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Find the quotient: −72 a 4 b 5 −8 a 9 b 5 .

9 a 5

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Find the quotient: 24 a 5 b 3 48 a b 4 .

Solution

24 a 5 b 3 48 a b 4
Use fraction multiplication. 24 48 a 5 a b 3 b 4
Simplify and use the Quotient Property. 1 2 a 4 1 b
Multiply. a 4 2 b
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Find the quotient: 16 a 7 b 6 24 a b 8 .

2 a 6 3 b 2

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Find the quotient: 27 p 4 q 7 −45 p 12 q .

3 q 6 5 p 8

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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.

Find the quotient: 14 x 7 y 12 21 x 11 y 6 .

Solution

14 x 7 y 12 21 x 11 y 6
Simplify and use the Quotient Property. 2 y 6 3 x 4

Be very careful to simplify 14 21 by dividing out a common factor, and to simplify the variables by subtracting their exponents.

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Find the quotient: 28 x 5 y 14 49 x 9 y 12 .

4 y 2 7 x 4

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Find the quotient: 30 m 5 n 11 48 m 10 n 14 .

5 8 m 5 n 3

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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.

Find the quotient: ( 3 x 3 y 2 ) ( 10 x 2 y 3 ) 6 x 4 y 5 .

Solution

Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.

( 3 x 3 y 2 ) ( 10 x 2 y 3 ) 6 x 4 y 5
Simplify the numerator. 30 x 5 y 5 6 x 4 y 5
Simplify, using the Quotient Rule. 5 x
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Find the quotient: ( 3 x 4 y 5 ) ( 8 x 2 y 5 ) 12 x 5 y 8 .

2 xy 2

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Find the quotient: ( −6 a 6 b 9 ) ( −8 a 5 b 8 ) −12 a 10 b 12 .

−4 ab 5

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Key concepts

  • Equivalent Fractions Property
    • If a , b , c are whole numbers where b 0 , c 0 , then
      a b = a · c b · c and a · c b · c = a b
  • Zero Exponent
    • If a is a non-zero number, then a 0 = 1 .
    • Any nonzero number raised to the zero power is 1 .
  • Quotient Property for Exponents
    • If a is a real number, a 0 , and m , n are whole numbers, then
      a m a n = a m n , m > n and a m a n = 1 a n m , n > m
  • Quotient to a Power Property for Exponents
    • If a and b are real numbers, b 0 , and m is a counting number, then
      ( a b ) m = a m b m
    • To raise a fraction to a power, raise the numerator and denominator to that power.

Practice makes perfect

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

  1. ( 10 p ) 0
  2. 10 p 0

  1. 1
  2. 10

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  1. ( −27 x 5 y ) 0
  2. −27 x 5 y 0

  1. 1
  2. −27 x 5

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  1. ( −92 y 8 z ) 0
  2. −92 y 8 z 0

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  1. 15 0
  2. 15 1

  1. 1
  2. 15

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Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

( a 4 · a 6 a 3 ) 2

a 14

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( x 3 ) 6 ( x 4 ) 7

1 x 10

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( 2 r 3 5 s ) 4

16 r 12 625 s 4

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( 3 y 2 · y 5 y 15 · y 8 ) 0

1

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( 15 z 4 · z 9 0.3 z 2 ) 0

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( r 2 ) 5 ( r 4 ) 2 ( r 3 ) 7

1 r 3

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( p 4 ) 2 ( p 3 ) 5 ( p 2 ) 9

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( 3 x 4 ) 3 ( 2 x 3 ) 2 ( 6 x 5 ) 2

3 x 8

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( −2 y 3 ) 4 ( 3 y 4 ) 2 ( −6 y 3 ) 2

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Divide Monomials

In the following exercises, divide the monomials.

36 x 3 ÷ ( −2 x 9 )

−18 x 6

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−35 x 7 −42 x 13

5 6 x 6

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18 r 5 s 3 r 3 s 9

6 r 2 s 8

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−12 x 4 y 9 15 x 6 y 3

4 y 6 5 x 2

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48 x 11 y 9 z 3 36 x 6 y 8 z 5

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64 x 5 y 9 z 7 48 x 7 y 12 z 6

4 z 3 x 2 y 3

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( 10 u 2 v ) ( 4 u 3 v 6 ) 5 u 9 v 2

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( 6 m 2 n ) ( 5 m 4 n 3 ) 3 m 10 n 2

10 n 2 m 4

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( 6 a 4 b 3 ) ( 4 a b 5 ) ( 12 a 8 b ) ( a 3 b )

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( 4 u 5 v 4 ) ( 15 u 8 v ) ( 12 u 3 v ) ( u 6 v )

5 u 4 v 3

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Mixed practice

  1. 24 a 5 + 2 a 5
  2. 24 a 5 2 a 5
  3. 24 a 5 2 a 5
  4. 24 a 5 ÷ 2 a 5

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  1. 15 n 10 + 3 n 10
  2. 15 n 10 3 n 10
  3. 15 n 10 3 n 10
  4. 15 n 10 ÷ 3 n 10

  1. 18 n 10
  2. 12 n 10
  3. 45 n 20
  4. 5

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  1. p 4 p 6
  2. ( p 4 ) 6

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  1. q 5 q 3
  2. ( q 5 ) 3

  1. q 8
  2. q 15

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  1. z 6 z 5
  2. z 5 z 6

  1. z
  2. 1 z

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( 8 x 5 ) ( 9 x ) ÷ 6 x 3

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( 4 y 5 ) ( 12 y 7 ) ÷ 8 y 2

6 y 6

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27 a 7 3 a 3 + 54 a 9 9 a 5

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32 c 11 4 c 5 + 42 c 9 6 c 3

15 c 6

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32 y 5 8 y 2 60 y 10 5 y 7

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48 x 6 6 x 4 35 x 9 7 x 7

3 x 2

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63 r 6 s 3 9 r 4 s 2 72 r 2 s 2 6 s

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56 y 4 z 5 7 y 3 z 3 45 y 2 z 2 5 y

y z 2

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Everyday math

Memory One megabyte is approximately 10 6 bytes. One gigabyte is approximately 10 9 bytes. How many megabytes are in one gigabyte?

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Memory One megabyte is approximately 10 6 bytes. One terabyte is approximately 10 12 bytes. How many megabytes are in one terabyte?

1,000,000

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Writing exercises

Vic thinks the quotient x 20 x 4 simplifies to x 5 . What is wrong with his reasoning?

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Mai simplifies the quotient y 3 y by writing y 3 y = 3 . What is wrong with her reasoning?

Answers will vary.

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When Dimple simplified 3 0 and ( −3 ) 0 she got the same answer. Explain how using the Order of Operations correctly gives different answers.

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Roxie thinks n 0 simplifies to 0 . What would you say to convince Roxie she is wrong?

Answers will vary.

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Self check

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

.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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
Practice Key Terms 1

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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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