a Power Property(ab)m=ambm,b0

Example 10.52

Simplify: (x2)3x5.(x2)3x5.

Try It 10.103

Simplify: (a4)5a9.(a4)5a9.

Try It 10.104

Simplify: (b5)6b11.(b5)6b11.

Example 10.53

Simplify: m8(m2)4.m8(m2)4.

Try It 10.105

Simplify: k11(k3)3.k11(k3)3.

Try It 10.106

Simplify: d23(d4)6.d23(d4)6.

Example 10.54

Simplify: (x7x3)2.(x7x3)2.

Try It 10.107

Simplify: (f14f8)2.(f14f8)2.

Try It 10.108

Simplify: (b6b11)2.(b6b11)2.

Example 10.55

Simplify: (p2q5)3.(p2q5)3.

Try It 10.109

Simplify: (m3n8)5.(m3n8)5.

Try It 10.110

Simplify: (t10u7)2.(t10u7)2.

Example 10.56

Simplify: (2x33y)4.(2x33y)4.

Try It 10.111

Simplify: (5b9c3)2.(5b9c3)2.

Try It 10.112

Simplify: (4p47q5)3.(4p47q5)3.

Example 10.57

Simplify: (y2)3(y2)4(y5)4.(y2)3(y2)4(y5)4.

Try It 10.113

Simplify: (y4)4(y3)5(y7)6.(y4)4(y3)5(y7)6.

Try It 10.114

Simplify: (3x4)2(x3)4(x5)3.(3x4)2(x3)4(x5)3.

Divide Monomials

We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials.

Example 10.58

Find the quotient: 56x5÷7x2.56x5÷7x2.

Try It 10.115

Find the quotient: 63x8÷9x4.63x8÷9x4.

Try It 10.116

Find the quotient: 96y11÷6y8.96y11÷6y8.

When we divide monomials with more than one variable, we write one fraction for each variable.

Example 10.59

Find the quotient: 42x2y3−7xy5.42x2y3−7xy5.

Try It 10.117

Find the quotient: −84x8y37x10y2.−84x8y37x10y2.

Try It 10.118

Find the quotient: −72a4b5−8a9b5.−72a4b5−8a9b5.

Example 10.60

Find the quotient: 24a5b348ab4.24a5b348ab4.

Try It 10.119

Find the quotient: 16a7b624ab8.16a7b624ab8.

Try It 10.120

Find the quotient: 27p4q7−45p12q.27p4q7−45p12q.

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

Example 10.61

Find the quotient: 14x7y1221x11y6.14x7y1221x11y6.

Try It 10.121

Find the quotient: 28x5y1449x9y12.28x5y1449x9y12.

Try It 10.122

Find the quotient: 30m5n1148m10n14.30m5n1148m10n14.

In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we'll first find the product of two monomials in the numerator before we simplify the fraction.

Example 10.62

Find the quotient: (3x3y2)(10x2y3)6x4y5.(3x3y2)(10x2y3)6x4y5.

Try It 10.123

Find the quotient: (3x4y5)(8x2y5)12x5y8.(3x4y5)(8x2y5)12x5y8.

Try It 10.124

Find the quotient: (−6a6b9)(−8a5b8)−12a10b12.(−6a6b9)(−8a5b8)−12a10b12.

Section 10.4 Exercises

Practice Makes Perfect

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

219.

48424842

220.

3123431234

221.

x12x3x12x3

222.

u9u3u9u3

223.

r5rr5r

224.

y4yy4y

225.

y4y20y4y20

226.

x10x30x10x30

227.

10310151031015

228.

r2r8r2r8

229.

aa9aa9

230.

225225

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

231.

5050

232.

100100

233.

a0a0

234.

x0x0

235.

7070

236.

4040

237.
  1. (10p)0(10p)0
  2. 10p010p0
238.
  1. (3a)0(3a)0
  2. 3a03a0
239.
  1. (−27x5y)0(−27x5y)0
  2. −27x5y0−27x5y0
240.
  1. (−92y8z)0(−92y8z)0
  2. −92y8z0−92y8z0
241.
  1. 150150
  2. 151151
242.
  1. 6060
  2. 6161
243.

2·x0+5·y02·x0+5·y0

244.

8·m04·n08·m04·n0

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

245.

(32)5(32)5

246.

(45)3(45)3

247.

(m6)3(m6)3

248.

(p2)5(p2)5

249.

(xy)10(xy)10

250.

(ab)8(ab)8

251.

(a3b)2(a3b)2

252.

(2xy)4(2xy)4

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

253.

(x2)4x5(x2)4x5

254.

(y4)3y7(y4)3y7

255.

(u3)4u10(u3)4u10

256.

(y2)5y6(y2)5y6

257.

y8(y5)2y8(y5)2

258.

p11(p5)3p11(p5)3

259.

r5r4·rr5r4·r

260.

a3·a4a7a3·a4a7

261.

(x2x8)3(x2x8)3

262.

(uu10)2(uu10)2

263.

(a4·a6a3)2(a4·a6a3)2

264.

(x3·x8x4)3(x3·x8x4)3

265.

(y3)5(y4)3(y3)5(y4)3

266.

(z6)2(z2)4(z6)2(z2)4

267.

(x3)6(x4)7(x3)6(x4)7

268.

(x4)8(x5)7(x4)8(x5)7

269.

(2r35s)4(2r35s)4

270.

(3m24n)3(3m24n)3

271.

(3y2·y5y15·y8)0(3y2·y5y15·y8)0

272.

(15z4·z90.3z2)0(15z4·z90.3z2)0

273.

(r2)5(r4)2(r3)7(r2)5(r4)2(r3)7

274.

(p4)2(p3)5(p2)9(p4)2(p3)5(p2)9

275.

(3x4)3(2x3)2(6x5)2(3x4)3(2x3)2(6x5)2

276.

(−2y3)4(3y4)2(−6y3)2(−2y3)4(3y4)2(−6y3)2

Divide Monomials

In the following exercises, divide the monomials.

277.

48b8÷6b248b8÷6b2

278.

42a14÷6a242a14÷6a2

279.

36x3÷(−2x9)36x3÷(−2x9)

280.

20u8÷(−4u6)20u8÷(−4u6)

281.

18x39x218x39x2

282.

36y94y736y94y7

283.

−35x7−42x13−35x7−42x13

284.

18x5−27x918x5−27x9

285.

18r5s3r3s918r5s3r3s9

286.

24p7q6p2q524p7q6p2q5

287.

8mn1064mn48mn1064mn4

288.

10a4b50a2b610a4b50a2b6

289.

−12x4y915x6y3−12x4y915x6y3

290.

48x11y9z336x6y8z548x11y9z336x6y8z5

291.

64x5y9z748x7y12z664x5y9z748x7y12z6

292.

(10u2v)(4u3v6)5u9v2(10u2v)(4u3v6)5u9v2

293.

(6m2n)(5m4n3)3m10n2(6m2n)(5m4n3)3m10n2

294.

(6a4b3)(4ab5)(12a8b)(a3b)(6a4b3)(4ab5)(12a8b)(a3b)

295.

(4u5v4)(15u8v)(12u3v)(u6v)(4u5v4)(15u8v)(12u3v)(u6v)

Mixed Practice

296.
  1. 24a5+2a524a5+2a5
  2. 24a52a524a52a5
  3. 24a52a524a52a5
  4. 24a5÷2a524a5÷2a5
297.
  1. 15n10+3n1015n10+3n10
  2. 15n103n1015n103n10
  3. 15n103n1015n103n10
  4. 15n10÷3n1015n10÷3n10
298.
  1. p4p6p4p6
  2. (p4)6(p4)6
299.
  1. q5q3q5q3
  2. (q5)3(q5)3
300.
  1. y3yy3y
  2. yy3yy3
301.
  1. z6z5z6z5
  2. z5z6z5z6
302.

(8x5)(9x)÷6x3(8x5)(9x)÷6x3

303.

(4y)(12y7)÷8y2(4y)(12y7)÷8y2

304.

27a73a3+54a99a527a73a3+54a99a5

305.

32c114c5+42c96c332c114c5+42c96c3

306.

32y58y260y105y732y58y260y105y7

307.

48x66x435x97x748x66x435x97x7

308.

63r6s39r4s272r2s26s63r6s39r4s272r2s26s

309.

56y4z57y3z345y2z25y56y4z57y3z345y2z25y

Everyday Math

310.

Memory One megabyte is approximately 106106 bytes. One gigabyte is approximately 109109 bytes. How many megabytes are in one gigabyte?

311.

Memory One megabyte is approximately 106106 bytes. One terabyte is approximately 10121012 bytes. How many megabytes are in one terabyte?

Writing Exercises

312.

Vic thinks the quotient x20x4x20x4 simplifies to x5.x5. What is wrong with his reasoning?

313.

Mai simplifies the quotient y3yy3y by writing y3y=3.y3y=3. What is wrong with her reasoning?

314.

When Dimple simplified 3030 and (−3)0(−3)0 she got the same answer. Explain how using the Order of Operations correctly gives different answers.

315.

Roxie thinks n0n0 simplifies to 0.0. What would you say to convince Roxie she is wrong?

Self Check

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?

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© Mar 3, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.