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5.6: Divide Monomials

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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

Note

Before you get started, take this readiness quiz.

  1. Simplify: 824.
    If you missed this problem, review Exercise 1.6.4.
  2. Simplify: (2m3)5.
    If you missed this problem, review Exercise 6.2.22.
  3. Simplify: 12x12y
    If you missed this problem, review Exercise 1.6.10.

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 a and b are real numbers, and m and n are whole numbers, then

 Product Property aman=am+n Power Property (am)n=amn Product 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. 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, and c are whole numbers where b0,c0.

thenab=acbcandacbc=ab

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

 Consider x5x2andx2x3 What do they mean? xxxxxxxxxxxx Use the Equivalent Fractions Property. x⋅̸x⋅̸xxxx⋅̸1x⋅̸x Simplify. x31x

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:

x5x2x2x3x521x32x31x

This leads to the Quotient Property for Exponents.

QUOTIENT PROPERTY FOR EXPONENTS

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

aman=amn,m>n and aman=1anm,n>m

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

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

Exercise 5.6.1

Simplify:

  1. x9x7
  2. 31032
Answer

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

1.

Since 9 > 7, there are more factors of x in the numerator. x to the ninth power divided by x to the seventh power.
Use the Quotient Property, aman=amn x to the power of 9 minus 7.
Simplify. x squared.

2.

Since 10 > 2, there are more factors of x in the numerator. 3 to the tenth power divided by 3 squared.
Use the Quotient Property, aman=amn 3 to the power of 10 minus 2.
Simplify. 3 to the eighth power.
Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

Exercise 5.6.2

Simplify:

  1. x15x10
  2. 61465
Answer
  1. x5
  2. 69

Exercise 5.6.3

Simplify:

  1. y43y37
  2. 1015107
Answer
  1. y6
  2. 108

Exercise 5.6.4

Simplify:

  1. b8b12
  2. 7375
Answer

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

1.

Since 12 > 8, there are more factors of b in the denominator. b to the eighth power divided b to the twelfth power.
Use the Quotient Property, aman=1anm 1 divided by b to the power of 12 minus 8.
Simplify. 1 divided by b to the fourth power.

2.

Since 5 > 3, there are more factors of 3 in the denominator. 7 cubed divided by 7 to the fifth power.
Use the Quotient Property, aman=1anm 1 divided by 7 to the power of 5 minus 3.
Simplify. 1 divided by 7 squared.
Simplify. 1 forty-ninth.
Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.

Exercise 5.6.5

Simplify:

  1. x18x22
  2. 12151230
Answer
  1. 1x4
  2. 11215

Exercise 5.6.6

Simplify:

  1. m7m15
  2. 98919
Answer
  1. 1m8
  2. 1911

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.

Exercise 5.6.7

Simplify:

  1. a5a9
  2. x11x7
Answer

1. Is the exponent of a larger in the numerator or denominator? Since 9 > 5, there are more a's in the denominator and so we will end up with factors in the denominator.

  a to the fifth power divided by a to the ninth power.
Use the Quotient Property, aman=1anm 1 divided by a to the power of 9 minus 5.
Simplify. 1 divided by a to the fourth power.

2. Notice there are more factors of xx in the numerator, since 11 > 7. So we will end up with factors in the numerator.

  x to the eleventh power divided by x to the seventh power.
Use the Quotient Property, aman=1anm x to the power of 11 minus 7.
Simplify. x to the fourth power.

Exercise 5.6.8

Simplify:

  1. b19b11
  2. z5z11
Answer
  1. b8
  2. 1z6

Exercise 5.6.9

Simplify:

  1. p9p17
  2. w13w9
Answer
  1. 1p8
  2. w4

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 amam. From your earlier work with fractions, you know that:

22=11717=14343=1

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

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

Consider 88, which we know is 1.

88=1 Write 8 as 23.2323=1 Subtract exponents. 233=1 Simplify. 20=1

Now we will simplify amam in two ways to lead us to the definition of the zero exponent. In general, for a0:

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 amam simplifies to a0 and to 1. So a0=1.

ZERO EXPONENT

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

Exercise 5.6.10

Simplify:

  1. 90
  2. n0
Answer

The definition says any non-zero number raised to the zero power is 1.

  1. 90Use the definition of the zero exponent.1
  2. n0Use the definition of the zero exponent.1

Exercise 5.6.11

Simplify:

  1. 150
  2. m0
Answer
  1. 1
  2. 1

Exercise 5.6.12

Simplify:

  1. k0
  2. 290
Answer
  1. 1
  2. 1

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. We can use the product to a power rule to rewrite this expression.

(2x)0 Use the product to a power rule. 20x0 Use the zero exponent property. 11 Simplify. 1

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

Exercise 5.6.13

Simplify:

  1. (5b)0
  2. (4a2b)0.
Answer
  1. (5b)0Use the definition of the zero exponent.1
  2. (4a2b)0Use the definition of the zero exponent.1

Exercise 5.6.14

Simplify:

  1. (11z)0
  2. (11pq3)0.
Answer
  1. 1
  2. 1

Exercise 5.6.15

Simplify:

  1. (6d)0
  2. (8m2n3)0.
Answer
  1. 1
  2. 1

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)3This means:xyxyxyMultiply the fractions.xxxyyyWrite with exponents.x3y3

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

 We see that (xy)3 is x3y3 We write: (xy)3x3y3

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

QUOTIENT TO A POWER PROPERTY FOR EXPONENTS

If a and b are real numbers, b0, and m is a counting number, then

(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

Exercise 5.6.16

Simplify:

  1. (37)2
  2. (b3)4
  3. (kj)3
Answer

1.

  3 sevenths squared.
Use the Quotient Property, (ab)m=ambm 3 squared divided by 7 squared.
Simplify. 9 forty-ninths.

2.

  b thirds to the fourth power.
Use the Quotient Property, (ab)m=ambm b to the fourth power divided by 3 to the fourth power.
Simplify. b to the fourth power divided by 81.

3.

  k divided by j, in parentheses, cubed.
Raise the numerator and denominator to the third power. k cubed divided by j cubed.

Exercise 5.6.17

Simplify:

  1. (58)2
  2. (p10)4
  3. (mn)7
Answer
  1. 2564
  2. p410,000
  3. m7n7

Exercise 5.6.18

Simplify:

  1. (13)3
  2. (2q)3
  3. (wx)4
Answer
  1. 127
  2. 8q3
  3. w4x4

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 and b are real numbers, and m and n are whole numbers, then

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

Exercise 5.6.19

Simplify: (y4)2y6

Answer

(y4)2y6Multiply the exponents in the numerator.y8y6Subtract the exponents.y2

Exercise 5.6.20

Simplify: (m5)4m7

Answer

m13

Exercise 5.6.21

Simplify: (k2)6k7

Answer

k5

Exercise 5.6.22

Simplify: b12(b2)6

Answer

b12(b2)6Multiply the exponents in the numerator.b12b12Subtract the exponents.b0Simplify1

Notice that after we simplified the denominator in the first step, the numerator and the denominator were equal. So the final value is equal to 1.

Exercise 5.6.23

Simplify:n12(n3)4

Answer

1

Exercise 5.6.24

Simplify:x15(x3)5

Answer

1

Exercise 5.6.25

Simplify: (y9y4)2

Answer

(y9y4)2Remember parentheses come before exponents.Notice the bases are the same, so we can simplify(y5)2inside the parentheses. Subtract the exponents.Multiply the exponents.y10

Exercise 5.6.25

Simplify: (r5r3)4

Answer

r8

Exercise 5.6.25

Simplify: (v6v4)3

Answer

v6

Exercise 5.6.26

Simplify:(j2k3)4

Answer

Here we cannot simplify inside the parentheses first, since the bases are not the same.

(j2k3)4Raise the numerator and denominator to the third powerusing the Quotient to a Power Property,(ab)m=ambm(j2)4(k3)4Use the Power Property and simplify.j8k12

Exercise 5.6.27

Simplify:(a3b2)4

Answer

a12b8

Exercise 5.6.28

Simplify: (q7r5)3

Answer

q21r15

Exercise 5.6.29

Simplify:(2m25n)4

Answer

(2m25n)4Raise the numerator and denominator to the fourth(2m2)4(5n)4power, using the Quotient to a Power Property,(ab)m=ambm24(m2)454n4Use the Power Property and simplify.16m8625n4

Exercise 5.6.30

Simplify:(7x39y)2

Answer

49x681y2

Exercise 5.6.31

Simplify: (3x47y)2

Answer

9x849v2

Exercise 5.6.32

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

Answer

(x3)4(x2)5(x6)5Use the Power Property,(am)n=amn(x12)(x10)(x30)Add the exponents in the numerator.x22x30Use the Quotient Property,aman=1anm1x8

Exercise 5.6.32

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

Answer

1a6

Exercise 5.6.33

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

Answer

1p15

Exercise 5.6.34

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

Answer

(10p3)2(5p)3(2p5)4 Use the Product to a Power Property, (ab)m=ambm(10)2(p3)2(5)3(p)3(2)4(p5)4 Use the Power Property, (am)n=amn100p6125p316p20 Add the exponents in the denominator. 100p612516p23 Use the Quotient Property, aman=1anm10012516p17 Simplify. 120p17

Exercise 5.6.35

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

Answer

9r18

Exercise 5.6.36

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

Answer

2x

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.

Exercise 5.6.37

Find the quotient: 56x7÷8x3

Answer

56x7÷8x3 Rewrite as a fraction. 56x78x3 Use fraction multiplication. 568x7x3 Simplify and use the Quotient Property. 7x4

Exercise 5.6.38

Find the quotient:42y9÷6y3

Answer

7y6

Exercise 5.6.39

Find the quotient:48z8÷8z2

Answer

6z6

Exercise 5.6.40

Find the quotient: 45a2b35ab5

Answer

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

45a2b35ab5 Use fraction multiplication. 455a2ab3b5 Simplify and use the Quotient Property. 9a1b2 Multiply. 9ab2

Exercise 5.6.41

Find the quotient: 72a7b38a12b4

Answer

9a5b

Exercise 5.6.42

Find the quotient: 63c8d37c12d2

Answer

9dc4

Exercise 5.6.43

Find the quotient: 24a5b348ab4

Answer

24a5b348ab4 Use fraction multiplication. 2448a5ab3b4 Simplify and use the Quotient Property. 12a41b Multiply. a42b

Exercise 5.6.44

Find the quotient: 16a7b624ab8

Answer

2a63b2

Exercise 5.6.45

Find the quotient: 27p4q745p12q

Answer

3q65p8

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.

Exercise 5.6.46

Find the quotient: 14x7y1221x11y6

Answer

Be very careful to simplify 1421 by dividing out a common factor, and to simplify the variables by subtracting their exponents.

14x7y1221x11y6 Simplify and use the Quotient Property. 2y63x4

Exercise 5.6.47

Find the quotient:28x5y1449x9y12

Answer

4y27x4

Exercise 5.6.48

Find the quotient:30m5n1148m10n14

Answer

58m5n3

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.

Exercise 5.6.49

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

Answer

(6x2y3)(5x3y2)(3x4y5) Simplify the numerator. 30x5y53x4y5 Simplify. 10x

Exercise 5.6.50

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

Answer

2ab2

Exercise 5.6.51

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

Answer

4xy5

Note

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

Key Concepts

  • Quotient Property for Exponents:
    • If a is a real number, a0, and m,n are whole numbers, then: aman=amn,m>n and aman=1amn,n>m
  • Zero Exponent
    • If a is a non-zero number, then a0=1.
  • Quotient to a Power Property for Exponents:
    • If a and b are real numbers, b0, and mm is a counting number, then: (ab)m=ambm
    • To raise a fraction to a power, raise the numerator and denominator to that power.
  • Summary of Exponent Properties
    • If a,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=ambmQuotient Propertyaman=amn,a0,m>nanan=1,a0,n>mZero Exponent Definitiona0=1,a0Quotient to a Power Property(ab)m=ambm,b0

This page titled 5.6: Divide Monomials is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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