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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

explain and give four Example hyperbolic function
Lukman Reply
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred 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
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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