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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

Equation in the form of a pending point y+2=1/6(×-4)
Jose Reply
definition of quadratic formula
Ahmed Reply
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
What Reply
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
Mckenzie Reply
x > $110,000
bruce
greater than $110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use?
nawal Reply
$1.38 worth of strawberries + $1.14 worth of blueberries which= $2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times $2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa
Vina has $4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
Mckenzie Reply
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
Crispy Reply
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce
Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs $3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound?
Vonna Reply
33.336
Robinson
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot?
Imaan Reply
Ivan has $8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have?
mikayla Reply
2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75
Melissa
John has $175 in $5 and $10 bills in his drawer. The number of $5 bills is three times the number of $10 bills. How many of each are in the drawer?
mikayla Reply
7-$10 21-$5
Robert
Enrique borrowed $23,500 to buy a car. He pays his uncle 2% interest on the $4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total $23,500? (Round your answer to the nearest tenth of a percent.)
Parker Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Markeice Reply
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert
what is the answer for this 3×9+28÷4-8
What Reply
315
lashonna
how do you do xsquard+7x+10=0
What
(x + 2)(x + 5), then set each factor to zero and solve for x. so, x = -2 and x = -5.
bruce
I skipped it
What
In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
Jenise Reply
how do i set this up
Jenise
25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil

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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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