Answers (Anticipation Guide and Lesson 7-1) - Mrs. Speer's Site

Glencoe Algebra 1

A1

Chapter 7

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

NAME

DATE

7 Anticipation Guide

Polynomials

PERIOD

Step 1

Before you begin Chapter 7

? Read each statement.

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

? Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

STEP 1 A, D, or NS

Statement

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

2. (k3)4 is equivalent to k12.

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

( ) 4.

2 5

3

is the same as

2 5

3

.

5. A polynomial may contain one or more monomials.

6. The degree of the polynomial 3x2y3- 5y2 + 8x3 is 3 because the highest exponent is 3.

7. The sum of the two polynomials (3x2y - 4xy2 + 2y3) and (6xy2 + 2x2y - 7) in simplest form is 5x2y + 2xy2 + 2y3 - 7.

8. (4m2 + 2m - 3) - (m2 - m + 3) is equal to 3m2 + m.

9. Because there are different exponents in each factor, the distributive property cannot be used to multiply 3n3 by (2n2 + 4n - 12).

10. The FOIL method of multiplying two binomials stands for First, Outer, Inner, Last.

11. The square of r + t, (r + t)2, will always equal r2 + t2.

12. The product of (x + y) and (x - y) will always equal x2 - y2.

STEP 2 A or D

D A A

D A D

A D D

A D A

Step 2

After you complete Chapter 7

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

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

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

Chapter 7

3

Answers

Glencoe Algebra 1

Chapter Resources

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NAME

DATE

7-1 Study Guide and Intervention

PERIOD

Multiplying Monomials

Monomials 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.

Product of Powers

For any number a and all integers m and n, am an = am + n.

Example 1 Simplify (3x6)(5x2). (3x6)(5x2) = (3)(5)(x6 x2) Group the coefficients

= (3 5)(x6 + 2) = 15x8 The product is 15x8.

and the variables Product of Powers Simplify.

Exercises

Simplify each expression.

1. y(y5)

y6

2. n2 n7

n9

Example 2 Simplify (-4a3b)(3a2b5). (-4a3b)(3a2b5) = (-4)(3)(a3 a2)(b b5)

= -12(a3 + 2)(b1 + 5) = -12a5b6 The product is -12a5b6.

3. (-7x2)(x4)

-7x6

4. x(x2)(x4)

x7

5. m m5

m6

6. (-x3)(-x4)

x7

7. (2a2)(8a)

16a3

10. 1(2a3b)(6b3) 3

4a3b4

( ) 13. (5a2bc3) 1abc4 5 a3b2c7

8. (rs)(rn3)(n2)

r2n6

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

20x10

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

-20x3y5

9. (x2y)(4xy3)

4x3y4

12. (-3j2k4)(2jk6)

-6j3k10

15. (10x3yz2)(-2xy5z)

-20x4y6z3

Chapter 7

5

Glencoe Algebra 1

Lesson 7-1

Answers (Anticipation Guide and Lesson 7-1)

Glencoe Algebra 1

A2

Chapter 7

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

NAME

7-1

DATE

PERIOD

Study Guide and Intervention (continued)

Multiplying Monomials

Simplify Expressions 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.

Power of a Power Power of a Product

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

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

Example 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.

Power of a Power Power of a Product Group the coefficients and the variables Product of Powers Power of a Power

Exercises

Simplify each expression.

1. (y5)2

y10

2. (n7)4

n 28

3. (x2)5(x3)

x13

4. -3(ab4)3

-3a3b12

5. (-3ab4)3

-27a3b12

6. (4x2b)3

64x6b3

7. (4a2)2(b3)

16a4b3

8. (4x)2(b3)

16x2b3

9. (x2y4)5

x10y20

10. (2a3b2)(b3)2

2a3b8

( ) 13. (25a2b)3 1 abf 2 5 625a8b5f2

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

512x9y3

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

-48x4y6

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

72j10k9

15. (2x3y2z2)3(x2z)4

8x17y6z10

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

12n12y10

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

-243a15n8

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

-768x14y2

Chapter 7

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Glencoe Algebra 1

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

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

Lesson 7-1

NAME

7-1 Skills Practice

DATE

PERIOD

Multiplying Monomials

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

1. 11 Yes; 11 is a real number and an example of a constant. 2. a - b No; this is the difference, not the product, of two variables. 3. p2 No; this is the quotient, not the product, of two variables.

r 2

4. y Yes; single variables are monomials. 5. j3k Yes; this is the product of two variables. 6. 2a + 3b No; this is the sum of two monomials.

Simplify.

7. a2(a3)(a6) a11 9. (y2z)(yz2) y3z3 11. (a2b4)(a2b2) a4b 6 13. (2x2)(3x5) 6x7 15. (4xy3)(3x3y5) 12x4y8 17. (-5m3)(3m8) -15m11 19. (102)3 106 or 1,000,000 21. (-6p)2 36p2 23. (3pr2)2 9p2r4

8. x(x2)(x7) x10 10. (2k2)(3k) 5k3 12. (cd2)(c3d2) c4d4 14. (5a7)(4a2) 20a9 16. (7a5b2)(a2b3) 7a7b5 18. (-2c4d)(-4cd) 8c5d2 20. (p3)12 p36 22. (-3y)3 -27y3 24. (2b3c4)2 4b6c8

GEOMETRY Express the area of each figure as a monomial.

25.

x5

x7

26.

x2

cd

cd

c2d2

27.

4p 9p3

18p4

Chapter 7

7

Glencoe Algebra 1

Answers (Lesson 7-1)

Glencoe Algebra 1

A3

Chapter 7

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

NAME

7-1 Practice

DATE

PERIOD

Multiplying Monomials

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

1. 21a2 No; this involves the quotient, not the product, of variables.

7b

2. b3c2 Yes; this is the product of a number, 1, and two variables.

2

2

Simplify each expression.

3. (-5x2y)(3x4) -15x6y

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

( ) 7.

(-15xy4)

-

1 3

x y

3

5x2y7

( ) 9.

(-18m2n)2

-

1 6

m n

2

-54m5n4

( ) 11. 2p 2 4p2 3 9

13. (0.4k3)3 0.064k9

4. (2ab2f 2)(4a3b2f 2) 8a4b4f4

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

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

10. (0.2a2b3)2 0.04a4b6

( ) 12. 1 ad3 2 1 a2d6

4

16

14. [(42)2]2 48 or 65,536

GEOMETRY Express the area of each figure as a monomial.

15.

3ab2

6a2b4

18a3b6

16.

5x3

(25x6)

17.

6ab 3

4a2b

12a3b4

GEOMETRY Express the volume of each solid as a monomial.

18.

3h2

27h6

3h2 3h2

19.

m3n

m4n5

n

20.

3g

mn3

7g2

(63g4)

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

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

Chapter 7

8

Glencoe Algebra 1

Answers

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

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

Lesson 7-1

NAME

DATE

7-1 Word Problem Practice

PERIOD

Multiplying Monomials

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

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

x

3 ft

4. SPORTS The volume of a sphere is given

by

the

formula

V

=

4 r3, 3

where

r

is

the

radius of the sphere. Find the volume of

air in three different basketballs. Use

= 3.14. Round your answers to the

nearest whole number.

Ball

Radius (in.) Volume (in3)

Child's

4

Women's

4.5

HTH

4.8

Source: WikiAnswers

268 382 463

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.

a. Find the power in a household circuit that has 20 amperes of current and

5 ohms of resistance. 2000 watts

3. PROBABILITY If you flip a coin 3 times in a row, there are 23 outcomes that can occur.

Outcomes HHH TTT HTT THH HTH TTH HHT THT

If you then flip the coin two more times, there are 23 ? 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

b. If the current is reduced by one half, what happens to the power?

The power is one-fourth the previous amount.

Chapter 7

9

Glencoe Algebra 1

Answers (Lesson 7-1)

Glencoe Algebra 1

A4

Chapter 7

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

NAME

7-1 Enrichment

DATE

PERIOD

An Wang

An Wang (1920?1990) 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.

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.)

10011012 = 1 ? 26 + 0 ? 25 + 0 ? 24+ 1 ? 23 + 1 ? 22 + 0 ? 21 + 1 ? 20 = 1 ? 64 + 0 ? 32 + 0 ? 16 + 1 ? 8 + 1 ? 4 + 0 ? 2 + 1 ? 1 = 64 + 0 + 0 + 8 + 4 + 0 + 1 = 77

Find the decimal value of each binary number.

1. 11112 15

2. 100002 16

3. 110000112 195

Write each decimal number as a binary number.

4. 101110012 185

5. 8 10002

6. 11 10112

7. 29 111012

8. 117 11101012

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.

The American Standard Guide for Information Interchange (ASCII) A 65 N 78 a 97 n 110 B 66 O 79 b 98 o 111 C 67 P 80 c 99 p 112 D 68 Q 81 d 100 q 113 E 69 R 82 e 101 r 114 F 70 S 83 f 102 s 115 G 71 T 84 g 103 t 116 H 72 U 85 h 104 u 117 I 73 V 86 i 105 v 118 J 74 W 87 j 106 w 119 K 75 X 88 k 107 x 120 L 76 Y 89 l 108 y 121 M 77 Z 90 m 109 z 122

Chapter 7

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Glencoe Algebra 1

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

Lesson 7-2

NAME

DATE

7-2 Study Guide and Intervention

PERIOD

Dividing Monomials

Quotients of Monomials To divide two powers with the same base, subtract the

exponents.

Quotient of Powers Power of a Quotient

For

all

integers

m

and

n

and

any

nonzero

number

a,

a m a n

=

am-n.

( ) For any integer m and any real numbers a and b, b 0,

a b

m

=

a m b m

.

Example 1 Simplify a4b7 . Assume ab 2

that no denominator equals zero.

( )( ) a4b7

ab 2

=

a 4 a

b 7 b 2

Group powers with the same base.

= (a4 - 1)(b7 - 2) Quotient of Powers

= a3b5

Simplify.

The quotient is a3b5 .

( ) Example 2

Simplify

2 a 3 b 5

3

. Assume

3 b 2

that no denominator equals zero.

( ) 2a3b5 3 = (2a3b5)3

3 b 2

(3b2)3

=

2 3 ( a 3 ) 3 ( b 5 ) 3 (3)3(b2)3

=

8a9b15 27b6

=

8 a 9 b 9 27

The quotient is 8a9b9 . 27

Power of a Quotient Power of a Product Power of a Power Quotient of Powers

Exercises

Simplify each expression. Assume that no denominator equals zero.

1. 55 53 or 125

5 2

2. m6 m2

m 4

3. p5n4 p3n3

p 2 n

4.

a 2 a

a

7. xy6 y2

y 4 x

5. x5y3 y

x 5 y 2

( ) 8.

2 a 2 b a

3

8a3b3

6. -2y7 - 1 y2

14y5

7

( ) 9. 4p4r4 3 64 p6r6 3p2r2 27

( ) 10.

2 r 5 w 3

4

16r 4

r 4 w 3

( ) 11. 3r6n3 4 81 r4n8 2r5n 16

12. r7n7t2 r4n4

n 3 r 3 t 2

Chapter 7

11

Glencoe Algebra 1

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

Chapter 7

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

Lesson 7-2

Answers (Lesson 7-2)

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

NAME

7-2

DATE

PERIOD

Study Guide and Intervention (continued)

Dividing Monomials

Negative Exponents 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

number raised to the opposite power; for example, 6-3 simplify expressions that have negative exponents.

=

1 . 6 3

These

definitions

can

be

used

to

Zero Exponent

For any nonzero number a, a0 = 1.

Negative Exponent Property

For

any

nonzero

number

a

and

any

integer

n,

a -n

=

1 a n

and

1 a -n

=

a n .

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

Example

Simplify 4a-3b6 . Assume that no denominator equals zero. 16a2b6c-5

( )( )( )( ) 4a-3b6

16a2b6c-5

=

4 16

a -3 a 2

b 6 b 6

1 c -5

Group powers with the same base.

=

1 4

(a-3-2)(b6-6)(c5)

Quotient of Powers and Negative Exponent Properties

=

1 4

a-5b0c5

Simplify.

( ) =

1 4

1 a 5

(1)c5

=

c 5 4 a 5

The solution is c5 . 4 a 5

Negative Exponent and Zero Exponent Properties Simplify.

Exercises

Simplify each expression. Assume that no denominator equals zero.

1. 22 25 or 32

2 -3

2. m m5

m -4

3. p-8 1 p3 p11

4. b-4 b

b -5

(-x-1y)0 5.

w

4w-1y2 4y2

6. (a2b3)2 a6b8

(ab)-2

7. x4y0 x6

x -2

8. (6a-1b)2 36 (b2)4 a2b6

9. (3rt)2u-4 9r3 r-1t2u7 u11

10. m-3t-5

1

(m2t3)-1 mt2

( ) 11. 1 4m2n2 0 8m-1

12.

(-2mn2)-3 4m-6n4

- m3 32n 10

NAME

7-2 Skills Practice

DATE

PERIOD

Dividing Monomials

Simplify each expression. Assume that no denominator equals zero.

1. 65 61 or 6

6 4

2. 912 94 or 6561

9 8

3. x4 x2

x 2

4. r3t2 1 r3t4 t2

5. m 1 m3 m2

7. 12n5 n4 36n 3

6. 9d7 3d

3 d 6

8. x w4x3 2

w 4 x

9. a3b5 a2b3

ab 2

10. m m7p2 4

m 3 p 2

11. -21w5x2 - 3w

7 w 4 x 5

x 3

( ) 13. 4p7 2 16p14

7 r 2

49r4

12. 32x3y2z5 -4x2yz3

-8xyz2

14. 4-4 1 or 1 44 256

15. 8-2 1 or 1 82 64

( ) 16. 5 -2 9 3 25

( ) 17. 9 -1 11 11 9

18. h3 h9

h -6

19. k0(k4)(k-6) 1 k 2

21. f-7 1 f 4 f 11

23. f-5g4 g4h2 h-2 f5

25. -15t0u-1 - 3

5 u 3

u 4

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

m 3 k 6

( ) 22.

16p5w2

0

1

2 p 3 w 3

24. 15x6y-9 3x5y2

5xy -11

26. 48x6y7z5 -6xy5z6

8x 5y 2 - z

A5

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Glencoe Algebra 1

Chapter 7

12

Answers

Glencoe Algebra 1

Chapter 7

13

Glencoe Algebra 1

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