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Simplify: ( 7 n ) 2 ( 2 n 12 ) .

98 n 14

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Simplify: ( 4 m ) 2 ( 3 m 3 ) .

48 m 5

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Simplify: ( 3 p 2 q ) 4 ( 2 p q 2 ) 3 .

Solution

( 3 p 2 q ) 4 ( 2 p q 2 ) 3
Use the Power of a Product Property. 3 4 ( p 2 ) 4 q 4 · 2 3 p 3 ( q 2 ) 3
Use the Power Property. 81 p 8 q 4 · 8 p 3 q 6
Use the Commutative Property. 81 · 8 · p 8 · p 3 · q 4 · q 6
Multiply the constants and add the exponents for
each variable.
648 p 11 q 10
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Simplify: ( u 3 v 2 ) 5 ( 4 u v 4 ) 3 .

64 u 18 v 22

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Simplify: ( 5 x 2 y 3 ) 2 ( 3 x y 4 ) 3 .

675 x 7 y 18

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Multiply monomials

Since a monomial    is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

Multiply: ( 4 x 2 ) ( 5 x 3 ) .

Solution

( 4 x 2 ) ( 5 x 3 )
Use the Commutative Property to rearrange the factors. 4 · ( −5 ) · x 2 · x 3
Multiply. 20 x 5
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Multiply: ( 7 x 7 ) ( 8 x 4 ) .

−56 x 11

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Multiply: ( 9 y 4 ) ( 6 y 5 ) .

54 y 9

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Multiply: ( 3 4 c 3 d ) ( 12 c d 2 ) .

Solution

( 3 4 c 3 d ) ( 12 c d 2 )
Use the Commutative Property to rearrange
the factors.
3 4 · 12 · c 3 · c · d · d 2
Multiply. 9 c 4 d 3
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Multiply: ( 4 5 m 4 n 3 ) ( 15 m n 3 ) .

12 m 5 n 6

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Multiply: ( 2 3 p 5 q ) ( 18 p 6 q 7 ) .

12 p 11 q 8

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Access additional online resources

Key concepts

  • Exponential Notation On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.

    This is read a to the m th power.

  • Product Property of Exponents
    • If a is a real number and m , n are counting numbers, then
      a m · a n = a m + n
    • To multiply with like bases, add the exponents.
  • Power Property for Exponents
    • If a is a real number and m , n are counting numbers, then
      ( a m ) n = a m n
  • Product to a Power Property for Exponents
    • If a and b are real numbers and m is a whole number, then
      ( a b ) m = a m b m

Practice makes perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression using the Product Property of Exponents.

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression using the Power Property of Exponents .

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

( 6 m ) 3

−216 m 3

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( 4 x y z ) 4

256 x 4 y 4 z 4

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Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

( x 2 ) 4 · ( x 3 ) 2

x 14

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( a 2 ) 6 · ( a 3 ) 8

a 36

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( 3 x ) 2 ( 5 x )

45 x 3

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( 5 a ) 2 ( 2 a ) 3

200 a 5

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( 10 x 2 y ) 3

1,000 x 6 y 3

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( −2 a 3 b 2 ) 4

16 a 12 b 8

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( 2 3 x 2 y ) 3

8 27 x 6 y 3

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( 8 a 3 ) 2 ( 2 a ) 4

1,024 a 10

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( 10 p 4 ) 3 ( 5 p 6 ) 2

25,000 p 24

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( 1 2 x 2 y 3 ) 4 ( 4 x 5 y 3 ) 2

x 18 y 18

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( 1 3 m 3 n 2 ) 4 ( 9 m 8 n 3 ) 2

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

144 m 8 n 22

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( 2 p q 4 ) 3 ( 5 p 6 q ) 2

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Multiply Monomials

In the following exercises, multiply the following monomials.

( 12 x 2 ) ( −5 x 4 )

−60 x 6

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( −8 u 6 ) ( −9 u )

72 u 7

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( −6 c 4 ) ( −12 c )

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( 1 5 r 8 ) ( 20 r 3 )

4 r 11

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( 4 a 3 b ) ( 9 a 2 b 6 )

36 a 5 b 7

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( 6 m 4 n 3 ) ( 7 m n 5 )

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( 4 7 x y 2 ) ( 14 x y 3 )

8 x 2 y 5

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( 5 8 u 3 v ) ( 24 u 5 v )

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( 2 3 x 2 y ) ( 3 4 x y 2 )

1 2 x 3 y 3

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( 3 5 m 3 n 2 ) ( 5 9 m 2 n 3 )

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Everyday math

Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is 6 2 , on the third round is 6 3 , as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.

Round Number of people
1 6
2 6 2
3 6 3
8 ?

1,679,616

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Salary Raul’s boss gives him a 5% raise every year on his birthday. This means that each year, Raul’s salary is 1.05 times his last year’s salary. If his original salary was $40,000 , his salary after 1 year was $40,000 ( 1.05 ) , after 2 years was $40,000 ( 1.05 ) 2 , after 3 years was $40,000 ( 1.05 ) 3 , as shown in the table below. What will Raul’s salary be after 10 years? Simplify the expression, to show Raul’s salary in dollars.

Year Salary
1 $40,000 ( 1.05 )
2 $40,000 ( 1.05 ) 2
3 $40,000 ( 1.05 ) 3
10 ?
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Writing exercises

Use the Product Property for Exponents to explain why x · x = x 2 .

Answers will vary.

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Explain why −5 3 = ( −5 ) 3 but −5 4 ( −5 ) 4 .

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Jorge thinks ( 1 2 ) 2 is 1 . What is wrong with his reasoning?

Answers will vary.

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Explain why x 3 · x 5 is x 8 , and not x 15 .

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Self check

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

.

After reviewing this checklist, what will you do to become confident for all objectives?

Questions & Answers

what is mutation
Janga Reply
what is a cell
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Sifune
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mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
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Linsy
how many types of solid did we have
Samuel Reply
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What is Atoms
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urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
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how is urine formed in human
Rahma
what is the diference between a cavity and a canal
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malaria is caused by an insect called mosquito.
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Malaria is cause by female anopheles mosquito
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Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
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a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
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Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
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endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
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Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
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biology is the study of life.
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Biology is the study of how living organisms live and survive in a specific environment
Sifune
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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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