Use Multiplication Properties of Exponents* - OpenStax CNX

OpenStax-CNX module: m60215 1

Use Multiplication Properties of

Exponents*

OpenStax

This work is produced by OpenStax-CNX and licensed under the

Creative Commons Attribution License 4.0�

Abstract

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

• Simplify expressions with exponents

• Simplify expressions using the Product Property for Exponents

• Simplify expressions using the Power Property for Exponents

• Simplify expressions using the Product to a Power Property

• Simplify expressions by applying several properties

• Multiply monomials

note: Before you get started, take this readiness quiz.

1.Simplify: 34 ·

34 .

If you missed this problem, review .2.Simplify: (−2) (−2) (−2) .If you missed this problem, review .

1 Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, 24 meansto multiply 2 by itself 4 times, so 24 means 2 · 2 · 2 · 2.

Let's review the vocabulary for expressions with exponents.

note:

This is read a to the mth power.

*Version 1.2: Jan 18, 2017 12:45 pm -0600�http://creativecommons.org/licenses/by/4.0/

http://cnx.org/content/m60215/1.2/


In the expression am, the exponentm tells us how many times we use the basea as a factor.

Before we begin working with variable expressions containing exponents, let's simplify a few expressionsinvolving only numbers.

Example 1

Simplify: a43b71c(56

)2d(0.63)

2.

Solution : Solutiona

43

Multiply three factors of 4. 4 · 4 · 4Simplify. 64

b

71

Multiply one factor of 7. 7

c (56

)2Multiply two factors.

(56

) (56

)Simplify. 25

36

d

(0.63)2

Multiply two factors. (0.63) (0.63)

Simplify. 0.3969

note:

Exercise 2 (Solution on p. 21.)


)2d(0.43)

2.

note:



)3d(0.218)

2.



Example 2Simplify: a(−5)4b−54.

Solution : Solution

1. a

(−5)4

Multiply four factors of − 5. (−5) (−5) (−5) (−5)Simplify. 625

2. b

−54

Multiply four factors of 5. − (5 · 5 · 5 · 5)Simplify. − 625

note:


Simplify: a(−3)4b−34.

note:


Simplify: a(−13)2b−132.

Notice the similarities and di�erences in Example 2a and Example 2b! Why are the answers di�erent? Aswe follow the order of operations in part a the parentheses tell us to raise the (−5) to the 4th power. In partb we raise just the 5 to the 4th power and then take the opposite.

2 Simplify Expressions Using the Product Property for Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same basewith the same exponent. But when you multiply and divide, the exponents may be di�erent, and sometimesthe bases may be di�erent, too.

We'll derive the properties of exponents by looking for patterns in several examples.First, we will look at an example that leads to the Product Property.

What does this mean?How many factors altogether?

So, we have

Notice that 5 is the sum of the exponents, 2 and 3.



Table 1

We write:

x2 · x3

x2+3

x5

(1)

The base stayed the same and we added the exponents. This leads to the Product Property for Expo-nents.

note: If a is a real number, and m and n are counting numbers, then

am · an = am+n (2)

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

22 · 23 ?= 22+3

4 · 8 ?= 25

32 = 32X

(3)

Example 3Simplify: y5 · y6.

Solution : Solution

Use the product property, am · an = am+n.

Simplify.

Table 2

note: Exercise 8 (Solution on p. 21.)

Simplify: b9 · b8.

note:


Simplify: x12 · x4.



Example 4Simplify: a25 · 29b3 · 34.

Solution : Solution

1. a


Simplify.

Table 3

2. b


Simplify.

Table 4


Simplify: a5 · 55b49 · 49.

note:


Simplify: a76 · 78b10 · 1010.

Example 5Simplify: aa7 · abx27 · x13.

Solution : Solution

1. a

Rewrite, a = a1.


Simplify.



Table 5

2. b

Notice, the bases are the same, so add the exponents.

Simplify.

Table 6


Simplify: ap5 · pby14 · y29.

note:


Simplify: az · z7bb15 · b34.

We can extend the Product Property for Exponents to more than two factors.

Example 6Simplify: d4 · d5 · d2.

Solution : Solution

Add the exponents, since bases are the same.

Simplify.

Table 7


Simplify: x6 · x4 · x8.

note:


Simplify: b5 · b9 · b5.



3 Simplify Expressions Using the Power Property for Exponents

Now let's look at an exponential expression that contains a power raised to a power. See if you can discovera general property.

What does this mean?How many factors altogether?

So we have

Notice that 6 is the product of the exponents, 2 and3.

Table 8

We write:

(x2

)3x2·3

x6

(4)

We multiplied the exponents. This leads to the Power Property for Exponents.

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

(am)n= am·n (5)

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

(32)3 ?

= 32·3

(9)3 ?

= 36

729 = 729X

(6)

Example 7

Simplify: a(y5)9b(44)7.



Solution : Solutiona

Use the power property, (am)n = am·n.

Simplify.

Table 9

b

Use the power property.

Simplify.

Table 10


Simplify: a(b7)5b(54)3.

note:


Simplify: a(z6)9b(37)7.

4 Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Can you �nd this pattern?

(2x)3

What does this mean? 2x · 2x · 2xWe group the like factors together. 2 · 2 · 2 · x · x · xHow many factors of 2 and of x? 23 · x3

Notice that each factor was raised to the power and (2x)3is 23 · x3.

We write: (2x)3

23 · x3

The exponent applies to each of the factors! This leads to the Product to a Power Property forExponents.



note: If a and b are real numbers and m is a whole number, then

(ab)m

= ambm (7)

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

(2 · 3)2 ?= 22 · 32

62?= 4 · 9

36 = 36X

(8)

Example 8Simplify: a(−9d)2b(3mn)

3.

Solution : Solution

1. a

Use Power of a Product Property, (ab)m = ambm.

Simplify.

Table 11

2. b

Use Power of a Product Property, (ab)m = ambm.

Simplify.

Table 12


Simplify: a(−12y)2b(2wx)5.

note:


Simplify: a(5wx)3b(−3y)3.



5 Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let's summarize them and thenwe'll do some examples that use more than one of the properties.

note: 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

= am·n

Product to a Power (ab)m

= ambm

All exponent properties hold true for any real numbers m and n. Right now, we only use whole numberexponents.

Example 9

Simplify: a(y3)6(

y5)4b(−6x4y5

)2.

Solution : Solution

1. a (y3)6(

y5)4

Use the Power Property. y15 · y20

Add the exponents. y35

2. b (−6x4y5

)2Use the Product to a Power Property. (−6)2

(x4

)2(y5)2

Use the Power Property. (−6)2(x8

) (y10

)Simplify. 36x8y10

note:


Simplify: a(a4)5(

a7)4b(−2c4d2

)3.

note:


Simplify: a(−3x6y7

)4b(q4)5(

q3)3.

Example 10

Simplify: a(5m)2 (

3m3)b(3x2y

)4(2xy2

)3.

Solution : Solution



1. a

(5m)2 (

3m3)

Raise 5m to the second power. 52m2 · 3m3

Simplify. 25m2 · 3m3

Use the Commutative Property. 25 · 3 ·m2 ·m3

Multiply the constants and add the exponents. 75m5

2. b (3x2y

)4(2xy2

)3Use the Product to a Power Property.

(34x8y4

) (23x3y6

)Simplify.

(81x8y4

) (8x3y6

)Use the Commutative Property. 81 · 8 · x8 · x3 · y4 · y6

Multiply the constants and add the exponents. 648x11y10

note:


Simplify: a(5n)2 (

3n10)b(c4d2

)5(3cd5

)4.

note:


Simplify: a(a3b2

)6(4ab3

)4b(2x)

3 (5x7

).

6 Multiply Monomials

Since amonomial is an algebraic expression, we can use the properties of exponents to multiply monomials.

Example 11Multiply:

(3x2

) (−4x3

).

Solution : Solution (3x2

) (−4x3

)Use the Commutative Property to rearrange the terms. 3 · (−4) · x2 · x3

Multiply. −12x5

note:


Multiply:(5y7

) (−7y4

).



note:


Multiply:(−6b4

) (−9b5

).

Example 12Multiply:

(56x

3y) (

12xy2).

Solution : Solution (56x

3y) (

12xy2)

Use the Commutative Property to rearrange the terms. 56 · 12 · x

3 · x · y · y2

Multiply. 10x4y3

note:


Multiply:(25a

4b3) (

15ab3).

note:


Multiply:(23r

5s) (

12r6s7).

note: Access these online resources for additional instruction and practice with using multiplica-tion properties of exponents:

• Multiplication Properties of Exponents1

7 Key Concepts

� Exponential Notation

� Properties of Exponents

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

Product Property am · an = am+n

Power Property (am)n

= am·n

Product to a Power (ab)m

= ambm

1https://openstax.org/l/25MultiPropExp



8

8.1 Practice Makes Perfect

Simplify Expressions with ExponentsIn the following exercises, simplify each expression with exponents.

Exercise 37

a 35

b 91

c(13

)2d (0.2)

4


a 104

b 171

c(29

)2d (0.5)

3

Exercise 39

a 26

b 141

c(25

)3d (0.7)

2


a 83

b 81

c(34

)3d (0.4)

3

Exercise 41

a (−6)4b −64


a (−2)6b −26

Exercise 43

a −(14

)4b(− 1

4

)4Exercise 44 (Solution on p. 22.)

a −(23

)2b(− 2

3

)2



Exercise 45

a −0.52b (−0.5)2


a −0.14b (−0.1)4

Simplify Expressions Using the Product Property for ExponentsIn the following exercises, simplify each expression using the Product Property for Exponents.

Exercise 47d3 · d6


x4 · x2

Exercise 49n19 · n12


q27 · q15

Exercise 51a 45 · 49 b 89 · 8


a 310 · 36 b 5 · 54

Exercise 53a y · y3 b z25 · z8


a w5 · w b u41 · u53

Exercise 55w · w2 · w3


y · y3 · y5

Exercise 57a4 · a3 · a9


c5 · c11 · c2

Exercise 59mx ·m3


ny · n2

Exercise 61ya · yb


xp · xq

Simplify Expressions Using the Power Property for ExponentsIn the following exercises, simplify each expression using the Power Property for Exponents.

Exercise 63a(m4

)2b(103

)6http://cnx.org/content/m60215/1.2/



a(b2)7

b(38)2

Exercise 65a(y3)x

b (5x)y


a(x2

)yb (7a)

b

Simplify Expressions Using the Product to a Power PropertyIn the following exercises, simplify each expression using the Product to a Power Property.

Exercise 67a (6a)

2b (3xy)

2


a (5x)2b (4ab)

2

Exercise 69a (−4m)

3b (5ab)

3


a (−7n)3 b (3xyz)4

Simplify Expressions by Applying Several PropertiesIn the following exercises, simplify each expression.

Exercise 71

a(y2)4 · (y3)2

b(10a2b

)3Exercise 72 (Solution on p. 22.)

a(w4

)3 · (w5)2

b(2xy4

)5Exercise 73

a(−2r3s2

)4b(m5

)3 · (m9)4


a(−10q2p4

)3b(n3

)10 · (n5)2

Exercise 75

a (3x)2(5x)

b(5t2

)3(3t)

2


a (2y)3(6y)

b(10k4

)3(5k6

)2



Exercise 77

a (5a)2(2a)

3

b(12y

2)3( 2

3y)2


a (4b)2(3b)

3

b(12j

2)5( 2

5j3)2

Exercise 79

a(25x

2y)3

b(89xy

4)2


a(2r2

)3(4r)

2

b(3x3

)3(x5

)4Exercise 81

a(m2n

)2(2mn5

)4b(3pq4

)2(6p6q

)2Multiply Monomials

In the following exercises, multiply the monomials.

Exercise 82 (Solution on p. 22.)(6y7

) (−3y4

)Exercise 83(−10x5

) (−3x3

)Exercise 84 (Solution on p. 22.)(−8u6

)(−9u)

Exercise 85(−6c4

)(−12c)

Exercise 86 (Solution on p. 22.)(15f

8) (

20f3)

Exercise 87(14d

5) (

36d2)

Exercise 88 (Solution on p. 23.)(4a3b

) (9a2b6

)Exercise 89(

6m4n3) (

7mn5)

Exercise 90 (Solution on p. 23.)(47rs

2) (

14rs3)

Exercise 91(58x

3y) (

24x5y)

Exercise 92 (Solution on p. 23.)(23x

2y) (

34xy

2)



Exercise 93(35m

3n2) (

59m

2n3)

Mixed PracticeIn the following exercises, simplify each expression.

Exercise 94 (Solution on p. 23.)(x2

)4 · (x3)2

Exercise 95(y4)3 · (y5)2

Exercise 96 (Solution on p. 23.)(a2)6 · (a3)8

Exercise 97(b7)5 · (b2)6

Exercise 98 (Solution on p. 23.)(2m6

)3Exercise 99(

3y2)4

Exercise 100 (Solution on p. 23.)(10x2y

)3Exercise 101(

2mn4)5

Exercise 102 (Solution on p. 23.)(−2a3b2

)4Exercise 103(−10u2v4

)3Exercise 104 (Solution on p. 23.)(

23x

2y)3

Exercise 105(79pq

4)2

Exercise 106 (Solution on p. 23.)(8a3

)2(2a)

4

Exercise 107(5r2

)3(3r)

2

Exercise 108 (Solution on p. 23.)(10p4

)3(5p6

)2Exercise 109(

4x3)3(

2x5)4

Exercise 110 (Solution on p. 23.)(12x

2y3)4(

4x5y3)2

Exercise 111(13m

3n2)4(

9m8n3)2

Exercise 112 (Solution on p. 23.)(3m2n

)2(2mn5

)4Exercise 113(

2pq4)3(

5p6q)2



8.2 Everyday Math


Email Kate emails a �yer to ten of her friends and tells them to forward it to ten of their friends,who forward it to ten of their friends, and so on. The number of people who receive the email onthe second round is 102, on the third round is 103, as shown in the table below. How many peoplewill receive the email on the sixth round? Simplify the expression to show the number of peoplewho receive the email.

Round Number of people

1 10

2 102

3 103

. . . . . .

6 ?

Table 13

Exercise 115Salary Jamal's boss gives him a 3% raise every year on his birthday. This means that each year,Jamal's salary is 1.03 times his last year's salary. If his original salary was $35,000, his salary after1 year was $35, 000 (1.03), after 2 years was $35, 000(1.03)

2, after 3 years was $35, 000(1.03)

3, as

shown in the table below. What will Jamal's salary be after 10 years? Simplify the expression, toshow Jamal's salary in dollars.

Year Salary

1 $35, 000 (1.03)

2 $35, 000(1.03)2

3 $35, 000(1.03)3

. . . . . .

10 ?

Table 14


Clearance A department store is clearing out merchandise in order to make room for new inven-tory. The plan is to mark down items by 30% each week. This means that each week the cost ofan item is 70% of the previous week's cost. If the original cost of a sofa was $1,000, the cost forthe �rst week would be $1, 000 (0.70) and the cost of the item during the second week would be

$1, 000(0.70)2. Complete the table shown below. What will be the cost of the sofa during the �fth

week? Simplify the expression, to show the cost in dollars.



Week Cost

1 $1, 000 (0.70)

2 $1, 000(0.70)2

3

. . . . . .

8 ?

Table 15

Exercise 117Depreciation Once a new car is driven away from the dealer, it begins to lose value. Each year,a car loses 10% of its value. This means that each year the value of a car is 90% of the previousyear's value. If a new car was purchased for $20,000, the value at the end of the �rst year would be$20, 000 (0.90) and the value of the car after the end of the second year would be $20, 000(0.90)

2.

Complete the table shown below. What will be the value of the car at the end of the eighth year?Simplify the expression, to show the value in dollars.

Week Cost

1 $20, 000 (0.90)

2 $20, 000(0.90)2

3

4 . . .

5 ?

Table 16

8.3 Writing Exercises


Use the Product Property for Exponents to explain why x · x = x2.

Exercise 119Explain why −53 = (−5)3 but −54 6= (−5)4.Exercise 120 (Solution on p. 23.)

Jorge thinks(12

)2is 1. What is wrong with his reasoning?

Exercise 121Explain why x3 · x5 is x8, and not x15.



8.4 Self Check

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

b Af-ter reviewing this checklist, what will you do to become con�dent for all goals?



Solutions to Exercises in this Module

Solution to Exercise (p. 2)a 216 b15c 9

49d 0.1849Solution to Exercise (p. 2)a32b 21 c 8

125d0.047524Solution to Exercise (p. 3)a 81 b −81Solution to Exercise (p. 3)a 169 b −169Solution to Exercise (p. 4)b17

Solution to Exercise (p. 4)x16

Solution to Exercise (p. 5)a 56 b 418


Solution to Exercise (p. 6)a p6 b y43

Solution to Exercise (p. 6)a z8 b b49


Solution to Exercise (p. 6)b19

Solution to Exercise (p. 8)a b35 b 512

Solution to Exercise (p. 8)a z54 b 349

Solution to Exercise (p. 9)a 144y2 b 32w5x5

Solution to Exercise (p. 9)a 125w3x3 b −27y3Solution to Exercise (p. 10)a a48 b −8c12d6Solution to Exercise (p. 10)a 81x24y28 b q29

Solution to Exercise (p. 11)a 75n12 b 81c24d30

Solution to Exercise (p. 11)a 256a22b24 b 40x10

Solution to Exercise (p. 11)−35y11Solution to Exercise (p. 12)54b9

Solution to Exercise (p. 12)6a5b6

Solution to Exercise (p. 12)8r11s8



Solution to Exercise (p. 13)a 10,000 b 17 c 4

81d 0.125Solution to Exercise (p. 13)a 512 b 8 c 2764d 0.064Solution to Exercise (p. 13)a 64 b−64Solution to Exercise (p. 13)a − 4

9b 4

9Solution to Exercise (p. 14)a−0.001b 0.001Solution to Exercise (p. 14)x6

Solution to Exercise (p. 14)q42


Solution to Exercise (p. 14)a w6 b u94

Solution to Exercise (p. 14)y9

Solution to Exercise (p. 14)c18

Solution to Exercise (p. 14)ny+2

Solution to Exercise (p. 14)xp+q

Solution to Exercise (p. 15)a b14 b 316

Solution to Exercise (p. 15)a x2y b 7ab

Solution to Exercise (p. 15)a 25x2 b 16a2b2

Solution to Exercise (p. 15)a −343n3 b 81x4y4z4

Solution to Exercise (p. 15)a w22 b 32x5y20

Solution to Exercise (p. 15)a −1000q6p12 b n40

Solution to Exercise (p. 15)a 48y4 b 25, 000k24

Solution to Exercise (p. 16)a 432b5 b 1

200j16

Solution to Exercise (p. 16)a 128r8 b 1

200j16

Solution to Exercise (p. 16)−18y11Solution to Exercise (p. 16)72u7



Solution to Exercise (p. 16)4f11


Solution to Exercise (p. 16)8r2s5

Solution to Exercise (p. 16)12x

3y3


Solution to Exercise (p. 17)a36

Solution to Exercise (p. 17)8m18

Solution to Exercise (p. 17)1000x6y3


Solution to Exercise (p. 17)827x

6y3

Solution to Exercise (p. 17)1024a10

Solution to Exercise (p. 17)25000p24

Solution to Exercise (p. 17)x18y18

Solution to Exercise (p. 17)144m8n22

Solution to Exercise (p. 18)1, 000, 000Solution to Exercise (p. 18)$168.07Solution to Exercise (p. 19)Answers will vary.Solution to Exercise (p. 19)Answers will vary.


Use Multiplication Properties of Exponents* - OpenStax CNX

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