8-1b lesson master...392 algebra name copyright © wright group/mcgraw-hill questions on spur...

392 Algebra

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Questions on SPUR ObjectivesSee pages 521–523 for objectives.

Lesson Master

USES Objective H

1. An art-supply store sells tubes of white paint in 4 sizes and

in 3 different brands. How many different choices of size/brand

are possible?

2. Edward wears jeans, a T-shirt, and a sweatshirt every day to

school. He has 6 pair of jeans, 9 T-shirts, and 4 sweatshirts.

How many different outfi ts can he wear?

3. Amy, Beth, Carlo, and Dion plan to run for the positions of

president, vice-president, secretary, and treasurer of the

student council. How many ways could the four offi ces be fi lled?

4. A combination lock has 50 numbers. The combination consists

of 3 numbers, each of which can repeat. How many different

combinations can be formed?

a. Write your answer in exponential form.

b. Write your answer in base 10.

5. Emma can choose from 50 types of freshwater fi sh for her new

aquarium. She can also choose from 12 types of artifi cial plants.

a. If she chooses just one type of plant and one type of fi sh,

how many different ways can she set up her tank?

b. Write your answer in scientifi c notation.

6. A math quiz has 8 true-false questions and 17

multiple-choice questions with 4 answer choices.

a. How many different answer sheets are possible?

b. What is the probability of getting all the answers on the quiz

correct by guessing?

Use this information for 7–9. Seven Chicago Bears football fans each

wrote a letter from the phrase “GO BEARS” on their chests to show their

team support. When they arrived at the game they sat next to each

other in random order.

7. How many different forms of the phrase are possible?

8. Write your answer in scientifi c notation.

9. What is the probability that the fans spelled the phrase

correctly when they fi rst sat down at the game?

8-1B

Answer PageBack to Lesson 8-1

Algebra 393

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A bakery offers the following breakfast options.

10. Gilbert always orders a bagel and a drink. How many different

choices are available?

11. Before Anna gets to school she orders one item to eat and one

item to drink. How many different choices can she make?

Your aunt orders 1 doughnut, 2 bagels, 2 muf! ns, and 5 drinks for her

and some co-workers.

12. Write the number of ways she can order

a. in exponential form.

b. in base 10.

c. in scientifi c notation.

Avery and his friends attend the Fall Frolic every year. There are

15 food booths, 20 game booths, 12 rides, and a haunted house.

13. Avery plans to eat some food and play some games. How

many different choices can he make?

14. Avery’s friend Melody wants to attend one food booth, one

game booth, one ride and the haunted house. How many

different choices does she have?

15. How does the answer to Question 14 change if Melody

wants to attend 2 food booths, 2 game booths, 2 rides, and

the haunted house? She can attend the same booth or ride

more than once.

16. Avery’s parents also attend the festival and intend to eat at

a food booth. What is the probability that they will eat at the

same food booth as Avery?

8-1B

Doughnuts Bagels Muf! ns Beverages

Plain Cinnamon/raisin Blueberry Orange juice

Chocolate Blueberry Apple Coffee

Pumpkin Oatmeal Chocolate Tea

Buttermilk Cinnamon Apple juice

Banana/nut

SMP08ALG_NA_TR2_C08.indd 393 5/31/07 10:07:36 AM

Answer PageBack to Lesson 8-1

Algebra 395

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Lesson Master8-2B

SKILLS Objective A

In 1–4, an expression is given.

a. Write the expression in expanded form.

b. Write the expression as a single power.

1. 53 · 5 2. x 3 · x 2

a. a.

b. b.

3. (−34)2 4. (x 3)5

a. a.

b. b.

In 5–25, simplify.

5. a5 · a 6. b2 · b4

7. c7 · −1c 3 8. 2d · 5d 9

9. e 3f 4 · e 2f 10. jk · 3jk2

11. −2m2n3 · −4m4n5 12. (6p0)3

13. 2q(q 3)4 14. 5(rs)2

15. 3t 0(t 5)5 16. b20 · b5

17. −u24 · uv 3 18. −10x5 · 0.2x 3

19. wx17 · w17x 20. 7y 6z 3 · 3y 2z 3

21. −12c16d 9 · 0.5c0d 7 22. 0.875ef 0 · −16e 2f

23. (9g 2)(9h3) 24. (10i )2(0.01i 9)

25. 16j 10k6 · (0.5j 5k2)3

Back to Lesson 8-2 Answer Page

396 Algebra

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PROPERTIES Objective G

In 26–33, solve for x and y and name the power property used to ! nd the solution.

26. 25 · 2 x = 215

27. 4x · 4x = 410

28. (63)x = 612

29. (7x)x = 7 9

30. (a3 · a x)2 = a6

31. c 4(c 3)x = c10

32. de 2 · 3d xey = 3d 3e5

33. ( fg)x · f 3g 4 = f 9g 10

34. Write a multiplication problem that uses the Product of Powers Property

to get an answer of k17.

35. Write a multiplication problem that uses the Power of a Power Property

to get an answer of m20.

8-2B



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Lesson Master

SKILLS Objective A

In 1 and 2, a fraction is given.

a. Write the numerator and denominator in expanded form.

b. Simplify the fraction.

1. a5

__

a2

2. 10j 5k3

_____ 12j 2k

a. a.

b. b.

In 3–13, simplify.

3. −c10

___

c7

4. 3.10 × 108

________ 2 × 103

5. 3.9 × 106

_______

3 × 104

6. 3d 3

___

6d

7. 22e5

____

11e3

8. 16f 6

____ −4f 6

9. 27g10

____

18g 4

10. −24h11

_____ 9h7

11. (−2)x

____

(−2)y

12. 30m8n4 ______

42m5n2

13. 81p4r

______

−36pr 4

8-3B


Algebra 399

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14. Write an algebraic fraction for which you can use the Quotient

of Powers Property to simplify to 3x 5.

In 15–19, use the Quotient of Powers Property to ! nd the value of x.

15. y x

__

y 3

= y11 16. 513

___

5x

= 54

17. 18mx

____

21m3

= 6m4

____ 7 18.

(−4)7

____

(−4)x = (−4)5

19. (2y)x

____

(2y)8

= (2y)12

20. Amber tried to simplify m9

___ m3

, and she got m3. Explain the error she made

in simplifying the fraction.

21. Multiple Choice. Which expression can be simplifi ed to 5m2x?

A 10mx

____

2mx

B 10m3x

_____

2mx

C 10m6x

_____ 2m3x

D 10m8x

_____ 2m4x

22. Multiple Choice. Consider the equation 5m

__ 5n

= 56. Which statement

accurately describes the values of m and n?

A m = n

B m = 4 and n = 2

C m + n = 6

D m - n = 6

8-3B



Algebra 401

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Lesson Master

SKILLS Objectives A and B

In 1 and 2, rewrite without negative exponents.

1. x −3 · y −4 2. 3a2b−5

In 3–8, rewrite: a. without fractions; b. without negative exponents.

3. 32c−4

____

16c2 4. −d

−2e2 _____

de

a. a.

b. b.

5. ( 1 __ f 4 ) −3

6. ( 2 ___ g −2 ) 4

a. a.

b. b.

7. 70h4j−2k3

_______

10h9k 8. −56m

−6n−1 ________

8m2n6

a. a.

b. b.

In 9–14, give the answer as a simple fraction.

9. 4−3 10. 9−2

11. ( 1 __ 4 ) −1

12. ( 5 __ 6 ) −2

13. (−2)−4 14. (75) 3 __ 5

15. Order the following numbers from least to greatest.

( 3 __ 4 ) −2 , 3 __ 16 , (−3)(−4), ( 2 __ 3 )

−4

8-4B


402 Algebra

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In 16–24, use the properties of negative exponents to ! nd the value of x.

16. p−3

___

p x = 1

__

p7 17. ( 3 __ 4 )

x

= 16 __ 9

18. r−2sx

____

s3

= 1 ___ r 2s6

19. t5

__

tx = t11

20. 2−3 · w x = 1 ___ 8w3

21. w x

____

2−3w 4 = 8 __

w 5

22. ( 2h−9

____

3hx

) -1

= 3h16

____ 2 23. −14j−4k2x = −14 ____

j 4k10

24. ( 1 __ 2 ) −3x

= 2

25. Justin simplifi ed ( 2 _ 5 ) −2

, and he got 4 __ 25

. Explain the error he made in

simplifying the fraction.

26. Multiple Choice. Which expression can be simplifi ed to x 4

__ 81

?

A ( x

__ 3 )

−4 B ( x

3

__

3 ) −1

C ( 3 __ x ) −4

D ( 3 __ x 3 ) −1

8-4B



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Lesson Master

SKILLS Objectives A, B and C

In 1–3, simplify and give the answer as a simple fraction.

1. 4 ( 1 __ 2 ) 4 2. 96 ( 1 __ 4 )

4

In 3 and 4, an expression is given.

a. Write the expression in expanded form.

b. Simplify the expression.

3. (−7x)2 4. ( 2a __ 5b ) 3

a. a.

b. b.

In 5–11, simplify and give the answer as a simple fraction.

5. (3y2)3 6. −(2m2)4

7. (−4n5)3 8. −(6pr2)2

9. ( c __ 4 ) 2 10. ( 9 __ 4 )

2 · ( 4 __ 3 )

5

11. ( 49 __ e ) · ( 3e2

___ 7 )

3

In 12–16, rewrite without parentheses and simplify.

12. (2a2b)5 13. 4(−3c3d4)2

14. ( −3e5f 7

_____

5e3f 9 )

3

15. −4

__

7 · ( h

5

___ 2h2

) 4

16. (6j 5k)2 · ( k ___ 2j 6 ) 3

8-5B


Algebra 405

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In 17–19, ! nd the area of the ! gure.

17. 18.

19.

Area of a trapezoid is found by

1

__ 2 · height · (base 1 + base 2).


In 20–22,

a. Tell what value of x will make the statement true for all values of the

variables.

b. Identify the property that justi! es the ! rst step in simplifying the

statement.

20. (−2m3n−1)x = −32m15

______ n5

21. 10 · ( 6m ___ 5n3 ) x =

72m2

____

5n6

a. a.

b. b.

22. x3 = −27m18n12

a.

b.

8-5B

40 cm

cm17x2

30

4x2y3 ft

17x in.

25x in.

x3 in.



Algebra 407

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Lesson Master

SKILLS Objectives D and E

1. The area of a square is 64 square units. What is the length of a side?

In 2–10, write the exact value or approximate the number to the nearest

hundredth.

2. √ ## 169 3. − √ ## 441

4. √ ## 225 5.

√ # 85

6. − √ ## 110 7. 81 1 __ 2

8. 34 1 __ 2 9. − (196)

1 __ 2

10. − (13) 1 __ 2

In 11–18, evaluate the expression. Write the exact value or

approximate to the nearest hundredth.

11. √ ##### 1,048 - 24 12.

√ ### 9 + 64

13. (16 + 81) 1 __ 2 14. (25 + 144)

1 __ 2

15. 3 √ # 13 · √ # 13 16. −2 √ # 6 ·

√ # 6

17. 4 ( 7 __ 10 ) 1 __ 2 · ( 7 __ 10 )

1 __ 2 18.

2

__

3 · ( 15 __ 8 )

1 __ 2 · ( 15 __ 8 )

1 __ 2

19. Find the length of the missing 20. Veronica needs a new pole for her kite.

side of the right triangle. What is the height, h, of the kite? Round

to the nearest whole number.

8-6B

9

15 x

15 cm

42.7 cm

25 cm

h


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21. If f (y) = 3 √ # y √ # y , what is f (8)?

22. Which of the expressions below are equal to (48)1/2?

A 4 √ # 3 B 2 √ # 24 C 2 √ # 12 D √ # 6 ·

√ # 8

In 23–32, write the exact value or approximate the number to the

nearest hundredth.

23. 3 √ ### 9.261 24.

3 √ ## 125

25. 3 √ ## −27 26. −

3 √ ## 343

27. 3 √ ## 512 28.

3 √ # 1 __ 4 ·

3 √ # 1 __ 4 ·

3 √ # 1 __ 4

29. 3 √ ## 1.2 ·

3 √ ## 1.2 ·

3 √ ## 1.2 30.

3 √ # 5 ·

3 √ # 5 ·

3 √ # 5

31. 3 √ # 64 ·

3 √ # −8 32.

3 √ # 27 ·

3 √ ## −216 ·

3 √ # −1

8-6B



410 Algebra

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Lesson Master

SKILLS Objective D

In 1–4, evaluate the expression.

1. √ # 18 · √ # 2 2.

√ ##### 16 · 25 · 225

3.

√ # 99

____

√ # 11

4.

√ # 73

____

√ # 7

In 5–7, simplify. Give the exact value. Assume all variables

are positive.

5.

√ # 72 6. 3 √ ## 160 7.

√ # 12 __ 9

In 8 and 9, write the exact value of the unknown in simpli! ed form.

8. 9.

10. A bowling ball manufacturer created a clear resin ball that can

contain any colored fi gure. The maximum length of the fi gure can

be found by the expression 2 √ ##

s

__ 4π where s represents the surface

area of the ball. What is the length of the fi gure that can be placed

in a ball with a surface area of 72.25π square inches?

11. Find the exact value of the area of a triangle with a base of

4 √ # 3 inches and a height of

√ # 6 inches.

8-7B

5

x 10

6

y3


Algebra 411

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In 12–23, simplify. Give the exact value. Assume all variables are positive.

12.

√ ### 48a2b2 13.

√ ### 56c2d5

14. − √ ### 250e 3f 6 15. −2 √ ## 40 x

16. 5 √ ## 32y7 17. 3

√ #### 112m4n8p5

18.

√ ## 216h

5

_____ 24h7

19. − √ ### 147j 9k2

_______ √ ### 12j 5k2

20.

√ ### 128s4t3

______ 50t 21.

√ ### 85xy10

_______

√ ### 5x5y−2

22. 4 √ ## 12n ·

√ ## 12n 23.

√ ### 20d4e3 ·

√ ## 5d 2e

8-7B



Algebra 413

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Lesson Master

REPRESENTATIONS Objectives I and J

1. A square has a diagonal with a length of 12 centimeters.

a. Find the length of a side of the square.

b. Find the area of the square.

2. The screen of a projection TV is 41.5 inches long and

28 inches tall. The length of the diagonal of the screen

represents the size of the television. What is the size of

the television?

In 3 and 4, ! nd the area of the ! gure.

3. 4.

In 5 and 6, ! nd the exact length of a side of the cube.

5. A cube with volume 1,728 cubic inches 6. A cube with volume 648 cubic inches

7. What is the volume of a cube with a side length of 8 inches?

8. What is the volume of a cube with a side whose diagonal

measures 5 √ # 2 inches?

8-8B

20

12

25

24


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In 9–13, ! nd the distance between the given points. Give the exact

simpli! ed value.

9. (2, 6) and (7, 1) 10. (5, −7) and (8, 2)

11. (10, −12) and (9, −14) 12. (−7, −8) and (1, 2)

13. (−9, −9) and (−11, 5)

14. Use the graph at the right to complete the following.

a. What is the exact value of the distance between A and B?

b. What is the exact value of the distance between C and D?

c. What is the exact value of the perimeter of the rectangle?

15. Use the graph below to complete the following.

a. What is the exact value of the distance between X and Z ?

b. What is the exact value of the area of △XYZ ?

8-8B

2

1

3

4

5

1

3

5

4

34 2 15 4 51 2 3

x

y

A(1, 2)

C (5, 2)

B (2, 1)

D (2, 5)

2

1

3

4

5

1

3

5

4

4 2 15 4 51 3

x

y

Y(3, 2)

Z (5, 2)

X (3, 2)



Algebra 415

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Lesson Master

SKILLS Objective C

In 1–6, rewrite without parentheses and without negative exponents.

1. (2a 3b−4)3 2. ( 5m2n _____

4m3n4 ) 4

3. (−7x −4y 8) · (2x 5y)6 4. ( 6m−1n2 _______

11m4n−7 ) −2

5. (6x 3y −6)−2 6. (−a 4b 2)−3 · (2a5b−6)

PROPERTIES Objective F

7. Tell whether the pattern 3x 2 = x 3 is true for the given instances.

a. x = 0 b. x = 1 c. x = 3

8. Tell whether the pattern x 1 _ 2 = √ $ x is true for the given instances.

a. x = 0 b. x = 1 c. x = 4

9. Find a counterexample for the pattern x 2 · x 4 = x 8.

10. Terri and Kandy both simplifi ed ( 3a7b2

____ a5b−6

) −2 . State which property

each student used for each step of their work.

Terri’s Work

Step 1: (3a2b8)−2 a.

Step 2: 3−2a−4b−16 b.

Step 3: 1 _____ 9a4b16

c.

Kandy’s Work

Step 1: (3a7b2)−2 _______

(a5b−6)−2 d.

Step 2: 3−2a−14b−4 ________

a−10b12 e.

Step 3: 3−2a−4b−16 f.

Step 4: 1 _____ 9a4b16

g.

8-9A


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