<< Chapter < Page Chapter >> Page >

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

98 n 14

Got questions? Get instant answers now!

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

48 m 5

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Simplify: ( u 3 v 2 ) 5 ( 4 u v 4 ) 3 .

64 u 18 v 22

Got questions? Get instant answers now!

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

675 x 7 y 18

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Multiply: ( 7 x 7 ) ( 8 x 4 ) .

−56 x 11

Got questions? Get instant answers now!

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

54 y 9

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Multiply: ( 4 5 m 4 n 3 ) ( 15 m n 3 ) .

12 m 5 n 6

Got questions? Get instant answers now!

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

12 p 11 q 8

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 4 x y z ) 4

256 x 4 y 4 z 4

Got questions? Get instant answers now!

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

( x 2 ) 4 · ( x 3 ) 2

x 14

Got questions? Get instant answers now!

( a 2 ) 6 · ( a 3 ) 8

a 36

Got questions? Get instant answers now!

( 3 x ) 2 ( 5 x )

45 x 3

Got questions? Get instant answers now!

( 5 a ) 2 ( 2 a ) 3

200 a 5

Got questions? Get instant answers now!

( 10 x 2 y ) 3

1,000 x 6 y 3

Got questions? Get instant answers now!

( −2 a 3 b 2 ) 4

16 a 12 b 8

Got questions? Get instant answers now!

( 2 3 x 2 y ) 3

8 27 x 6 y 3

Got questions? Get instant answers now!

( 8 a 3 ) 2 ( 2 a ) 4

1,024 a 10

Got questions? Get instant answers now!

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

25,000 p 24

Got questions? Get instant answers now!

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

x 18 y 18

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

144 m 8 n 22

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Multiply Monomials

In the following exercises, multiply the following monomials.

( 12 x 2 ) ( −5 x 4 )

−60 x 6

Got questions? Get instant answers now!

( −8 u 6 ) ( −9 u )

72 u 7

Got questions? Get instant answers now!

( −6 c 4 ) ( −12 c )

Got questions? Get instant answers now!

( 1 5 r 8 ) ( 20 r 3 )

4 r 11

Got questions? Get instant answers now!

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

36 a 5 b 7

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

8 x 2 y 5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

1 2 x 3 y 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 ?
Got questions? Get instant answers now!

Writing exercises

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

Answers will vary.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Answers will vary.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Prealgebra' conversation and receive update notifications?

Ask