Answers (Anticipation Guide and Lesson 7-1)

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter 7

Before you begin Chapter 7

Polynomials

Anticipation Guide

DATE

PERIOD

A1

Statement

A

12. The product of (x + y) and (x - y) will always equal x 2 - y 2.

After you complete Chapter 7

D

11. The square of r + t, (r + t) 2, will always equal r 2 + t 2.

Glencoe Algebra 1

Answers

3

Glencoe Algebra 1

? For those statements that you mark with a D, use a piece of paper to write an

example of why you disagree.

? Did any of your opinions about the statements change from the first column?

Chapter 7

A

D

D

A

7. The sum of the two polynomials (3x 2y - 4xy 2 + 2y 3) and

(6xy 2 + 2x 2y - 7) in simplest form is 5x 2y + 2xy 2 + 2y 3 - 7.

8. (4m 2 + 2m - 3) - (m 2 - m + 3) is equal to 3m 2 + m.

9. Because there are different exponents in each factor, the

distributive property cannot be used to multiply 3n 3 by

(2n 2 + 4n - 12).

10. The FOIL method of multiplying two binomials stands for

First, Outer, Inner, Last.

D

D

6. The degree of the polynomial 3x 2y 3- 5y 2 + 8x 3 is 3 because the

highest exponent is 3.

5

23

is the same as ?

.

A

3

A

A

D

STEP 2

A or D

5. A polynomial may contain one or more monomials.

2

4. ?

(5)

3. To divide two powers that have the same base, subtract

the exponents.

1. When multiplying two powers that have the same base,

multiply the exponents.

2. (k 3)4 is equivalent to k 12.

? Reread each statement and complete the last column by entering an A or a D.

Step 2

STEP 1

A, D, or NS

? Write A or D in the first column OR if you are not sure whether you agree or

disagree, write NS (Not Sure).

? Decide whether you Agree (A) or Disagree (D) with the statement.

? Read each statement.

Step 1

7

NAME

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

DATE

PERIOD

and the variables

3

Chapter 7

a3b2c7

(5

)

1

13. (5a 2bc 3) ?

abc 4

4a3b4

1

10. ?

(2a 3b)(6b 3)

16a3

7. (2a2)(8a)

x7

4. x(x2)(x4)

y6

1. y(y5)

-20x3y5

5

14. (-5xy)(4x2)(y4)

20x10

11. (-4x3)(-5x7)

r2n6

8. (rs)(rn3)(n2)

m6

5. m ? m5

n9

2. n2 ? n7

Simplify.

Product of Powers

Simplify each expression.

Exercises

= (3 ? 5)(x6 + 2)

= 15x8

The product is 15x8.

? a n = a m + n.

Glencoe Algebra 1

-20x4y6z3

15. (10x3yz2)(-2xy5z)

-6j3k10

12. (-3j2k4)(2jk6)

4x3y4

9. (x2y)(4xy3)

x7

6. (-x3)(-x4)

-7x6

3. (-7x2)(x4)

Example 2

Simplify (-4a3b)(3a2b5).

(-4a3b)(3a2b5) = (-4)(3)(a3 ? a2)(b ? b5)

= -12(a3 + 2)(b1 + 5)

= -12a5b6

The product is -12a5b6.

For any number a and all integers m and n, am

Example 1

Simplify (3x6)(5x2).

(3x6)(5x2) = (3)(5)(x6 ? x2)

Group the coefficients

Product of Powers

A monomial is a number, a variable, or the product of a number and one or

more variables with nonnegative integer exponents. An expression of the form xn is called a

power and represents the product you obtain when x is used as a factor n times. To multiply

two powers that have the same base, add the exponents.

Multiplying Monomials

Study Guide and Intervention

Monomials

7-1

NAME

Answers (Anticipation Guide and Lesson 7-1)

Lesson 7-1

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter Resources

A2

Glencoe Algebra 1

Multiplying Monomials

Study Guide and Intervention

DATE

For any number a and all integers m and n, (ab)m = ambm.

Power of a Product

16x2b3

16a4b3

2

8. (4x)2(b3)

Chapter 7

12n12y10

16. (-2n6y5)(-6n3y2)(ny)3

625a8b5f2

1

13. (25a 2b) 3 ?

abf

6

-243a15n8

17. (-3a3n4)(-3a3n)4

-48x4y6

14. (2xy)2(-3x2)(4y4)

512x9y3

2a3b8

(5 )

11. (-4xy)3(-2x2)3

12

10. (2a3b2)(b3)2

-27a b

7. (4a2)2(b3)

-3a b

3

5. (-3ab4)3

n 28

2. (n )

7 4

12

3

4. -3(ab4)3

y 10

1. (y )

5 2

Simplify each expression.

Exercises

Power of a Power

Product of Powers

2 5

3

3

Glencoe Algebra 1

-768x14y2

18. -3(2x)4(4x5y)2

8x17y6z10

15. (2x3y2z2)3(x2z)4

72j10k9

12. (-3j2k3)2(2j2k)3

x10y20

9. (x2y4)5

64x b

6

6. (4x2b)3

x13

3. (x ) (x )

Group the coefficients and the variables

Power of a Product

Power of a Power

Simplify (-2ab2)3(a2)4.

(-2ab2)3(a2)4 = (-2ab2)3(a8)

= (-2)3(a3)(b2)3(a8)

= (-2)3(a3)(a8)(b2)3

= (-2)3(a11)(b2)3

= -8a11b6

The product is -8a11b6.

Example

We can combine and use these properties to simplify expressions involving monomials.

For any number a and all integers m and n, (am)n = amn.

Power of a Power

An expression of the form (xm)n is called a power of a power

and represents the product you obtain when xm is used as a factor n times. To find the

power of a power, multiply exponents.

(continued)

PERIOD

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter 7

Simplify Expressions

7-1

NAME

Multiplying Monomials

Skills Practice

DATE

PERIOD

8

11

22. (-3y)3 -27y3

24. (2b3c4)2 4b6c8

23. (3pr ) 9p r

x7

Chapter 7

25.

x5

x2

26.

c2d2

7

cd

cd

27.

GEOMETRY Express the area of each figure as a monomial.

2 2

20. (p3)12 p36

2 4

21. (-6p) 36p

2

9p3

4p

18p4

18. (-2c4d)(-4cd) 8c5d2

16. (7a5b2)(a2b3) 7a7b5

2

19. (102)3 106 or 1,000,000

3

3 5

17. (-5m )(3m ) -15m

3

15. (4xy )(3x y ) 12x y

13. (2x2)(3x5) 6x7

14. (5a7)(4a2) 20a9

12. (cd2)(c3d2) c4d4

4 8

10. (?2k2)(?3k) 5k3

6

4

11. (a2b4)(a2b2) a b

2

9. (y z)(yz ) y z

2

8. x(x2)(x7) x10

3 3

7. a2(a3)(a6) a11

Simplify.

6. 2a + 3b No; this is the sum of two monomials.

5. j3k Yes; this is the product of two variables.

4. y Yes; single variables are monomials.

p2

r

3. ?2 No; this is the quotient, not the product, of two variables.

Glencoe Algebra 1

2. a - b No; this is the difference, not the product, of two variables.

1. 11 Yes; 11 is a real number and an example of a constant.

Determine whether each expression is a monomial. Write yes or no. Explain.

7-1

NAME

Answers (Lesson 7-1)

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Lesson 7-1

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter 7

Multiplying Monomials

Practice

DATE

PERIOD

(

3

6

)

(3 )

4 2

?

p

9

)

2

1

16

? a 2d 6

8

A3

18a b

3

6a2b4

6

3ab2

16.

(25x )¦Ð

6

5x3

17.

27h6

3h2

3h2

3h2

19.

m4n5

m3n

mn3

n

20.

(63g4)¦Ð

7g2

Chapter 7

Glencoe Algebra 1

Answers

8

Glencoe Algebra 1

22. HOBBIES Tawa wants to increase her rock collection by a power of three this year and

then increase it again by a power of two next year. If she has 2 rocks now, how many

rocks will she have after the second year? 26 or 64

21. COUNTING A panel of four light switches can be set in 24 ways. A panel of five light

switches can set in twice this many ways. In how many ways can five light switches

be set? 25 or 32

18.

3g

12a3b4

GEOMETRY Express the volume of each solid as a monomial.

15.

4a2b

14. [(42)2]2 4 or 65,536

(4 )

1

12. ?

ad 3

10. (0.2a2b3)2 0.04a4b6

GEOMETRY Express the area of each figure as a monomial.

13. (0.4k3)3 0.064k9

2

11. ?

p

2

1

9. (-18m 2n) 2 - ?

mn 2 -54m5n4

(

6ab 3

4 4

8. (-xy)3(xz) -x4y3z

4

1 3

7. (-15xy 4) - ?

xy 5x2y7

3 2 2

4. (2ab f )(4a b f ) 8a b f

4

6. (4g3h)(-2g5) -8g8h

2

5. (3ad4)(-2a2) -6a3d4

2 2

2

3. (-5x y)(3x ) -15x y

6

Simplify each expression.

2

1

b 3c 2

2. ?

Yes; this is the product of a number, ?

, and two variables.

7b

21a 2

1. ?

No; this involves the quotient, not the product, of variables.

Determine whether each expression is a monomial. Write yes or no. Explain your

reasoning.

7-1

NAME

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Multiplying Monomials

3 ft

TTT

Chapter 7

If you then flip the coin two more times,

there are 23 ¡Á 22 outcomes that can

occur. How many outcomes can occur if

you flip the quarter as mentioned above

plus four more times? Write your answer

in the form 2x. 29

TTH

THT

HHT

THH

HTH

HTT

HHH

Outcomes

3. PROBABILITY If you flip a coin 3 times

in a row, there are 23 outcomes that can

occur.

x

2. CIVIL ENGINEERING A developer is

planning a sidewalk for a new

development. The sidewalk can be

installed in rectangular sections that

have a fixed width of 3 feet and a length

that can vary. Assuming that each

section is the same length, express the

area of a 4-section sidewalk as a

monomial. 12x

9

DATE

PERIOD

4

4.5

4.8

Women¡¯s

HTH

268

382

463

Volume (in3)

Glencoe Algebra 1

The power is one-fourth the

previous amount.

b. If the current is reduced by one half,

what happens to the power?

a. Find the power in a household circuit

that has 20 amperes of current and

5 ohms of resistance. 2000 watts

5. ELECTRICITY An electrician uses the

formula W = I2R , where W is the power

in watts, I is the current in amperes, and

R is the resistance in ohms.

Source: WikiAnswers

Radius (in.)

Ball

Child¡¯s

4. SPORTS The volume of a sphere is given

4 3

by the formula V = ?

¦Ðr , where r is the

3

radius of the sphere. Find the volume of

air in three different basketballs. Use

¦Ð = 3.14. Round your answers to the

nearest whole number.

Word Problem Practice

1. GRAVITY An egg that has been falling

for x seconds has dropped at an average

speed of 16x feet per second. If the egg is

dropped from the top of a building, its

total distance traveled is the product of

the average rate times the time. Write a

simplified expression to show the

distance the egg has traveled after x

seconds. 16x2

7-1

NAME

Answers (Lesson 7-1)

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Lesson 7-1

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

A4

Glencoe Algebra 1

Enrichment

PERIOD

2. 100002 16

3. 110000112 195

6. 11 10112

Chapter 7

10

8. 117 11101012

4. 101110012 185

69

70

71

72

73

74

75

76

77

G

H

I

J

K

L

M

68

D

F

67

C

E

65

66

A

B

Z

Y

X

W

V

U

T

S

R

Q

P

O

N

90

89

88

87

86

85

84

83

82

81

80

79

78

97

108

107

106

105

104

103

102

101

100

99

98

n

z

y

x

w

v

u

t

s

r

q

p

o

110

122

121

120

119

118

117

116

115

114

113

112

111

Glencoe Algebra 1

m 109

l

k

j

i

h

g

f

e

d

c

b

a

The American Standard Guide for

Information Interchange (ASCII)

7. 29 111012

9. The chart at the right shows a set of decimal

code numbers that is used widely in storing

letters of the alphabet in a computer¡¯s memory.

Find the code numbers for the letters of your

name. Then write the code for your name

using binary numbers. Answers will vary.

5. 8 10002

Write each decimal number as a binary number.

1. 11112 15

Find the decimal value of each binary number.

10011012 = 1 ¡Á 26 + 0 ¡Á 25 + 0 ¡Á 24+ 1 ¡Á 23 + 1 ¡Á 22 + 0 ¡Á 21 + 1 ¡Á 20

= 1 ¡Á 64 + 0 ¡Á 32 + 0 ¡Á 16 + 1 ¡Á 8 + 1 ¡Á 4 + 0 ¡Á 2 + 1 ¡Á 1

= 64 + 0

+

0 + 8 + 4 + 0 + 1

= 77

Digital computers store information as numbers. Because the electronic circuits of a

computer can exist in only one of two states, open or closed, the numbers that are stored can

consist of only two digits, 0 or 1. Numbers written using only these two digits are called

binary numbers. To find the decimal value of a binary number, you use the digits to write

a polynomial in 2. For instance, this is how to find the decimal value of the number

10011012. (The subscript 2 indicates that this is a binary number.)

An Wang (1920¨C1990) was an Asian-American who became one of the pioneers of the

computer industry in the United States. He grew up in Shanghai, China, but came to the

United States to further his studies in science. In 1948, he invented a magnetic pulse

controlling device that vastly increased the storage capacity of computers. He later founded

his own company, Wang Laboratories, and became a leader in the development of desktop

calculators and word processing systems. In 1988, Wang was elected to the National

Inventors Hall of Fame.

DATE

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter 7

An Wang

7-1

NAME

4

ab

7

ab

Simplify ?

. Assume

2

4-1

7-2

( )( b )

Group powers with the same base.

Chapter 7

(rw )

2r 5w 3

10. ?

4 3

xy 6

yx

7. ?

y2

4

2

a

4. ?

a a

5

4

16r 4

55 3

1. ?

5 or 125

2

3

( 2r n )

)

r 6n 3

11. 3?

5

(

2a 2b

8. ?

a

x 5y 3

xy

4

5. ?

y

5 2

m

m6

2. ?

m2

4

2

5

3

=?

2 3

(2a 3b 5) 3

(3b )

2 3(a 3) 3(b 5) 3

=?

(3) 3(b 2) 3

8a 9b 15

=?

27b 6

8a 9b 9

=?

27

8a 9b 9

.

The quotient is ?

27

3

2a b

(?

3b )

11

81 4 8

?

rn

8a3b3

16

m

( 3b )

Quotient of Powers

Power of a Power

Power of a Product

Power of a Quotient

nrt

3

27

64 6 6

?

pr

1

7

Glencoe Algebra 1

7 7 2

nt

12. r?

r 4n4

3 3 2

4p 4 r 4

3p r

( )

9. ?

2 2

-2y 7

14y

6. ?5 - ?y 2

3. ?

p3n3

2

p 5n 4

pn

Simplify each expression. Assume that no denominator equals zero.

Exercises

b

a

=?

m .

3

2a 3b 5

Simplify ?

. Assume

2

m

that no denominator equals zero.

Example 2

(b)

a

For any integer m and any real numbers a and b, b ¡Ù 0, ?

= (a )(b ) Quotient of Powers

= a3b5

Simplify.

The quotient is a3b5 .

ab

a 4b 7

a4 b7

?

= ?

2

a ?2

m

a

a

m-n

.

For all integers m and n and any nonzero number a, ?

n = a

that no denominator equals zero.

Example 1

Power of a Quotient

PERIOD

To divide two powers with the same base, subtract the

Dividing Monomials

Quotient of Powers

exponents.

DATE

Study Guide and Intervention

Quotients of Monomials

7-2

NAME

Answers (Lesson 7-1 and Lesson 7-2)

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Lesson 7-2

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter 7

DATE

Dividing Monomials

Study Guide and Intervention

(continued)

PERIOD

A5

1

(a -3-2)(b 6-6)(c 5)

=?

( )

Simplify.

Negative Exponent and Zero Exponent Properties

Simplify.

Quotient of Powers and Negative Exponent Properties

Group powers with the same base.

Chapter 7

(m t )

mt

-3 -5

1

t

?2

?

10. m

2 3 -1

7. ?

x6

-2

xy

x

4 0

b

b -4

4. ?

b

-5

2

2

1. ?

25 or 32

-3

2

0

w

4y

(6a b)

(b )

2

36

ab

Glencoe Algebra 1

0

12

1

Answers

4m 2 n 2

11. ?

-1

( 8m " )

?

8. ?

2 4

2 6

-1

4w y

?2

5. ?

-1 2

-1

(-x y)

m

m

2. ?

m5

-4

1

p

(3rt) u

r tu

-4

9r

u

3

m3

32n

Glencoe Algebra 1

12. ?

-?

-6 4

10

(-2mn )

4m n

2 -3

?

9. ?

-1 2 7

11

2

(a 2b 3) 2

(ab)

6. ?

a6b8

-2

?

3. ?

3

11

p

p

-8

Simplify each expression. Assume that no denominator equals zero.

Exercises

6

16a b c

( 16 )( a )( b )( c )

4

1 -5 0 5

=?

a bc

4

1 1

= ? ?5 (1)c 5

4 a

c5

=?

4a 5

c5

The solution is ?

.

4a 5

16a b c

-3

4a b

Simplify ?

. Assume that no denominator equals zero.

2 6 -5

4a -3b 6

4 a -3 b 6 1

?

?

?6 ?

= ?

2 6 -5

2

-5

Example

The simplified form of an expression containing negative exponents must contain only

positive exponents.

a

1

1

n

For any nonzero number a and any integer n, a -n = ?

n and ?

-n = a .

Negative Exponent Property

a

For any nonzero number a, a0 = 1.

Zero Exponent

Any nonzero number raised to the zero power is 1; for example,

(-0.5)0 = 1. Any nonzero number raised to a negative power is equal to the reciprocal of the

1

. These definitions can be used to

number raised to the opposite power; for example, 6 -3 = ?

63

simplify expressions that have negative exponents.

Negative Exponents

7-2

NAME

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Dividing Monomials

Skills Practice

DATE

PERIOD

1

m

4

a2b3

3

16p 14

49r

1

64

1

8

-1

11

9

?

1

k

Chapter 7

5u

u

3

-15t 0u -1

25. ?

-?

3

4

4 2

f -5g 4 g h

h

f

?

23. ?

-2

5

f -7 1

f f

?

21. ?

4

11

19. k0(k4)(k-6) ?2

( 11 )

9

17. ?

15. 8-2 ?2 or ?

?

4

2

4p 7

7r

( )

13. ?2

x

7w x

3w

-21w 5x 2

11. ?

-?

4 5

3

a 3b 5

9. ?

ab 2

36n

12n 5 n

?

7. ?

m

m

?2

5. ?

3

x

wx

-2

9

25

?

1

256

3

0

1

13

48x 6y 7z 5

-6xy z

8x 5 y 2

z

26. ?

-?

5 6

15x 6y -9

5xy

16p 5w 2

2p 3w 3

(?)

24. ?

3x5y2

-11

22.

k"

m

20. k-1(?-6)(m3) ?6

h3

18. ?

h9

-6

h

(3)

5

16. ?

1

4

14. 4-4 ?4 or ?

32x 3y 2z 5

-8xyz

12. ?

-4x2yz3

2

m 7p 2

10. ?

m4

m 3p 2

w 4x 3 2

8. ?

x

4

3d

t

9d

6. ?

3d

6

rt

7

r 3t 2 1

?

4. ?

3 4

2

x4 2

3. ?

x

2

12

9

9

2. ?

94 or 6561

8

6

6

61 or 6

1. ?

4

5

Simplify each expression. Assume that no denominator equals zero.

7-2

NAME

Glencoe Algebra 1

Answers (Lesson 7-2)

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Lesson 7-2

Copyright ? Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

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