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

2 x

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Divide monomials

You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.

Find the quotient: 56 x 7 ÷ 8 x 3 .

Solution

56 x 7 ÷ 8 x 3 Rewrite as a fraction. 56 x 7 8 x 3 Use fraction multiplication. 56 8 x 7 x 3 Simplify and use the Quotient Property. 7 x 4

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Find the quotient: 42 y 9 ÷ 6 y 3 .

7 y 6

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Find the quotient: 48 z 8 ÷ 8 z 2 .

6 z 6

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Find the quotient: 45 a 2 b 3 −5 a b 5 .

Solution

When we divide monomials with more than one variable, we write one fraction for each variable.

45 a 2 b 3 −5 a b 5 Use fraction multiplication. 45 −5 · a 2 a · b 3 b 5 Simplify and use the Quotient Property. −9 · a · 1 b 2 Multiply. 9 a b 2

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Find the quotient: −72 a 7 b 3 8 a 12 b 4 .

9 a 5 b

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Find the quotient: −63 c 8 d 3 7 c 12 d 2 .

−9 d c 4

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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

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

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

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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. This follows the order of operations. Remember, a fraction bar is a grouping symbol.

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

Solution

( 6 x 2 y 3 ) ( 5 x 3 y 2 ) ( 3 x 4 y 5 ) Simplify the numerator. 30 x 5 y 5 3 x 4 y 5 Simplify. 10 x

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Find the quotient: ( 6 a 4 b 5 ) ( 4 a 2 b 5 ) 12 a 5 b 8 .

2 a b 2

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Find the quotient: ( −12 x 6 y 9 ) ( −4 x 5 y 8 ) −12 x 10 y 12 .

−4 x y 5

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Access these online resources for additional instruction and practice with dividing monomials:

Key concepts

  • 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 m n , n > m
  • Zero Exponent
    • If a is a non-zero number, then a 0 = 1 .

  • 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.

  • Summary of Exponent Properties
    • If a , b are real numbers and m , n are whole numbers, then
      Product Property a m · a n = a m + n Power Property ( a m ) n = a m · n Product to a Power ( a b ) m = a m b m Quotient Property a m b m = a m n , a 0 , m > n a m a n = 1 a n m , a 0 , n > m Zero Exponent Definition a o = 1 , a 0 Quotient to a Power Property ( a b ) m = a m b m , b 0

Practice makes perfect

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

x 18 x 3 5 12 5 3

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y 20 y 10 7 16 7 2

y 10 7 14

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p 21 p 7 4 16 4 4

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u 24 u 3 9 15 9 5

u 21 9 10

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q 18 q 36 10 2 10 3

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t 10 t 40 8 3 8 5

1 t 30 1 64

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x x 7 10 10 3

1 x 6 1 100

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Simplify Expressions with Zero Exponents

In the following exercises, simplify.


13 0
k 0

1 1

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27 0
( 27 0 )

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15 0
( 15 0 )

−1 −1

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( 25 x ) 0
25 x 0

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( 6 y ) 0
6 y 0

1 6

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( 12 x ) 0
( −56 p 4 q 3 ) 0

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7 y 0 ( 17 y ) 0
( −93 c 7 d 15 ) 0

7 1

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12 n 0 18 m 0
( 12 n ) 0 ( 18 m ) 0

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15 r 0 22 s 0
( 15 r ) 0 ( 22 s ) 0

−7 0

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Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

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

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( 2 5 ) 2 ( x 3 ) 4 ( a b ) 5

4 25 x 4 81 a 5 b 5

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( a 3 b ) 4 ( 5 4 m ) 2

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( x 2 y ) 3 ( 10 3 q ) 4

x 3 8 y 3 10,000 81 q 4

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Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

( y 4 z 10 ) 5

y 20 z 50

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( 3 m 5 5 n ) 3

27 m 15 125 n 3

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( 5 u 7 2 v 3 ) 4

625 u 28 16 v 12

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( t 2 ) 5 ( t 4 ) 2 ( t 3 ) 7

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( q 3 ) 6 ( q 2 ) 3 ( q 4 ) 8

1 q 8

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( −2 p 2 ) 4 ( 3 p 4 ) 2 ( −6 p 3 ) 2

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( −2 k 3 ) 2 ( 6 k 2 ) 4 ( 9 k 4 ) 2

64 k 6

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

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( −10 n 2 ) 3 ( 4 n 5 ) 2 ( 2 n 8 ) 2

−4,000

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Divide Monomials

In the following exercises, divide the monomials.

−72 u 12 ÷ 1 2 u 4

−6 u 8

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45 a 6 b 8 −15 a 10 b 2

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54 x 9 y 3 −18 x 6 y 15

3 x 3 y 12

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20 m 8 n 4 30 m 5 n 9

−2 m 3 3 n 5

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18 a 4 b 8 −27 a 9 b 5

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45 x 5 y 9 −60 x 8 y 6

−3 y 3 4 x 3

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64 q 11 r 9 s 3 48 q 6 r 8 s 5

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65 a 10 b 8 c 5 42 a 7 b 6 c 8

65 a 3 b 2 42 c 3

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( 10 m 5 n 4 ) ( 5 m 3 n 6 ) 25 m 7 n 5

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( −18 p 4 q 7 ) ( −6 p 3 q 8 ) −36 p 12 q 10

−3 q 5 p 5

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( 6 a 4 b 3 ) ( 4 a b 5 ) ( 12 a 2 b ) ( a 3 b )

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( 4 u 2 v 5 ) ( 15 u 3 v ) ( 12 u 3 v ) ( u 4 v )

5 v 4 u 2

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Mixed Practice


24 a 5 + 2 a 5
24 a 5 2 a 5
24 a 5 · 2 a 5
24 a 5 ÷ 2 a 5

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15 n 10 + 3 n 10
15 n 10 3 n 10
15 n 10 · 3 n 10
15 n 10 ÷ 3 n 10

18 n 10
12 n 10
45 n 20
5

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p 4 · p 6
( p 4 ) 6

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q 5 · q 3
( q 5 ) 3

q 8
q 15

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z 6 z 5
z 5 z 6

z 1 z

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( 8 x 5 ) ( 9 x ) ÷ 6 x 3

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( 4 y ) ( 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 gigabyte is approximately 10 9 bytes. One terabyte is approximately 10 12 bytes. How many gigabytes are in one terabyte?

10 3

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Writing exercises

Jennifer thinks the quotient a 24 a 6 simplifies to a 4 . What is wrong with her reasoning?

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Maurice simplifies the quotient d 7 d by writing d 7 d = 7 . What is wrong with his reasoning?

Answers will vary.

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

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Robert thinks x 0 simplifies to 0. What would you say to convince Robert he 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.

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “simplify expressions using the Quotient Property for Exponents,” “simplify expressions with zero exponents,” “simplify expressions using the Quotient to a Power Property,” “simplify expressions by applying several properties,” and “divide monomials.” The rest of the cells are blank.

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

How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Kendra Reply
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
Malik Reply
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
Adda Reply
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Adda
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
Adda
(×+1)(×-4) = x^2-3×-4
Gee
yeah
Asamoah
Quadratic equations involving factorization
Winner Reply
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
Please help me solve this 10x+14=-2x+38
Erica
the previous equation should be 3x = 1/6 x=1/18
Sriram
for the new one 10x + 2x = 38 - 14
Sriram
12x = 24 x=2
Sriram
10x + 14 = -2x +38 10x + 2x = 38 - 14 12x = 24 divide both sides by the coefficient of x, which is 12 therefore × = 2
vida
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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