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7.2: Division with Exponents

  • Page ID
    137930
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    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:

    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
    Answer

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

    Since 10 > 8, there are more factors of xx in the numerator. x10x8x10x8
    Use the quotient property with m>n,aman=amnm>n,aman=amn. .
    Simplify. x2x2
    Since 9 > 2, there are more factors of 2 in the numerator. 29222922
    Use the quotient property with m>n,aman=amn.m>n,aman=amn. .
    Simplify. 2727

    Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

    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
    Answer

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

    Since 15 > 10, there are more factors of bb in the denominator. b10b15b10b15
    Use the quotient property with n>m,aman=1anm.n>m,aman=1anm. .
    Simplify. 1b51b5
    Since 5 > 3, there are more factors of 3 in the denominator. 33353335
    Use the quotient property with n>m,aman=1anm.n>m,aman=1anm. .
    Simplify. 132132
    Apply the exponent. 1919

    Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and 11 in the numerator.

    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
    Answer

    Since 9 > 5, there are more aa's in the denominator and so we will end up with factors in the denominator. a5a9a5a9
    Use the Quotient Property for n>m,aman=1anm.n>m,aman=1anm. .
    Simplify. 1a41a4
    Notice there are more factors of xx in the numerator, since 11 > 7. So we will end up with factors in the numerator. x11x7x11x7
    Use the Quotient Property for m>n,aman=anm.m>n,aman=anm. .
    Simplify. x4x4

    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
    Answer

    The definition says any non-zero number raised to the zero power is 1.1.

    120120
    Use the definition of the zero exponent. 1
    y0y0
    Use the definition of the zero exponent. 1

    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.

    Answer

    (7z)0(7z)0
    Use the definition of the zero exponent. 1

    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
    Answer

    The product is raised to the zero power. (−3x2y)0(−3x2y)0
    Use the definition of the zero exponent. 11
    Notice that only the variable yy is being raised to the zero power. −3x2y0−3x2y0
    Use the definition of the zero exponent. −3x21−3x21
    Simplify. −3x2−3x2

    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
    Answer

    .
    Use the Quotient to a Power Property, (ab)m=ambm(ab)m=ambm. .
    Simplify. .
    .
    Use the Quotient to a Power Property, (ab)m=ambm(ab)m=ambm. .
    Simplify. .
    .
    Raise the numerator and denominator to the third power. .

    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.

    Answer

    (x2)3x5(x2)3x5
    Multiply the exponents in the numerator, using the
    Power Property.
    x6x5x6x5
    Subtract the exponents. xx

    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.

    Answer

    m8(m2)4m8(m2)4
    Multiply the exponents in the numerator, using the
    Power Property.
    m8m8m8m8
    Subtract the exponents. m0m0
    Zero power property 11

    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.

    Answer

    (x7x3)2(x7x3)2
    Remember parentheses come before exponents, and the
    bases are the same so we can simplify inside the
    parentheses. Subtract the exponents.
    (x73)2(x73)2
    Simplify. (x4)2(x4)2
    Multiply the exponents. x8x8

    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.

    Answer

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

    (p2q5)3(p2q5)3
    Raise the numerator and denominator to the third power
    using the Quotient to a Power Property, (ab)m=ambm(ab)m=ambm
    (p2)3(q5)3(p2)3(q5)3
    Use the Power Property, (am)n=amn.(am)n=amn. p6q15p6q15

    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.

    Answer

    (2x33y)4(2x33y)4
    Raise the numerator and denominator to the fourth
    power using the Quotient to a Power Property.
    (2x3)4(3y)4(2x3)4(3y)4
    Raise each factor to the fourth power, using the Power
    to a Power Property.
    24(x3)434y424(x3)434y4
    Use the Power Property and simplify. 16x1281y416x1281y4

    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.

    Answer

    (y2)3(y2)4(y5)4(y2)3(y2)4(y5)4
    Use the Power Property. (y6)(y8)y20(y6)(y8)y20
    Add the exponents in the numerator, using the Product Property. y14y20y14y20
    Use the Quotient Property. 1y61y6

    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.

    Answer

    56x5÷7x256x5÷7x2
    Rewrite as a fraction. 56x57x256x57x2
    Use fraction multiplication to separate the number
    part from the variable part.
    567x5x2567x5x2
    Use the Quotient Property. 8x38x3

    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.

    Answer

    42x2y3−7xy542x2y3−7xy5
    Use fraction multiplication. 42−7x2xy3y542−7x2xy3y5
    Simplify and use the Quotient Property. −6x1y2−6x1y2
    Multiply. 6xy26xy2

    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.

    Answer

    24a5b348ab424a5b348ab4
    Use fraction multiplication. 2448a5ab3b42448a5ab3b4
    Simplify and use the Quotient Property. 12a41b12a41b
    Multiply. a42ba42b

    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.

    Answer

    14x7y1221x11y614x7y1221x11y6
    Simplify and use the Quotient Property. 2y63x42y63x4

    Be very careful to simplify 14211421 by dividing out a common factor, and to simplify the variables by subtracting their exponents.

    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.

    Answer

    Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.

    (3x3y2)(10x2y3)6x4y5(3x3y2)(10x2y3)6x4y5
    Simplify the numerator. 30x5y56x4y530x5y56x4y5
    Simplify, using the Quotient Rule. 5x5x

    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.

    Media

    Section 10.4 Exercises

    Practice Makes Perfect

    Simplify Expressions Using the Quotient Property of Exponents

    In the following exercises, simplify.

    219.

    4 8 4 2 4 8 4 2

    220.

    3 12 3 4 3 12 3 4

    221.

    x 12 x 3 x 12 x 3

    222.

    u 9 u 3 u 9 u 3

    223.

    r 5 r r 5 r

    224.

    y 4 y y 4 y

    225.

    y 4 y 20 y 4 y 20

    226.

    x 10 x 30 x 10 x 30

    227.

    10 3 10 15 10 3 10 15

    228.

    r 2 r 8 r 2 r 8

    229.

    a a 9 a a 9

    230.

    2 2 5 2 2 5

    Simplify Expressions with Zero Exponents

    In the following exercises, simplify.

    231.

    5 0 5 0

    232.

    10 0 10 0

    233.

    a 0 a 0

    234.

    x 0 x 0

    235.

    7 0 7 0

    236.

    4 0 4 0

    237.
    1. ( 10 p ) 0 ( 10 p ) 0
    2. 10 p 0 10 p 0
    238.
    1. ( 3 a ) 0 ( 3 a ) 0
    2. 3 a 0 3 a 0
    239.
    1. ( −27 x 5 y ) 0 ( −27 x 5 y ) 0
    2. −27 x 5 y 0 −27 x 5 y 0
    240.
    1. ( −92 y 8 z ) 0 ( −92 y 8 z ) 0
    2. −92 y 8 z 0 −92 y 8 z 0
    241.
    1. 15 0 15 0
    2. 15 1 15 1
    242.
    1. 6 0 6 0
    2. 6 1 6 1
    243.

    2 · x 0 + 5 · y 0 2 · x 0 + 5 · y 0

    244.

    8 · m 0 4 · n 0 8 · m 0 4 · n 0

    Simplify Expressions Using the Quotient to a Power Property

    In the following exercises, simplify.

    245.

    ( 3 2 ) 5 ( 3 2 ) 5

    246.

    ( 4 5 ) 3 ( 4 5 ) 3

    247.

    ( m 6 ) 3 ( m 6 ) 3

    248.

    ( p 2 ) 5 ( p 2 ) 5

    249.

    ( x y ) 10 ( x y ) 10

    250.

    ( a b ) 8 ( a b ) 8

    251.

    ( a 3 b ) 2 ( a 3 b ) 2

    252.

    ( 2 x y ) 4 ( 2 x y ) 4

    Simplify Expressions by Applying Several Properties

    In the following exercises, simplify.

    253.

    ( x 2 ) 4 x 5 ( x 2 ) 4 x 5

    254.

    ( y 4 ) 3 y 7 ( y 4 ) 3 y 7

    255.

    ( u 3 ) 4 u 10 ( u 3 ) 4 u 10

    256.

    ( y 2 ) 5 y 6 ( y 2 ) 5 y 6

    257.

    y 8 ( y 5 ) 2 y 8 ( y 5 ) 2

    258.

    p 11 ( p 5 ) 3 p 11 ( p 5 ) 3

    259.

    r 5 r 4 · r r 5 r 4 · r

    260.

    a 3 · a 4 a 7 a 3 · a 4 a 7

    261.

    ( x 2 x 8 ) 3 ( x 2 x 8 ) 3

    262.

    ( u u 10 ) 2 ( u u 10 ) 2

    263.

    ( a 4 · a 6 a 3 ) 2 ( a 4 · a 6 a 3 ) 2

    264.

    ( x 3 · x 8 x 4 ) 3 ( x 3 · x 8 x 4 ) 3

    265.

    ( y 3 ) 5 ( y 4 ) 3 ( y 3 ) 5 ( y 4 ) 3

    266.

    ( z 6 ) 2 ( z 2 ) 4 ( z 6 ) 2 ( z 2 ) 4

    267.

    ( x 3 ) 6 ( x 4 ) 7 ( x 3 ) 6 ( x 4 ) 7

    268.

    ( x 4 ) 8 ( x 5 ) 7 ( x 4 ) 8 ( x 5 ) 7

    269.

    ( 2 r 3 5 s ) 4 ( 2 r 3 5 s ) 4

    270.

    ( 3 m 2 4 n ) 3 ( 3 m 2 4 n ) 3

    271.

    ( 3 y 2 · y 5 y 15 · y 8 ) 0 ( 3 y 2 · y 5 y 15 · y 8 ) 0

    272.

    ( 15 z 4 · z 9 0.3 z 2 ) 0 ( 15 z 4 · z 9 0.3 z 2 ) 0

    273.

    ( r 2 ) 5 ( r 4 ) 2 ( r 3 ) 7 ( r 2 ) 5 ( r 4 ) 2 ( r 3 ) 7

    274.

    ( p 4 ) 2 ( p 3 ) 5 ( p 2 ) 9 ( p 4 ) 2 ( p 3 ) 5 ( p 2 ) 9

    275.

    ( 3 x 4 ) 3 ( 2 x 3 ) 2 ( 6 x 5 ) 2 ( 3 x 4 ) 3 ( 2 x 3 ) 2 ( 6 x 5 ) 2

    276.

    ( −2 y 3 ) 4 ( 3 y 4 ) 2 ( −6 y 3 ) 2 ( −2 y 3 ) 4 ( 3 y 4 ) 2 ( −6 y 3 ) 2

    Divide Monomials

    In the following exercises, divide the monomials.

    277.

    48 b 8 ÷ 6 b 2 48 b 8 ÷ 6 b 2

    278.

    42 a 14 ÷ 6 a 2 42 a 14 ÷ 6 a 2

    279.

    36 x 3 ÷ ( −2 x 9 ) 36 x 3 ÷ ( −2 x 9 )

    280.

    20 u 8 ÷ ( −4 u 6 ) 20 u 8 ÷ ( −4 u 6 )

    281.

    18 x 3 9 x 2 18 x 3 9 x 2

    282.

    36 y 9 4 y 7 36 y 9 4 y 7

    283.

    −35 x 7 −42 x 13 −35 x 7 −42 x 13

    284.

    18 x 5 −27 x 9 18 x 5 −27 x 9

    285.

    18 r 5 s 3 r 3 s 9 18 r 5 s 3 r 3 s 9

    286.

    24 p 7 q 6 p 2 q 5 24 p 7 q 6 p 2 q 5

    287.

    8 m n 10 64 m n 4 8 m n 10 64 m n 4

    288.

    10 a 4 b 50 a 2 b 6 10 a 4 b 50 a 2 b 6

    289.

    −12 x 4 y 9 15 x 6 y 3 −12 x 4 y 9 15 x 6 y 3

    290.

    48 x 11 y 9 z 3 36 x 6 y 8 z 5 48 x 11 y 9 z 3 36 x 6 y 8 z 5

    291.

    64 x 5 y 9 z 7 48 x 7 y 12 z 6 64 x 5 y 9 z 7 48 x 7 y 12 z 6

    292.

    ( 10 u 2 v ) ( 4 u 3 v 6 ) 5 u 9 v 2 ( 10 u 2 v ) ( 4 u 3 v 6 ) 5 u 9 v 2

    293.

    ( 6 m 2 n ) ( 5 m 4 n 3 ) 3 m 10 n 2 ( 6 m 2 n ) ( 5 m 4 n 3 ) 3 m 10 n 2

    294.

    ( 6 a 4 b 3 ) ( 4 a b 5 ) ( 12 a 8 b ) ( a 3 b ) ( 6 a 4 b 3 ) ( 4 a b 5 ) ( 12 a 8 b ) ( a 3 b )

    295.

    ( 4 u 5 v 4 ) ( 15 u 8 v ) ( 12 u 3 v ) ( u 6 v ) ( 4 u 5 v 4 ) ( 15 u 8 v ) ( 12 u 3 v ) ( u 6 v )

    Mixed Practice

    296.
    1. 24 a 5 + 2 a 5 24 a 5 + 2 a 5
    2. 24 a 5 2 a 5 24 a 5 2 a 5
    3. 24 a 5 2 a 5 24 a 5 2 a 5
    4. 24 a 5 ÷ 2 a 5 24 a 5 ÷ 2 a 5
    297.
    1. 15 n 10 + 3 n 10 15 n 10 + 3 n 10
    2. 15 n 10 3 n 10 15 n 10 3 n 10
    3. 15 n 10 3 n 10 15 n 10 3 n 10
    4. 15 n 10 ÷ 3 n 10 15 n 10 ÷ 3 n 10
    298.
    1. p4p6p4p6
    2. ( p 4 ) 6 ( p 4 ) 6
    299.
    1. q5q3q5q3
    2. ( q 5 ) 3 ( q 5 ) 3
    300.
    1. y3yy3y
    2. y y 3 y y 3
    301.
    1. z6z5z6z5
    2. z 5 z 6 z 5 z 6
    302.

    (8x5)(9x)÷6x3(8x5)(9x)÷6x3

    303.

    (4y)(12y7)÷8y2(4y)(12y7)÷8y2

    304.

    27 a 7 3 a 3 + 54 a 9 9 a 5 27 a 7 3 a 3 + 54 a 9 9 a 5

    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?


    This page titled 7.2: Division with Exponents is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

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