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Elementary Algebra 2e

6.5 Divide Monomials

Elementary Algebra 2e6.5 Divide Monomials
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 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
  13. Index

Learning Objectives

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

  • Simplify expressions using the Quotient Property for 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 6.9

Before you get started, take this readiness quiz.

Simplify: 824.824.
If you missed this problem, review Example 1.65.

Be Prepared 6.10

Simplify: (2m3)5.(2m3)5.
If you missed this problem, review Example 6.23.

Be Prepared 6.11

Simplify: 12x12y.12x12y.
If you missed this problem, review Example 1.67.

Simplify Expressions Using the Quotient Property for Exponents

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

Summary of Exponent Properties for Multiplication

If aandbaandb are real numbers, and mandnmandn are whole numbers, then

Product Property am·an=am+nam·an=am+n
Power Property (am)n=am·n(am)n=am·n
Product to a Power (ab)m=ambm(ab)m=ambm

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

Equivalent Fractions Property

If a,b,andca,b,andc are whole numbers where b0,c0b0,c0,

thenab=a·cb·canda·cb·c=abthenab=a·cb·canda·cb·c=ab

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

Consider x5x2x5x2 and x2x3x2x3
What do they mean? x·x·x·x·xx·xx·x·x·x·xx·x x·xx·x·xx·xx·x·x
Use the Equivalent Fractions Property. x·x·x·x·xx·xx·x·x·x·xx·x x·x·1x·x·xx·x·1x·x·x
Simplify. x3x3 1x1x

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

When the larger exponent was in the numerator, we were left with factors in the numerator.

When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.

We write:

x5x2x2x3x521x32x31xx5x2x2x3x521x32x31x

This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If aa is a real number, a0a0, and mandnmandn 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

Example 6.59

Simplify: x9x7x9x7 31032.31032.

Try It 6.117

Simplify: x15x10x15x10 61465.61465.

Try It 6.118

Simplify: y43y37y43y37 1015107.1015107.

Example 6.60

Simplify: b8b12b8b12 7375.7375.

Try It 6.119

Simplify: x18x22x18x22 12151230.12151230.

Try It 6.120

Simplify: m7m15m7m15 98919.98919.

Notice the difference in the two previous examples:

  • If we start with more factors in the numerator, we will end up with factors in the numerator.
  • If we start with more factors in the denominator, we will end up with factors in the denominator.


The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.

Example 6.61

Simplify: a5a9a5a9 x11x7.x11x7.

Try It 6.121

Simplify: b19b11b19b11 z5z11.z5z11.

Try It 6.122

Simplify: p9p17p9p17 w13w9.w13w9.

Simplify Expressions with an Exponent of Zero

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

22=11717=1−43−43=122=11717=1−43−43=1

In words, a number divided by itself is 1. So, xx=1xx=1, for any x(x0)x(x0), since any number divided by itself is 1.

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

Consider 8888, which we know is 1.

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

Now we will simplify amamamam in two ways to lead us to the definition of the zero exponent. In general, for a0a0:

This figure is divided into two columns. At the top of the figure, the left and right columns both contain a to the m power divided by a to the m power. In the next row, the left column contains a to the m minus m power. The right column contains the fraction m factors of a divided by m factors of a, represented in the numerator and denominator by a times a followed by an ellipsis. All the as in the numerator and denominator are canceled out. In the bottom row, the left column contains a to the zero power. The right column contains 1.

We see amamamam simplifies to a0a0 and to 1. So a0=1a0=1.

Zero Exponent

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

Any nonzero number raised to the zero power is 1.

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

Example 6.62

Simplify: 9090 n0.n0.

Try It 6.123

Simplify: 150150 m0.m0.

Try It 6.124

Simplify: k0k0 290.290.

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. 1·11·1
Simplify. 11
Table 6.1

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

Example 6.63

Simplify: (5b)0(5b)0 (−4a2b)0.(−4a2b)0.

Try It 6.125

Simplify: (11z)0(11z)0 (−11pq3)0.(−11pq3)0.

Try It 6.126

Simplify: (−6d)0(−6d)0 (−8m2n3)0.(−8m2n3)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: xy·xy·xyxy·xy·xy
Multiply the fractions. x·x·xy·y·yx·x·xy·y·y
Write with exponents. x3y3x3y3

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

We write: (xy)3(xy)3
x3y3x3y3

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

Quotient to a Power Property for Exponents

If aa and bb are real numbers, b0b0, 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=233323·23·23=827827=827(23)3=233323·23·23=827827=827

Example 6.64

Simplify: (37)2(37)2 (b3)4(b3)4 (kj)3.(kj)3.

Try It 6.127

Simplify: (58)2(58)2 (p10)4(p10)4 (mn)7.(mn)7.

Try It 6.128

Simplify: (13)3(13)3 (−2q)3(−2q)3 (wx)4.(wx)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 aandbaandb are real numbers, and mandnmandn are whole numbers, then

Product Property am·an=am+nam·an=am+n
Power Property (am)n=am·n(am)n=am·n
Product to a Power (ab)m=ambm(ab)m=ambm
Quotient Property aman=amn,a0,m>naman=1anm,a0,n>maman=amn,a0,m>naman=1anm,a0,n>m
Zero Exponent Definition ao=1,a0ao=1,a0
Quotient to a Power Property (ab)m=ambm,b0(ab)m=ambm,b0

Example 6.65

Simplify: (y4)2y6.(y4)2y6.

Try It 6.129

Simplify: (m5)4m7.(m5)4m7.

Try It 6.130

Simplify: (k2)6k7.(k2)6k7.

Example 6.66

Simplify: b12(b2)6.b12(b2)6.

Try It 6.131

Simplify: n12(n3)4.n12(n3)4.

Try It 6.132

Simplify: x15(x3)5.x15(x3)5.

Example 6.67

Simplify: (y9y4)2.(y9y4)2.

Try It 6.133

Simplify: (r5r3)4.(r5r3)4.

Try It 6.134

Simplify: (v6v4)3.(v6v4)3.

Example 6.68

Simplify: (j2k3)4.(j2k3)4.

Try It 6.135

Simplify: (a3b2)4.(a3b2)4.

Try It 6.136

Simplify: (q7r5)3.(q7r5)3.

Example 6.69

Simplify: (2m25n)4.(2m25n)4.

Try It 6.137

Simplify: (7x39y)2.(7x39y)2.

Try It 6.138

Simplify: (3x47y)2.(3x47y)2.

Example 6.70

Simplify: (x3)4(x2)5(x6)5.(x3)4(x2)5(x6)5.

Try It 6.139

Simplify: (a2)3(a2)4(a4)5.(a2)3(a2)4(a4)5.

Try It 6.140

Simplify: (p3)4(p5)3(p7)6.(p3)4(p5)3(p7)6.

Example 6.71

Simplify: (10p3)2(5p)3(2p5)4.(10p3)2(5p)3(2p5)4.

Try It 6.141

Simplify: (3r3)2(r3)7(r3)3.(3r3)2(r3)7(r3)3.

Try It 6.142

Simplify: (2x4)5(4x3)2(x3)5.(2x4)5(4x3)2(x3)5.

Divide Monomials

You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.

Example 6.72

Find the quotient: 56x7÷8x3.56x7÷8x3.

Try It 6.143

Find the quotient: 42y9÷6y3.42y9÷6y3.

Try It 6.144

Find the quotient: 48z8÷8z2.48z8÷8z2.

Example 6.73

Find the quotient: 45a2b3−5ab5.45a2b3−5ab5.

Try It 6.145

Find the quotient: −72a7b38a12b4.−72a7b38a12b4.

Try It 6.146

Find the quotient: −63c8d37c12d2.−63c8d37c12d2.

Example 6.74

Find the quotient: 24a5b348ab4.24a5b348ab4.

Try It 6.147

Find the quotient: 16a7b624ab8.16a7b624ab8.

Try It 6.148

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 6.75

Find the quotient: 14x7y1221x11y6.14x7y1221x11y6.

Try It 6.149

Find the quotient: 28x5y1449x9y12.28x5y1449x9y12.

Try It 6.150

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. This follows the order of operations. Remember, a fraction bar is a grouping symbol.

Example 6.76

Find the quotient: (6x2y3)(5x3y2)(3x4y5).(6x2y3)(5x3y2)(3x4y5).

Try It 6.151

Find the quotient: (6a4b5)(4a2b5)12a5b8.(6a4b5)(4a2b5)12a5b8.

Try It 6.152

Find the quotient: (−12x6y9)(−4x5y8)−12x10y12.(−12x6y9)(−4x5y8)−12x10y12.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with dividing monomials:

Section 6.5 Exercises

Practice Makes Perfect

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

356.

x18x3x18x3 5125351253

357.

y20y10y20y10 7167271672

358.

p21p7p21p7 4164441644

359.

u24u3u24u3 9159591595

360.

q18q36q18q36 102103102103

361.

t10t40t10t40 83858385

362.

bb9bb9 446446

363.

xx7xx7 1010310103

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

364.

200200 b0b0

365.

130130 k0k0

366.

270270 (270)(270)

367.

150150 (150)(150)

368.

(25x)0(25x)0 25x025x0

369.

(6y)0(6y)0 6y06y0

370.

(12x)0(12x)0 (−56p4q3)0(−56p4q3)0

371.

7y07y0(17y)0(17y)0 (−93c7d15)0(−93c7d15)0

372.

12n018m012n018m0 (12n)0(18m)0(12n)0(18m)0

373.

15r022s015r022s0 (15r)0(22s)0(15r)0(22s)0

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

374.

(34)3(34)3 (p2)5(p2)5 (xy)6(xy)6

375.

(25)2(25)2 (x3)4(x3)4 (ab)5(ab)5

376.

(a3b)4(a3b)4 (54m)2(54m)2

377.

(x2y)3(x2y)3 (103q)4(103q)4

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

378.

(a2)3a4(a2)3a4

379.

(p3)4p5(p3)4p5

380.

(y3)4y10(y3)4y10

381.

(x4)5x15(x4)5x15

382.

u6(u3)2u6(u3)2

383.

v20(v4)5v20(v4)5

384.

m12(m8)3m12(m8)3

385.

n8(n6)4n8(n6)4

386.

(p9p3)5(p9p3)5

387.

(q8q2)3(q8q2)3

388.

(r2r6)3(r2r6)3

389.

(m4m7)4(m4m7)4

390.

(pr11)2(pr11)2

391.

(ab6)3(ab6)3

392.

(w5x3)8(w5x3)8

393.

(y4z10)5(y4z10)5

394.

(2j33k)4(2j33k)4

395.

(3m55n)3(3m55n)3

396.

(3c24d6)3(3c24d6)3

397.

(5u72v3)4(5u72v3)4

398.

(k2k8k3)2(k2k8k3)2

399.

(j2j5j4)3(j2j5j4)3

400.

(t2)5(t4)2(t3)7(t2)5(t4)2(t3)7

401.

(q3)6(q2)3(q4)8(q3)6(q2)3(q4)8

402.

(−2p2)4(3p4)2(−6p3)2(−2p2)4(3p4)2(−6p3)2

403.

(−2k3)2(6k2)4(9k4)2(−2k3)2(6k2)4(9k4)2

404.

(−4m3)2(5m4)3(−10m6)3(−4m3)2(5m4)3(−10m6)3

405.

(−10n2)3(4n5)2(2n8)2(−10n2)3(4n5)2(2n8)2

Divide Monomials

In the following exercises, divide the monomials.

406.

56b8÷ 7b256b8÷ 7b2

407.

63v10÷ 9v263v10÷ 9v2

408.

−88y15÷ 8y3−88y15÷ 8y3

409.

−72u12÷ 12u4−72u12÷ 12u4

410.

45a6b8−15a10b245a6b8−15a10b2

411.

54x9y3−18x6y1554x9y3−18x6y15

412.

15r4s918r9s215r4s918r9s2

413.

20m8n430m5n920m8n430m5n9

414.

18a4b8−27a9b518a4b8−27a9b5

415.

45x5y9−60x8y645x5y9−60x8y6

416.

64q11r9s348q6r8s564q11r9s348q6r8s5

417.

65a10b8c542a7b6c865a10b8c542a7b6c8

418.

(10m5n4)(5m3n6)25m7n5(10m5n4)(5m3n6)25m7n5

419.

(−18p4q7)(−6p3q8)−36p12q10(−18p4q7)(−6p3q8)−36p12q10

420.

(6a4b3)(4ab5)(12a2b)(a3b)(6a4b3)(4ab5)(12a2b)(a3b)

421.

(4u2v5)(15u3v)(12u3v)(u4v)(4u2v5)(15u3v)(12u3v)(u4v)

Mixed Practice

422.

24a5+2a524a5+2a5 24a52a524a52a5 24a5·2a524a5·2a5 24a5÷2a524a5÷2a5

423.

15n10+3n1015n10+3n10 15n103n1015n103n10 15n10·3n1015n10·3n10 15n10÷3n1015n10÷3n10

424.

p4·p6p4·p6 (p4)6(p4)6

425.

q5·q3q5·q3 (q5)3(q5)3

426.

y3yy3y yy3yy3

427.

z6z5z6z5 z5z6z5z6

428.

(8x5)(9x)÷6x3(8x5)(9x)÷6x3

429.

(4y)(12y7)÷8y2(4y)(12y7)÷8y2

430.

27a73a3+54a99a527a73a3+54a99a5

431.

32c114c5+42c96c332c114c5+42c96c3

432.

32y58y260y105y732y58y260y105y7

433.

48x66x435x97x748x66x435x97x7

434.

63r6s39r4s272r2s26s63r6s39r4s272r2s26s

435.

56y4z57y3z345y2z25y56y4z57y3z345y2z25y

Everyday Math

436.

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

437.

Memory One gigabyte is approximately 109109 bytes. One terabyte is approximately 10121012 bytes. How many gigabytes are in one terabyte?

Writing Exercises

438.

Jennifer thinks the quotient a24a6a24a6 simplifies to a4a4. What is wrong with her reasoning?

439.

Maurice simplifies the quotient d7dd7d by writing d7d=7d7d=7. What is wrong with his reasoning?

440.

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

441.

Robert thinks x0x0 simplifies to 0. What would you say to convince Robert he is wrong?

Self Check

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

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “simplify expressions using the Quotient Property for Exponents,” “simplify expressions with zero exponents,” “simplify expressions using the Quotient to a Power Property,” “simplify expressions by applying several properties,” and “divide monomials.” The rest of the cells are blank.

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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