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Simplify: ( 5 n ) 2 ( 3 n 10 ) ( c 4 d 2 ) 5 ( 3 c d 5 ) 4 .

75 n 12 81 c 24 d 30

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Simplify: ( a 3 b 2 ) 6 ( 4 a b 3 ) 4 ( 2 x ) 3 ( 5 x 7 ) .

256 a 22 b 24 40 x 10

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Multiply monomials

Since a monomial    is an algebraic expression, we can use the properties of exponents to multiply monomials.

Multiply: ( 3 x 2 ) ( −4 x 3 ) .

Solution

( 3 x 2 ) ( −4 x 3 ) Use the Commutative Property to rearrange the terms. 3 · ( −4 ) · x 2 · x 3 Multiply. −12 x 5

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Multiply: ( 5 y 7 ) ( −7 y 4 ) .

−35 y 11

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Multiply: ( −6 b 4 ) ( −9 b 5 ) .

54 b 9

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Multiply: ( 5 6 x 3 y ) ( 12 x y 2 ) .

Solution

( 5 6 x 3 y ) ( 12 x y 2 ) Use the Commutative Property to rearrange the terms. 5 6 · 12 · x 3 · x · y · y 2 Multiply. 10 x 4 y 3

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Multiply: ( 2 5 a 4 b 3 ) ( 15 a b 3 ) .

6 a 5 b 6

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Multiply: ( 2 3 r 5 s ) ( 12 r 6 s 7 ) .

8 r 11 s 8

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Access these online resources for additional instruction and practice with using multiplication properties of exponents:

Key concepts

  • Exponential Notation
    This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m powder means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.
  • Properties of Exponents
    • 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

Practice makes perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.


3 5
9 1
( 1 3 ) 2
( 0.2 ) 4

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10 4
17 1
( 2 9 ) 2
( 0.5 ) 3

10,000 17 4 81 0.125

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2 6
14 1
( 2 5 ) 3
( 0.7 ) 2

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8 3
8 1
( 3 4 ) 3
( 0.4 ) 3

512 8 27 64
0.064

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( −6 ) 4
6 4

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( −2 ) 6
2 6

64 −64

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( 1 4 ) 4
( 1 4 ) 4

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( 2 3 ) 2
( 2 3 ) 2

4 9
4 9

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0.5 2
( −0.5 ) 2

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0.1 4
( −0.1 ) 4

−0.001 0.001

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Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

4 5 · 4 9 8 9 · 8

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3 10 · 3 6 5 · 5 4

3 16 5 5

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y · y 3 z 25 · z 8

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w 5 · w u 41 · u 53

w 6 u 94

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Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression using the Power Property for Exponents.

( m 4 ) 2 ( 10 3 ) 6

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( b 2 ) 7 ( 3 8 ) 2

b 14 3 16

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( y 3 ) x ( 5 x ) y

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( x 2 ) y ( 7 a ) b

x 2 y 7 a b

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Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

( 6 a ) 2 ( 3 x y ) 2

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( 5 x ) 2 ( 4 a b ) 2

25 x 2 16 a 2 b 2

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( −4 m ) 3 ( 5 a b ) 3

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( −7 n ) 3 ( 3 x y z ) 4

−343 n 3 81 x 4 y 4 z 4

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Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.


( y 2 ) 4 · ( y 3 ) 2
( 10 a 2 b ) 3

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( w 4 ) 3 · ( w 5 ) 2
( 2 x y 4 ) 5

w 22 32 x 5 y 20

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( −2 r 3 s 2 ) 4
( m 5 ) 3 · ( m 9 ) 4

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( −10 q 2 p 4 ) 3
( n 3 ) 10 · ( n 5 ) 2

−1000 q 6 p 12 n 40

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( 3 x ) 2 ( 5 x )
( 5 t 2 ) 3 ( 3 t ) 2

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( 2 y ) 3 ( 6 y )
( 10 k 4 ) 3 ( 5 k 6 ) 2

48 y 4 25,000 k 24

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( 5 a ) 2 ( 2 a ) 3
( 1 2 y 2 ) 3 ( 2 3 y ) 2

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( 4 b ) 2 ( 3 b ) 3
( 1 2 j 2 ) 5 ( 2 5 j 3 ) 2

432 b 5 1 200 j 16

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( 2 5 x 2 y ) 3
( 8 9 x y 4 ) 2

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( 2 r 2 ) 3 ( 4 r ) 2
( 3 x 3 ) 3 ( x 5 ) 4

128 r 8 1 200 j 16

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

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Multiply Monomials

In the following exercises, multiply the monomials.

( 6 y 7 ) ( −3 y 4 )

−18 y 11

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( −10 x 5 ) ( −3 x 3 )

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( −8 u 6 ) ( −9 u )

72 u 7

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( −6 c 4 ) ( −12 c )

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( 1 5 f 8 ) ( 20 f 3 )

4 f 11

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

36 a 5 b 7

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

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( 4 7 r s 2 ) ( 14 r s 3 )

8 r 2 s 5

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( 5 8 x 3 y ) ( 24 x 5 y )

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

1 2 x 3 y 3

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

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Mixed Practice

In the following exercises, simplify each expression.

( x 2 ) 4 · ( x 3 ) 2

x 14

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( a 2 ) 6 · ( a 3 ) 8

a 36

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( 10 x 2 y ) 3

1000 x 6 y 3

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( −2 a 3 b 2 ) 4

16 a 12 b 8

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( 2 3 x 2 y ) 3

8 27 x 6 y 3

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( 8 a 3 ) 2 ( 2 a ) 4

1024 a 10

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

25000 p 24

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

x 18 y 18

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

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

144 m 8 n 22

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

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Everyday math

Email Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is 10 2 , on the third round is 10 3 , as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.

Round Number of people
1 10
2 10 2
3 10 3
6 ?

1,000,000

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Salary Jamal’s boss gives him a 3% raise every year on his birthday. This means that each year, Jamal’s salary is 1.03 times his last year’s salary. If his original salary was $35,000, his salary after 1 year was $ 35,000 ( 1.03 ) , after 2 years was $ 35,000 ( 1.03 ) 2 , after 3 years was $ 35,000 ( 1.03 ) 3 , as shown in the table below. What will Jamal’s salary be after 10 years? Simplify the expression, to show Jamal’s salary in dollars.

Year Salary
1 $ 35,000 ( 1.03 )
2 $ 35,000 ( 1.03 ) 2
3 $ 35,000 ( 1.03 ) 3
10 ?
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Clearance A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by 30% each week. This means that each week the cost of an item is 70% of the previous week’s cost. If the original cost of a sofa was $1,000, the cost for the first week would be $ 1,000 ( 0.70 ) and the cost of the item during the second week would be $ 1,000 ( 0.70 ) 2 . Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.

Week Cost
1 $ 1,000 ( 0.70 )
2 $ 1,000 ( 0.70 ) 2
3
8 ?

$168.07

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Depreciation Once a new car is driven away from the dealer, it begins to lose value. Each year, a car loses 10% of its value. This means that each year the value of a car is 90% of the previous year’s value. If a new car was purchased for $20,000, the value at the end of the first year would be $ 20,000 ( 0.90 ) and the value of the car after the end of the second year would be $ 20,000 ( 0.90 ) 2 . Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.

Week Cost
1 $ 20,000 ( 0.90 )
2 $ 20,000 ( 0.90 ) 2
3
4
5 ?
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Writing exercises

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

Answers will vary.

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

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

Answers will vary.

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

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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 seven 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 with exponents,” “simplify expressions using the Product Property for Exponents,” “simplify expressions using the Power Property for Exponents,” “simplify expressions using the Product to a Power Property,” “simplify expressions by applying several properties,” and “multiply monomials.” The rest of the cells are blank.

After reviewing this checklist, what will you do to become confident for all goals?

Questions & Answers

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
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin Reply
y=2×+9
Jacob Reply
Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?
Hector Reply
Mayra has $124 in her checking account. She writes a check for $152. What is the New Balance in her checking account?
REVOLUTION Reply

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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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