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.
- Simplify: 824.
If you missed this problem, review Exercise 1.6.4. - Simplify: (2m3)5.
If you missed this problem, review Exercise 6.2.22. - 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 am⋅an=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 b≠0,c≠0.
thenab=a⋅cb⋅canda⋅cb⋅c=ab
As before, we’ll try to discover a property by looking at some examples.
Consider x5x2andx2x3 What do they mean? x⋅x⋅x⋅x⋅xx⋅xx⋅xx⋅x⋅x Use the Equivalent Fractions Property. x⋅̸x⋅̸x⋅x⋅xx⋅̸x̸x̸⋅x̸⋅1x⋅̸x̸⋅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:
x5x2x2x3x5−21x3−2x31x
This leads to the Quotient Property for Exponents.
QUOTIENT PROPERTY FOR EXPONENTS
If a is a real number, a≠0, and m and n are whole numbers, then
aman=am−n,m>n and aman=1an−m,n>m
A couple of examples with numbers may help to verify this property.
3432=34−25253=153−2819=3225125=1519=9✓15=15✓
Exercise 5.6.1
Simplify:
- x9x7
- 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. Use the Quotient Property, aman=am−n Simplify. 2.
Since 10 > 2, there are more factors of x in the numerator. Use the Quotient Property, aman=am−n Simplify.
Exercise 5.6.2
Simplify:
- x15x10
- 61465
- Answer
-
- x5
- 69
Exercise 5.6.3
Simplify:
- y43y37
- 1015107
- Answer
-
- y6
- 108
Exercise 5.6.4
Simplify:
- b8b12
- 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. Use the Quotient Property, aman=1an−m Simplify. 2.
Since 5 > 3, there are more factors of 3 in the denominator. Use the Quotient Property, aman=1an−m Simplify. Simplify.
Exercise 5.6.5
Simplify:
- x18x22
- 12151230
- Answer
-
- 1x4
- 11215
Exercise 5.6.6
Simplify:
- m7m15
- 98919
- Answer
-
- 1m8
- 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:
- a5a9
- 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.
Use the Quotient Property, aman=1an−m Simplify. 2. Notice there are more factors of xx in the numerator, since 11 > 7. So we will end up with factors in the numerator.
Use the Quotient Property, aman=1an−m Simplify.
Exercise 5.6.8
Simplify:
- b19b11
- z5z11
- Answer
-
- b8
- 1z6
Exercise 5.6.9
Simplify:
- p9p17
- w13w9
- Answer
-
- 1p8
- 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=1−43−43=1
In words, a number divided by itself is 1. So, xx=1, for any x(x≠0), 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. 23−3=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 a≠0:
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:
- 90
- n0
- Answer
-
The definition says any non-zero number raised to the zero power is 1.
- 90Use the definition of the zero exponent.1
- n0Use the definition of the zero exponent.1
Exercise 5.6.11
Simplify:
- 150
- m0
- Answer
-
- 1
- 1
Exercise 5.6.12
Simplify:
- k0
- 290
- Answer
-
- 1
- 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. 1⋅1 Simplify. 1
This tells us that any nonzero expression raised to the zero power is one.
Exercise 5.6.13
Simplify:
- (5b)0
- (−4a2b)0.
- Answer
-
- (5b)0Use the definition of the zero exponent.1
- (−4a2b)0Use the definition of the zero exponent.1
Exercise 5.6.14
Simplify:
- (11z)0
- (−11pq3)0.
- Answer
-
- 1
- 1
Exercise 5.6.15
Simplify:
- (−6d)0
- (−8m2n3)0.
- Answer
-
- 1
- 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:xy⋅xy⋅xyMultiply the fractions.x⋅x⋅xy⋅y⋅yWrite 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, b≠0, 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=233323⋅23⋅23=827827=827✓
Exercise 5.6.16
Simplify:
- (37)2
- (b3)4
- (kj)3
- Answer
-
1.
Use the Quotient Property, (ab)m=ambm Simplify. 2.
Use the Quotient Property, (ab)m=ambm Simplify. 3.
Raise the numerator and denominator to the third power.
Exercise 5.6.17
Simplify:
- (58)2
- (p10)4
- (mn)7
- Answer
-
- 2564
- p410,000
- m7n7
Exercise 5.6.18
Simplify:
- (13)3
- (−2q)3
- (wx)4
- Answer
-
- 127
- −8q3
- 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 Propertyam⋅an=am+nPower Property(am)n=am⋅nProduct to a Power(ab)m=ambmQuotient Propertyaman=am−n,a≠0,m>nanan=1,a≠0,n>mZero Exponent Definitiona0=1,a≠0Quotient to a Power Property(ab)m=ambm,b≠0
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=am⋅n(x12)(x10)(x30)Add the exponents in the numerator.x22x30Use the Quotient Property,aman=1an−m1x8
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=am⋅n100p6125p3⋅16p20 Add the exponents in the denominator. 100p6125⋅16p23 Use the Quotient Property, aman=1an−m100125⋅16p17 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. 568⋅x7x3 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: 45a2b3−5ab5
- Answer
-
When we divide monomials with more than one variable, we write one fraction for each variable.
45a2b3−5ab5 Use fraction multiplication. 45−5⋅a2a⋅b3b5 Simplify and use the Quotient Property. −9⋅a⋅1b2 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. 2448⋅a5a⋅b3b4 Simplify and use the Quotient Property. 12⋅a4⋅1b Multiply. a42b
Exercise 5.6.44
Find the quotient: 16a7b624ab8
- Answer
-
2a63b2
Exercise 5.6.45
Find the quotient: 27p4q7−45p12q
- 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, a≠0, and m,n are whole numbers, then: aman=am−n,m>n and aman=1am−n,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, b≠0, 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 Propertyam⋅an=am+nPower Property(am)n=am⋅nProduct to a Power(ab)m=ambmQuotient Propertyaman=am−n,a≠0,m>nanan=1,a≠0,n>mZero Exponent Definitiona0=1,a≠0Quotient to a Power Property(ab)m=ambm,b≠0