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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply

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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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