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Section 6.2 Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

( 10 x y z ) 3

1000 x 3 y 3 z 3

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Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

( 4 a 3 b 2 ) 3

64 a 9 b 6

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( 2 q 3 ) 4 ( 3 q ) 2

48 q 14

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( 1 3 x 2 ) 2 ( 1 2 x ) 3

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( 2 5 m 2 n ) 3

8 125 m 6 n 3

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Multiply Monomials

In the following exercises 8, multiply the monomials.

( −9 n 7 ) ( −16 n )

144 n 8

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( 7 p 5 q 3 ) ( 8 p q 9 )

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( 5 9 a b 2 ) ( 27 a b 3 )

15 a 2 b 5

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Section 6.3 Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

−4 ( y + 13 )

−4 y 52

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−6 u ( 2 u + 7 )

−12 u 2 42 u

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3 q 2 ( q 2 7 q + 6 ) 3

3 q 4 21 q 3 + 18 q 2

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( b 4 ) · 11

11 b 44

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Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using: the Distributive Property, the FOIL method, the Vertical Method.

( 6 y 7 ) ( 2 y 5 )

12 y 2 44 y + 35 12 y 2 44 y + 35 12 y 2 44 y + 35

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In the following exercises, multiply the binomials. Use any method.

( y 4 ) ( y 8 )

y 2 12 y + 32

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( q + 16 ) ( q 3 )

q 2 + 13 q 48

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( 5 m 8 ) ( 12 m + 1 )

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( u 2 + 6 ) ( u 2 5 )

u 4 + u 2 30

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( 9 x y ) ( 6 x 5 )

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( 8 m n + 3 ) ( 2 m n 1 )

16 m 2 n 2 2 m n 3

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Multiply a Trinomial by a Binomial

In the following exercises, multiply using the Distributive Property, the Vertical Method.

( n + 1 ) ( n 2 + 5 n 2 )

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( 3 x 4 ) ( 6 x 2 + x 10 )

18 x 3 21 x 2 34 x + 40 18 x 3 21 x 2 34 x + 40

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In the following exercises, multiply. Use either method.

( y 2 ) ( y 2 8 y + 9 )

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( 7 m + 1 ) ( m 2 10 m 3 )

7 m 3 69 m 2 31 m 3

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Section 6.4 Special Products

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

( q 15 ) 2

q 2 30 q + 225

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( 8 u + 1 ) 2

64 u 2 + 16 u + 1

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( 4 a 3 b ) 2

16 a 2 24 a b + 9 b 2

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Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

( y + 2 5 ) ( y 2 5 )

y 2 4 25

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( 12 c + 13 ) ( 12 c 13 )

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( 6 r ) ( 6 + r )

36 r 2

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( u + 3 4 v ) ( u 3 4 v )

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( 5 p 4 4 q 3 ) ( 5 p 4 + 4 q 3 )

25 p 8 16 q 6

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Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

( 6 a + 11 ) ( 6 a 11 )

36 a 2 121

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( 5 x + y ) ( x 5 y )

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( c 4 + 9 d ) 2

c 8 + 18 c 4 d + 81 d 2

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( p 5 + q 5 ) ( p 5 q 5 )

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( a 2 + 4 b ) ( 4 a b 2 )

4 a 3 + 3 a 2 b 4 b 3

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Section 6.5 Divide Monomials

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

( 12 0 ) ( −12 ) 0

1

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( 19 n ) 0 ( 25 m ) 0

0

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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.

( 3 x 4 2 y 2 ) 5

343 x 20 32 y 10

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( 3 n 2 ) 4 ( −5 n 4 ) 3 ( −2 n 5 ) 2

10,125 n 10 4

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Divide Monomials

In the following exercises, divide the monomials.

64 a 5 b 9 −16 a 10 b 3

4 b 6 a 5

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144 x 15 y 8 z 3 18 x 10 y 2 z 12

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( 8 p 6 q 2 ) ( 9 p 3 q 5 ) 16 p 8 q 7

9 p 2

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Section 6.6 Divide Polynomials

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

( 35 x 2 75 x ) ÷ 5 x

7 x 15

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550 p 6 300 p 4 10 p 3

55 p 3 30 p

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( 63 x y 3 + 56 x 2 y 4 ) ÷ ( 7 x y )

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96 a 5 b 2 48 a 4 b 3 56 a 2 b 4 8 a b 2

12 a 4 6 a 3 b 7 a b 2

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57 m 2 12 m + 1 −3 m

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105 y 5 + 50 y 3 5 y 5 y 3

21 y 2 + 10 1 y 2

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Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

( k 2 2 k 99 ) ÷ ( k + 9 )

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( v 2 16 v + 64 ) ÷ ( v 8 )

v 8

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( 3 x 2 8 x 35 ) ÷ ( x 5 )

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( n 2 3 n 14 ) ÷ ( n + 3 )

n 6 + 4 n + 3

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( 4 m 3 + m 5 ) ÷ ( m 1 )

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( u 3 8 ) ÷ ( u 2 )

u 2 + 2 u + 4

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Section 6.7 Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

( c −2 d ) ( c −3 d −2 )

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Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

The thickness of a dime is about 0.053 inches.

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In 2015, the population of the world was about 7,200,000,000 people.

7.2 × 10 9

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Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

1.5 × 10 10

15,000,000,000

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Multiply and Divide Using Scientific Notation

In the following exercises, multiply and write your answer in decimal form.

( 2 × 10 5 ) ( 4 × 10 −3 )

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( 3.5 × 10 −2 ) ( 6.2 × 10 −1 )

0.0217

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In the following exercises, divide and write your answer in decimal form.

9 × 10 −5 3 × 10 2

0.0000003

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Chapter practice test

For the polynomial 10 x 4 + 9 y 2 1
Is it a monomial, binomial, or trinomial?
What is its degree?

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In the following exercises, simplify each expression.

( 12 a 2 7 a + 4 ) + ( 3 a 2 + 8 a 10 )

15 a 2 + a 6

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( 9 p 2 5 p + 1 ) ( 2 p 2 6 )

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( 4 a 3 b 5 ) 2

16 a 6 b 10

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( −9 r 4 s 5 ) ( 4 r s 7 )

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3 k ( k 2 7 k + 13 )

3 k 3 21 k 2 + 39 k

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( v 9 ) ( 9 v 5 )

9 v 2 86 v + 45

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( 4 c 11 ) ( 3 c 8 )

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( n 6 ) ( n 2 5 n + 4 )

n 3 11 n 2 + 34 n 24

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( 2 x 15 y ) ( 5 x + 7 y )

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( 7 p 5 ) ( 7 p + 5 )

49 p 2 25

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( 87 x 15 y 3 z 22 ) 0

1

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12 x 2 + 42 x 6 2 x

6 x + 21 3 x

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( 70 x y 4 + 95 x 3 y ) ÷ 5 x y

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64 x 3 1 4 x 1

16 x 2 + 4 x + 1

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( y 2 5 y 18 ) ÷ ( y + 3 )

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Convert 83,000,000 to scientific notation.

8.3 × 10 7

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Convert 6.91 × 10 −5 to decimal form.

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In the following exercises, simplify, and write your answer in decimal form.

( 3.4 × 10 9 ) ( 2.2 × 10 −5 )

74,800

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8.4 × 10 −3 4 × 10 3

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A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial −16 t 2 + 1000 gives the height of the package t seconds a after it was dropped. Find the height when t = 6 seconds.

424 feet

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Questions & Answers

At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
Andrew Reply
divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny 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 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray 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
Practice Key Terms 2

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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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