68 6 +5^ ! 71. 2-1 -^o+ sU - Miami Arts Charter...2012/12/11 · 10by1. Wj Problem 5 Using...

Jy Lesson CheckDo you know HOW?Simplify each expression.

1. 2"5

2. m°

3. 5s2f_14

4..-3

\£ Got It? 4. Apopulation of insects triples every week. The number of insects ismodeled by the expression 5400 • 3W, where w is the number of weeks afterthe population was measured. Evaluate the expression for w = -2, w = 0,and w = 1. What does each value of the expression represent in the situation?

<0

Do you UNDERSTAND?

7. Vocabulary A positive exponent shows repeatedmultiplication. What repeated operation does anegative exponent show?

y/l

8. Error Analysis A student incorrectly simplified ' 0as shown below. Find and correct the student's error.

Evaluate each expression for a — 2 and b =

5. a3b~l

6. 2a-4b°

-4.^x^ _ anxna-nb° ~

">

">^3p? Practice and Problem-Solving ExercisesWjp Practice Simplify each expression. •<P See Problem 1.

9.3"2 10. (-4.25)° 11. (-5)-2

12. -5"2 13. (-4)~2 14. 2-6

15. -3° 16. -12-1 17.1

2°

18. 58_1 19. 1.5~2 20. (-5)"3

Simplify each expression

21. 4ab° 22-4,x '

23. 5x~4 24.

^_225 -— 26. fc"4/ 27.

3x~2y

28.

29. c~5d-7 30. c~5d7 31.8

2s~332.

,, 6a_1c-334. 2-3x2z~7 35. 12°t7u-n 36.

^D See Problem 2.

PowerAlaebra.com-

c"1

7ab~23w

7s

sr3

7s°r5

2-^2

Lesson 7-1 Zero and Negative Exponents 417

©Apply

Evaluate each expression for r = —3 and 5 = 5.

37. r~3 38. s~3

41. 4s_1 42. r°s~2

39.^s

43. r-*s2

4p See Problem 3.

40.-2

44. 2_4r3s-2

45. Internet Traffic The number ofvisitors to a certain Web site triples every month. 4^i See Problem 4.The number of visitors is modeled by the expression 8100 • 3"', where m is thenumber of months after the number ofvisitors was measured.

Evaluate the expression m = -4. What does the value of the expressionrepresent in the situation?

46. Population Growth A Galapagos cactus finch population increasesby half every decade. The number of finches is modeled by theexpression 45 • 1.5 , where d is the number of decades after thepopulation was measured. Evaluate the expression for d = —2, d = 0,

and d — 1. What does each value of the expression represent in thesituation?

Mental Math Is the value of each expression positive or negative"?

47. -22 48. (-2): 49. (-2)3 50. (-2)

Write each number as a power of 10 using negative exponents.

"•A 52.100

53.1000

55. a. Patterns Complete the pattern using powers of 5.

"?=• -T=" \=*52 51 5°

b. Write —; using a positive exponent.

c. Rewrite ^=77 as a powerofa.

Rewrite each fraction with all the variables in the numerator.

56..-2

57.4g

IP58.

5nf

3n

54.l

10,000

1

5-2

59.8c°

»„-2UeTe

60. Think About a Plan Suppose your drama club's budget doubles every year. Thisyear die budget is $500. How much was the club's budget 2 yr ago?• What expression models what the budget of the club will be in 1 yr?

In2yr? In y years?• What value ofy can you substitute into your expression to find the budget of the

club 2 yr ago?

61. Copy and complete the table at the right.

62. a. Simplify an • a~'\b. Reasoning What is the mathematical

relationship between a" and a~n? Explain.I e 7 4H0.5

418 Chapter 7 Exponents and Exponential Functions

63. Open-Ended Choose a fraction to use as a value for the variable a. Find the valuesof a-1, a2, and a~2.

64. Manufacturing A company is making metal rods with a target diameter of 1.5 mm.Arod is acceptablewhen its diameter iswithin 10-3 mm ofthe target diameter.Write an inequality for the acceptable range of diameters.

65. Reasoning Are 3x~2 and 3.v2 reciprocals? Explain.

\W Challenge Simplify each expression.r-7fc-8\0S3 68 6 + 5^_!

m2 3 3

71. 2-1 -^o + sU3"2 V22

66.

69. 23(5° - 6m2)

f-i,„i67. (-5)2 - (0.5)-2

2x~5y3 r2y570.

2n

72. For what value or values ofnis n 3= (-) ?

rSunshine State Standards Practice

"\

MA.912.A.1.1

MA.912.A.3.9

MA.912.A.3.1

MA.912.A.1.1

MA.912.A.3.9

SRIDDED RESPONSE

73. Whatis the simplified form of -6(-6)_1?

74. Segment CD represents the flight of a bird that passes through the points(1, 2) and (5, 4). What is the slope of a line that represents the flight of asecond bird that flew perpendicular to the first bird?

75. What is the solution of the equation 1.5(x - 2.5) = 3?

76. What is the simplified form of 13.5 - 4.71 + 5.6?

77. What is they-intercept of the graph of 3x - 2y = -8?

y

*D

cX

0 2 4

rMixed Review

A

Solve each system by graphing.

78. y > 3x + 4y < -3.v + 1

79. y < -2.Y + 1y < 2x - 1

4fj See Lesson 6-6.

80. y > 0.5xy < x + 2

Write an equation in slope-intercept form for the line with the given slope mandy-intercept b.

4& See Lesson 5-3.

81. m = -l,b = 4

84. m= -f^.b = -17

CPowerAlgebra.com

82. m = 5, b = -2

Km-!*-}

83. m= |, b = -3

86. m = 1.25, b = -3.79

Get Ready! To prepare for Lesson 7-2, do Exercises 87-91.

Simplify each expression.

87. 6 • 104 88. 7 • 10~2 89. 8.2 • 105 90. 3 • 10"3

^D See Lesson 7-1.

91. 3.4 • 105

Lesson 7-1 Zero and Negative Exponents 419

How do you write49.7 x 10 in scientific

notation?

Move the decimal pointone place to the left.Increase the exponent of10 by1.

Wj Problem 5 Using Scientific Notation to Order Numbers

What is the order of 49.7 X 10, 4.17 x 107, 0.047 X 109, and 495 from leastto greatest?

Know49.7 X 10, 4.17 x 107,0.047 x 109, and 495

The order of the numbers

from least to greatestWrite each number in scientific

notation. Then compare them.

Step 1 Write each number in scientific notation.

49.7 X 10 4.17 X 107 0.047 X 109

4- 4- «r

4.97 X 102 4.17 X 107 4.7 X 107

495

4

4.95 X 102

Step 2 Order the numbers by the powers of 10. Arrange the numbers with the samepower of 10 in order by their decimal parts.

4.95 X 102 4.97 X 102 4.17 X 107 4.7 X 107

Step 3 Write the original numbers in order.

495 49.7 X 10 4.17 X 107 0.047 X 10*

-5S& Got It? 5. What isthe orderof24.8 X 10~4, 28 X 103, 0.025 X 104, and 258 X 10from least to greatest?

&y Lesson Check

Do you know HOW?Write each number in scientific notation.

1. 0.0007 2. 32,000,000

Write each number in standard notation.

3. 3.5 X 106 4. 1.27 X 10~4

Order the numbers in each list from least to greatest.

5. 105, 10~3,10°;10_1, 101

6. 5 X 10~3,2 X 104,3 X 10°,7 X 10_1

7. 2.5 X 107, 2.1 X 107, 3.5 X 106, 3.6 X 106

PowerAlqebra.comrs

Do you UNDERSTAND?

8. Open-Ended Describe a situation in which it is easierto use numbers written in scientific notation than to

use numbers written in standard form.

9. Error Analysis Astudent wrote 1.88 x 10-5 instandard notation as shown below. Describe and

correct the student's mistake.

10. Reasoning Astudent claims that 3.5 X 1011isgreater than 1.4 X 1013 because 3.5 > 1.4.Is the student correct? Explain.

I Lesson 7-2 Scientific Notation 423

n||b^ Practice and Problem-Solving Exercises

Vjp Practice Is the number written in scientific notation? Ifnot, explain.

1«J

©Apply

11. 44 X 108

15. 7.3 X 10"5

12. 3.2 X 106

16. 1.12 X 101

Write each number in scientific notation.

19. 9,040,000,000 20. 0.02

23. 0.00325 24. 8,003,000


27. 5 X 102 28. 7.45 X 102

31. 8.97 X 10_1 32. 1.3 X 10°

13. 0.9 X 10~2

17. 457 X 107

21. 9.3 million

25. 0.00092

29. 2.04 X 103

33. 2.74 X 105

Order the numbers in each list from least to greatest.

35. 9 X 10~7,8 X 10-8, 7 X 10~6,6 X 10_1()

36. 8.2 X 105, 7.9 X 105, 2.7 X 105, 8.1 X 105

37. 50.1 X 10~3,4.8 X 10~3, 0.52 X 10~3, 56 X 10~3

38. 0.53 X 107, 5300 X 10_1,5.3 X 105, 530 X 108

4fJ See Problem 1.

14. 6.7 X 10009

18.9.54 X 1015

4p See Problem 2.

22. 21,700

26. 0.0156

4j* See Problem 3.

30. 7.2 X 106

34. 4.8 X 10-3

4p See Problems 4 and 5.

39. Physics The half-life of a radioactive isotope is the amount oftime it takes one halfof a sample of the isotope to decay. What is the order of the following radioactiveisotopes of uranium from shortest to longest half-life?

234u

235u

236uIsotope

Half-life (years)

232U

68.9 2.45 X 105 7.04 X 108 2.34 X 107

Simplify. Write each answer using scientific notation.

40. 4(2 X 10~3) 41. 8(3 X 1014)

43. 6(5.3 X 10"4) 44. 0.3(8.2 X 10"3)

46. Think About a Plan Alight-year is the distance light travels inone year. One light-year is about 5,878,000,000,000 mi. The tableshows the estimated distance, in light-years, of several stars fromEarth. How many miles is each star from Earth?• How do you convert from light-years to miles?• What notation is easier to use for writing long distances?

47. Writing You write 1billionin scientific notation as 109. Explainwhyyouwrite 436 billionin scientific notation as 4.36 X 1011rather than as 436 X 109.

42. 0.2(3 X 102)

45. 0.5(6.8 X 105)

Distance From Earth

Distance

Star

Proxima Centauri

Sirius

Vega

Polaris

vsamammSOURCE: NASA

(light-years)

4.2

8.7

27

431


48. Physics The radius of a water molecule is about 1.4angstroms. One angstromis 0.00000001 cm. What is the diameter of a water molecule in centimeters?

Use scientific notation.

49. Reasoning Explain how the exponent of 10changes when you multiply a numberwritten in scientific notation by 100.Show an example.

Q Challenge 50. Economics Gross domestic product (GDP) is ameasure of the economic output ofacountry. The GDP oftheUnited States was about 1.2 X 1013 dollars in 2005. Thisis about 3 times the U.S. GDP in 1985. What was the U.S. GDP in 1985?

51. Write 3^ in scientific notation.

«


MA.912.A.1.1

MA.912.D.7.1

MA.912.A.3.10

MA.912.A.2.1

52. A lab sample has a mass of 0.000345 g. What is this amount written inscientific notation?

CA> 0.345 X 103 GD 0.345 X 10"3 GD 3.45 X 104

CD {1,3,4,5,7}

53. What is the union of {1,3, 5, 7} and {3, 4, 5}?

CD{} <X>{1,7}

54. What is the equation of the graph at the right?

C£> y= 2x-i CD y= \x - \

CD 3.45 X 10"4

CE> {3,5}

(X> y = ~2x - ± C&y= -kx + 2

55. Short Response A student is collecting cans to raisemoney for a class trip. Each week the student collects150 cans. Make a graph of the situation. How manyweeks will it take the student to collect 1200 cans?

y

0 X

) 2

rMixed Review

^


56. cd~6 57. a°b3 -358. 9u> 59.4 m„-5

Graph each linear inequality.

61. y< -\x + 2 62. y> \x 63. y< 3x - 4


Rewrite each expression using exponents.

65. t • t' t • t • t • t' t 66. (6 - m)(6 - m)(6 - m)

68. 5 • 5 • 5 • s • s • s 69. 2 • 2 • 2 • 2 • 2 • x • x • x

4D See Lesson 7-1.

60.1=30 See Lesson 6-5.

64. y> -3x + \

[ PowerAaebra.com

4p See page 716.

67. (r + 2)(r + 2)(r + 2)(r + 2)

70. 8 • 8 • (x - l)(.v - l)(.v - 1)

I Lesson 7-2 Scientific Notation 425

a^ Lesson CheckDo you know HOW?

1. Whatis 84 • 88 written usingeach base onlyonce?

2. Whatis the simplified form of 2n3 • 3rc~2?

3. What is (3 X 105)(8 X 104) written inscientific notation?

4. Measurement The diameter of a penny isaboutl.9 X 10-5 km. It would take about

2.1 X 109 pennies placed end to end to circletheequator once. What is the approximate length ofthe equator?

Do you UNDERSTAND?

5. Writing Can x8 • y3 bewritten as a single power?Explain your reasoning.

6. Reasoning Suppose a X 10'" and b X 10" aretwo numbers in scientific notation. Is their productab X 10'"+" always, sometimes, or never a numberin scientific notation? Justify your answer.

7. Error Analysis Yourfriend says 4a2 • 3a5 = 7a7. Doyou agree with your friend? Explain.

'

1^p? Practice and Problem-Solving ExercisesW£ Practice Rewrite each expression using each base only once. 4f^ See Problem 1.

8. 73 • 74

11. 22 • 27 • 2°


14. m3m4

17. (7*5)(8.v)

20. V2 • b4 • b

23. (xY-)(x-Gy)

26. x6 • y2 • x4

9. (-6)12 • (-6)5 • (-6)2

12. 5~2 • 5-4 • 58

15. 5c' • cfi

18. 3X2 • x2

21. (-2m3)(3.5/n-3)

24. (5.v5)(3y6)(3x2)

27. a6b3 • a2b~2

10. 9( «-4 »-2

13. (-8)5 • (-8) -5

^D See Problem

16. 4t~5 • 2r3

19. (-2.4n4)(2/2-1)

22. (l5a3)(-3a)

25. (4c4)(flc3)(-3rt5c)

28. -m2 • 4r3 • 12r~4 • 5m

4jy See Problem

31. (4 X 106) • 10~3

34. (5 X 107)(3 X 1014)

2.

3.Simplify each expression. Write each answer in scientific notation.

29. (2 X 103)(3 X 102) 30. (2 X 106)(3 X 103)

32. (l X 103)(3.4 X 10~8) 33. (8 X 10~5)(7 X 10~3)

Write each answer in scientific notation. 4f* See Problem 4.

35. Astronomy Tlie distance light travels in one second (one light-second) is about1.86 X 105 mi.Saturnisabout 475 light-seconds from the sun. About howmanymiles from the sun is Saturn?

36. Biology Ahuman bodycontains about 2.7 X 104 microliters {/xL) of blood foreach pound ofbodyweight. Eachmicroliter ofblood contains about 7 X 104 whiteblood cells. About how many white blood cells are in the body of a 140-lb person?

Lesson 7-3 Multiplying Powers With the Same Base 429

© Apply Complete each equation.

37. 52 • 5 = 511

40. c~5 • c = c6

43. a a4 = 1

38. 57 • 5" = 53 39. 2 • 24 = 21

41. m9 • m~4 = m~9 42. a • a • a3 = am

44. a12 • a = a12 45. x3y -x =f

46. Think About a Plan A liter ofwater contains about 3.35 X 1025 molecules. TheMississippi River dischargesabout 1.7 X 107 Lofwater everysecond. Abouthowmany molecules does the Mississippi River dischargeeveryminute? Writeyouranswer in scientific notation.

• How can you use unit analysis to help you find the answer?• What properties can you use to make the calculation easier?

Geometry Find the area of each figure.

47.

3x2 +

y5 + 2

48.

--2x2

Simplify each expression. Write each answer in scientific notation.

51. (9 X 107)(3 X 10"16) 52. (8 X10_3)(0.1 X 109) 53. (0.7 X 10~12)(0.3 X 108)

54. (0.4 X 10°)(3 X 10"4) 55. (0.2 X 105)(4 X 10-12) 56. (0.5 X 1013)(0.3 X 10~4)

57. Chemistry In chemistry, a moleis a unit of measure equal to 6.02 X 1023 atoms ofa substance. Themass ofa singleneon atom isabout 3.35 X 10-23g.What is themass of 2 moles of neon atoms? Write your answer in scientific notation.


58. V-^7X9 • X '

61. 2a?(3a + 1)

59.l

/l . n-3a' • a

62. 8m3(m4 + 2)

60.,-5

63. -4x3(3.y3 - 10x)

64. a. Open-Ended Write y6 as a productoftwo powers withthe same base in fourdifferent ways. Use only positive exponents.

b. Write y6 as a productoftwo powers with the samebase in four different ways,using negative or zero exponents in each product.

c. Reasoning Howmany ways can youwrite j'6 as the product of twopowers?Explain your reasoning.


!€• Challenge Simplify each expression.

65. 3* • 32_v • 32

68. {a + b)2(a + b)~3

*266. 2" • 2" + 2 • 2

69. {t + 3)7(f + 3)

67. 3* • 2^' • 32 • 2X

70. 5V+1 -51"*

71. Nature A book shows an enlarged photo of a carpenter bee. A carpenter bee isabout 6 X 10-3 m long.Thephoto is 13.5 cm long.Abouthowmany times as longas a carpenter bee is the photo?

r

Sunshine State Standards Practice

MA.912.A.1.3

MA.912.A.3.9

MA.912.A.1.1

MA.912.A.3.1

MA.912.A.10.1

ix3y72. What is the simplified form of (2x2y3)(4xy 2) ?

C5D 6x3y5 OD 6x2y6 CO 8x2y

73. What is the ^-intercept of the graph of 5x - 3y = 30?

CD-10 <X> -6 CED6 CD 10

74. Atthe Athens Olympics, the winning time for the women's 100-m hurdles was2.06 X 10-1 min. Which number is another way to express this time in minutes?

CA} 0.206 OD 20.6 <3D 206 X 101 CO 206 X 10-2

75. What is the solution of 4x - 5 = 2x + 13?

CD3 <X>4 CH)9 CD 32

76. Extended Response Bill'scompany packages its circular mirrors in boxes withsquare bottoms, as shown at the right. Show your work for each answer.a. What is an expression for the area of the bottom of the box?b. If the mirror has a radius of 4 in., what is the area of the bottom of the box?

c. The area of the bottom of a second box is 196 in.2. What is the diameter of thelargest mirror the box can hold?

rMixed Review


77. 2,358,000 78. 0.00465 79. 0.00007

^D See Lesson 7-2.

80. 5.1 billion

Findthe third, seventh, and tenth terms of the sequence described by each rule. 4j) See Lesson 4-7.

81. A{n) = 10 + (n - 1)(4) 82. A[n) - -5 + (n - 1)(2) 83. A{n) = 1.2 + (n - l)(-4)


Simplify eachexpression. ^ See Lesson 7-1,

84. (-2)~4 85. 5xj'° S6.4m~ln2 87.-3x3y~2z6

^

i PowerAlgebra.com Lesson 7-3 Multiplying Powers With the Same Base 431

Plan

How do you raise anumber in scientific

notation to a power?A number written in

scientific notation is a

product. Use the propertyfor raising a product toa power.

Problem 5 Raising a Number in Scientific Notation to a Power

Aircraft The expression ipnv2 gives the kinetic energy, in joules, ofan object with amass of m kg traveling at a speed of vmeters per second. What is the kinetic energyof an experimental unmanned jet with a mass of 1.3 x 103 kg traveling at a speed ofabout 3.1 x 103m/s?

\mv2 4.(1.3X 103)(3.1 X103)- Sul?itUte *evalues for m* - and v into the expression.

= 2 ' 1-3 • 103 • 3.1- • (103)! Raise the two factors to the second power.

= \ •1.3 • 103 •3.12 • 106= \ • 1.3 •3.12 • 103 • 106= | •1.3 •3.12 • 103 +6

Multiply the exponents of a power raised to a power.

Use the Commutative Property of Multiplication.

Addexponents of powers with the same base.

= 6.2465 X 109 Simplify. Write in scientific notation.

The aircraft has a kinetic energy of about 6.2 x 109 joules.

\£ Got It? 5. What is the kinetic energy of an aircraft with a mass of2.5 X 10° kgtraveling at a speedof3 x 102 m/s?

^5^ Lesson CheckDo you know HOW?Simplify each expression.

1. {n3)6 2. (b~7)3

3. (3«)4 4. (9x5)2(x2)5

Simplify each expression. Write each answer inscientific notation.

Do you UNDERSTAND?

7. Vocabulary Compare and contrast the propertyfor raising a power to a power and the property formultiplying powers with the same base.

,58. Error Analysis One student simplified x x5 to.10x . A second student simplified xb + x° to 2xb.

Which student is correct? Explain.

9. Open-Ended Write four different expressions that areequivalent to (x4)3.

5-v25. (4 X 105) 6. (2 X 10~3)5

J">

^$r Practice and Problem-Solving Exercises

f^y Practice Simplify each expression.10. (/z8)4 11. (n4)8

14. (w7)'1 15. (x3r5

18. (a5)3c4 19. (c3)5(d3)0

«'12. (c2)5

16. d{d-2r9

20. (f2r2(f2rs

^P See Problems1 and 2.

13. (410)10

17. (z8)°z5

21. (m3rl{x2)5


©Apply


22. (4m)5

26. (3/z-6r4

30. (2x)3x2

34. (3b-2)2{a2b4)3

23. (7«)-2

27. (2y4r3

31. (y2z-3)5(y3)2

35. c-l2(c-2d)3d5

Simplify. Write each answer in scientific notation.

38. (3 X 105)2 39. (4 X 102)5

42. (7.4 X 104)2 43. (6.25 X 10"12)"2

24. (5y)4

28. (xy)°

32. (mg4)"W)

36. (2/2A:4)-5(A:-V7)6

40. (2 X 10-10)3

44. (3.5 X 10"4)3

4^ See Problems 3 and 4.

25. [Ug4)-1

29. (r25)5

33. p(p-7q3r2q-3

37.4fk6(2ju)3k5

4fj See Problem 5.

41. (2 X 10~3)3

45. (2.37 X 108)3

46. Geometry Theradius of a cylinderis 7.8 X 10~4 m. Theheightof the cylinder is3.4 X 10-2 m. What is the volume of the cylinder?Writeyour answer in scientificnotation. (Hint: V= nr2h)

Complete each equation.

47. {b2) = b8

50. (;z9) = 1

53. (5x )2 = 25x~4

48. (m )3 = m"12

51. Cy-4) =y12

54. (3x3y )3 = 27x9

v - v649. (x )' =x

h52. 7{cl) 7cf

2„3-\ _55. (m2n3) =

56. Think About a Plan How many times the volume of the small cube isthe volume of the large cube?• What expression can you write for the volume of the small cube? For

the volume of the large cube?• What property of exponents can you use to simplify the volume

expressions?


57. 32(3x)3

60. (-5x)2 + 5x2

63. 4xy204(-y)-3

/;>-

3x

/ /\ /)1

/

/

/

58. (4.1)5(4.1)-5

61. {-2a2b)\ab)3

64. (103)4(4.3 X 10~8)

59. {bsfb2

62. (2x~3)2(0.2x)2

65. (37)2(3-4)3

66. Reasoning Simplify (x~)3 and x2 . Arethe expressions equivalent? Explain.

67. Measurement How many cubic millimeters are in a cubic meter? Write youranswer as a power of 10.

68. Wind Energy The power generated by a wind turbine depends on the wind speed.Theexpression 800u3 gives the powerin wattsfora certain wind turbine at windspeed v in meters per second. If the wind speed triples, by what factor does thepower generated by the wind turbine increase?

69. Can youwritethe expression 49x2j'2z2 using onlyoneexponent? Show howorexplain why not.

j

CPowerAlqebra.com Lesson 7-4 More Multiplication Properties of Exponents 437

70. a. Geography Earth has a radius of about 6.4 X 106m. What is theapproximate surface area of Earth? Use the formula for the surface area

ofa sphere, S.A. = 47rr2. Writeyour answer in scientific notation.b. Oceanscoverabout 70% ofthe surfaceof the Earth.Abouthow many

square meters of Earth's surface are covered by ocean water?c. The oceans have an average depth of 3790 m. Estimate the volume of

water in Earth's oceans.

fjj Challenge Solve each equation. Use the fact that if ax ay, then x = y.

71. 56 = 25x

74. 4X = 26

72. 3* = 274

75. 32v = 94

73.

76. 2X =

2 = 2x

J_32

77. Reasoning How manydifferent ways are there to rewrite the expression 16x4 usingonly the property of raising a product to a power? Show the ways.


MA.912.A.4.1

MA.912.A.3.13

MA.912.A.1.5

78. Whichexpressiondoes NOT equal 25/z12?

<X> (5«6)2 <X> (5az3)(5zz9) CO 25(zz3)9

79. One morning an employee washed 4 cars and 2 vans in less than 115 min.That afternoon it took the employee more than 150 min to wash 3 cars and5 vans. The information can be represented by the inequalities4x + 2y < 115 and 3x + 5y > 150, where x is the time it takes to washone car and y is the time it takes to wash one van. Which region of thegraph at the right represents the possible numbers of minutes it takes towash a car and a van?

(DA COB CrD C CDD

80. Short Response Asnail travels at a speed of3 x 10~2 mi/h. What is the snail'sspeed in inches per minute? Show your work.

rMixed Review


81. bc~6b3 82. (a2b3){a6) 83. 9/zz3(6/?z2/24)

Find the slope of the line that passes through each pair of points.

85. (0, 3), (4, 0) 86. (2, -5), (3,1) 87. (-3, 6), (1, 0)


Write each fraction in simplest form.

12489. | 90.


91 -S-*'• 15 92 ?2**• 15x

2\652(/z2)

601Washing Times

^_

I 40 AA\_

£ 20CD

>

">s

v\

B Un VC>.

20 40 60

Car, x(min)

"

4jj See Lesson 7-3.

84. 2r(-2f4)

4p See Lesson 5-1.

88.(0,0), (1, -9)

4& See page 711.

Q3 3acy;J- 12a

**W\ J Simplifying an Exponential Expression(9jfi\~3

What is the simplified form of (—j-1 ?

/ov6\ :i / v''1 \3 pj.6( 4 ) = (fn J Rewrite using the reciprocal of "V.

How do you writean expression insimplified form?Use the properties ofexponents to write eachvariable with a singlepositiveexponent.

6\3(2.vb)

„12

Raise the numerator and denominator to the third power.

= ^i8 simP|ify-

^ Got It? 4. What is the simplified form of (^) ?

tf Lesson Check

Do you know HOW?Simplify each expression.

1 43,io4\3

*• (f)

4-(S)5y4

5. A large cube is made up of many small cubes. Thevolume ofthelarge cube is 7.506 X L05 mm3. Thevolume of each small cube is 2.78 X 104 mm3. Howmany small cubes make up the large cube?

Do you UNDERSTAND?

6. Vocabulary How is the property for raising aquotient to a power similar to the property for raisinga product to a power?

7. a. Reasoning Ross simplifies%as shown at the right.a' °Explain why Ross smethod works.

1,7-3

1

b. Open-Ended Write a quotient of powers and useRoss's method to simplify it.

Practice and Problem-Solving Exercises

f^r Practice Copy and complete each equation.

">

*>

4p See Problem 1.

8.^ =5'Simplify each expression.

0812.^

15.

18.

21.

c3d~5c4d~l

32m5t635m7r5

9 — = 223

06

13. ^

16.

19.

22.

-755

10s-9

I0m6n35m2n7

.v5y~823-4,3xy

10. I =3

14 ^

c2,-3

11.^A = 5'5J52

xny

3 2

— 1 3

I2g-lb6c~34a5b_1c5

17.

20.

23.

1 PowerAlgebra.com I Lesson 7-5 Division Properties of Exponents 443

©Apply

Simplify each quotient. Write each answer in scientific notation. 4p See Problem 2.

24.5.2 X 1013

27.

1.3 X 10'

- 53.4 X 10

2 X 10~8

25.

28.

3.6 X 1Q~10

9 X 10~6

-44.65 X 10

3.1 X 102

26.

29.

6.5 X 104

5 X 106

3.5 X 106

5 X 108

30. Computers The average time it takes a computer to execute one instruction ismeasured in picoseconds. There are 3.6 X 10lD picoseconds per hour. Whatfraction of a second is a picosecond?

31. Wildlife Data from a deer count in a forested area show that an estimated

3.16 X 103 deer inhabit 7.228 X 104 acres ofland. Whatis the density of thedeer population?

32. Astronomy The sun's massis 1.998 x 1030 kg. Saturn's massis 5.69 X I026kg.How many times as great as the mass of Saturn is the mass of the sun?

Simplify each expression 4f^ See Proble

33-(I)2 «• & «• m M%f"•(I)3 * (I)5 *(*}• 40. m

«• (§r «• (f)"" - m2 -(-sr45- («)2 *«f7

3

- »)- •cayExplain why each expression is not in simplest form .

49. 53m3 50. X5y-2 51. (2c)4 52. x°y *•£54. Think About a Plan During one year, about 163 million adults over 18 years old in

the United States spent a total of about 93 billion hours online at home. On average,how many hours per day did each adult spend online at home?• How do you write each number in scientific notation?• How do you convert the units to hours per day?

55. Television During one year, people in the United States older than 18 years oldwatched a total of 342 billion hours of television. The population of the UnitedStates older than 18 years old was about 209 million people.a. On average, how many hours of television did each person older than 18 years

old watch that year? Round to the nearest hour.b. On average, how many hours per week did each person older than 18 years old

watch that year? Round to the nearest hour.

Which property or properties of exponents would you use to simplify eachexpression?

56. 2~3 57. p 58.l

2~42759.

4-v3(24)o!5



60.

* (i)3

62.3.v3

(3-v)3

2x?Y5X3* as

63.

67.

(2fl6)(4a)8fl3

4.v-y8.v3(y-2)3

68. a. Open-Ended Write three numbers greater than 1000 in scientific notation.b. Divide each number by 2.c. Reasoning Is the exponent of the power of 10 divided by 2 when you divide a

number in scientific notation by 2? Explain.

69. Simplify the expression (*)-'in three different ways. Justify each step.

70. Geometry The area of the rectangle is 72a3b4. What is the lengthof the rectangle?

3M^a^ 12

C3D 12a2b3 dD I2a3b4 3 ^ L

--6ao


71. (^ 72.

75.a2fr4

76.

4 3 —3m inJp °

m~2n7p~8

(-4)2 V(-3)"3

73.(i)'

-3

77.(4x)zy\-2

^ z4

74.0.23 • 0.24

0.27

(6a3)(8fr4)(2fl4)(36/7_1)

78.

79. Physics The wavelength of a radio wave is defined as speed divided by frequency.AnFM radio station has a frequencyof9 X 107 waves per second. Thespeed of thewaves isabout 3 X 10° meterspersecond. What isthe wavelength ofthe station?

80. a. Error Analysis What mistake did the student make in simplifying theexpression at the right?

b. What is the correct simplified form of the expression?

81. Writing Suppose^ = a3 and \y = a_D. Find the values of.randy. Explainhow you found your answer.

82. a. Finance In 2000, the United States government owed about $5.63 trillion toits creditors. Tlie population of the United States was 282.4 million people. Howmuch did the government owe per person in 2000? Round to the nearest dollar.

b. In 2005, the debt had grown to S7.91 trillion, with a population of 296.9 million.How much did the government owe per person? Round to the nearest dollar.

c. What was the percent increase in the average amount owed per person from2000 to 2005?

Write each expression with only one exponent. You may need to useparentheses.

83.m'

n784.

107 • 10

!() -3

[*l.\\Mh\UU.\M*.UM

85.27.v;

8y386.

Am1

169m4

| Lesson 7-5 Division Properties of Exponents 445

87. a. Use the property for dividing powers with the same base to write ^r, as apower of a.

b. Usethe definitionofa zero exponent to simplify ^77.c. Reasoning Explain how your results from parts (a) and (b) justifythe definition

of a negative exponent.

15Challenge 88. Reasoning Use thedivision property toshow why 0° isundefined.

Simplify each expression. tm*

89. nx+2 - nx 90. n** + nx 91. (-j-^)3 92. -^

93. Astronomy Tliedensity of an object is the ratio of its mass to its volume. Neptunehas a mass of 1.02 X 1026 kg. Theradius ofNeptune is 2.48 X 104 km.What is thedensity ofNeptune in grams per cubic meter? (Hint: V= f /rr3)

*—& Sunshine State Standards Practice

(2v)5MA.912.A.4.1 94. Which expression is equivalent to *-y-?

<X> 2x2 CD 32x2 <3D 2x8 CD 32x~2

MA.912.A.3.10 95. Which equation is an equation of the line that contains the point (8, -3) and isperpendicular to the line y = -4x + 5?

CD y = -\x - 1 <®> y =\x + f CE> y =\x - 5 CD y = 4* - 35

MA.912.A.3.14 96. What is the solution of the system of equations y = -3x + 5 and y = -Ax - 1?

CK) (23,6) CD (6,23) CCD (-6, 23) CD (-6, -23)

MA.912.A.2.4 97. Short Response You have 8 bags ofgrass seed.Each bagcovers 1200 ft2 ofground.Tlie function A{b) = 1200/? represents the area A{b), in square feet, that b bagscover. What domain and range are reasonable for the function? Explain.

r —Mixed Review


98. (2ra-7)3 99. 2(3s-2r 3 100. (43c2)-l 101 (--3)2(/ 5)2 102.

See Lesson 7-4.

(7°72-3)2(H7)3

Solve each system by graphing.

103. y = 3.v 104. y -y = -2.v y =

= 2x + 1

= X - 3

105. y =x —

5

3

106 y =

y =

See Lesson 6-1.

7

8

Get Ready! To prepare

Graph each function.

107. y = Ax 108.

for Lesson 7-6,

y = 5x

do Exercises 107-

109. y = -3.v

11C .

110. y =

See Lesson 4-4.

1.5x


WF Lesson Check

Do you know HOW?Evaluate each function for the given value.

1. f(x) - 6 • 2* for* = 3

2. g(w) =45-3"' for w = -2

Graph each function.

3. y = 3X

4. fix) =a(±J

Do you UNDERSTAND?

5. Vocabulary Describe the differences between alinear function and an exponential function.

6. Reasoning Isy = (-2)* an exponential function?Justify your answer.

7. Error Analysis A student evaluatedthe function f(x) = 3 • Axfor x = -1 as shown at the

right. Describe and correct thestudent's mistake.

^p? Practice and Problem-Solving ExercisesWff Practice

*

1Determine whether each table or rule represents an exponential function.Explain why or why not.

^D See Problem 1.

8.

a32

«128

10. y = A • 5X

12. y = -5 • 0.25*

Evaluate each function for the given value.

14. fix) = 6*for a = 2

16. y = 20 • 0.5* for* = 3

9.1

18 33\11. y = 12 • x2

13. y = 7x + 3

4& See Problem 2.

15. git) = 2 • 0.4' for* = -2

17. h(w) - -0.5 • Awforw = 18

18. Finance An investment of $5000doubles in value every decade. The functionfix) = 5000 • 2X, where .vis the number of decades, models the growth of the valueof the investment. How much is the investment worth after 30 yr?

19. Wildlife Management A population of 75 foxes in a wildlife preserve quadruplesin size every 15 yr.The function y = 75 • Ax, where x is the number of 15-yr periods,models the population growth. How many foxes will there be after 45 yr?

Graph each exponential function.

20. y = Ax 21. y - -Ax

24. y=10 •(f)' 25. y = 0.1 • 2X


22. y

26. y=\ • 2X

l\x

4jj See Problem 3.

».y=-(r27. y = 1.25J

©Apply

28. Admissions Anew museum had 7500 visitorsthis year. The museum curators 4fJ See Problem 4.expect the number of visitors to grow by 5%each year. The functiony = 7500 • 1.05v models the predicted number of visitors each year after x years.Graph the function.

29. Environment A solid waste disposal plan proposes to reduce the amount ofgarbage each person throws out by 2% each year. This year, each person threwout an average of 1500 lb of garbage. Tlie function y = 1500 • 0.98* models theaverage amount of garbage each person will throw out each year after x years. Graphthe function.

Evaluate each function over the domain {—2, —1, 0,1,2,3}. As the values of the

domain increase, do the values of the range increase or decrease"?

30. fix) = 5X 31. y = 2.5* 32. h(x) = 0.1* 33. fix) = 5 • 4V

34. y = 0.5-v 35. y = 8V 36. g(x) = A • 10* 37. v = 100 • 0.3*

38. Think About a Plan Hydra are small freshwater animals. Theycan double in number every two days in a laboratory tank.Suppose one tank has an initial population of 60 hydra.When will there be more than 5000 hydra?• How can a table help you identify a pattern?• What function models the situation?

39. a. Graph y —2X, y = Ax, and y = 0.25* on the same axes.b. What point is on all three graphs?c. Does the graph of an exponential function intersect the x-axis?

Explain.

d. Reasoning How does the graph of y = bx change as thebase b increases or decreases?

Hydra

Which function has the greater value for the given value ofx"?

40. y = Axory = x4 for x = 2 41. f(x) = 10 • 2X orf(x) = 200 • x2 forx = 7

42. y - 3*or y = x3 for x = 5 43. f(x) = 2x or f(x) = lOO.v2 forx = 10

44. Computers A computer valued at $1500 loses 20% of its value each year.a. Write a function rule that models the value of the computer.b. Find the value of the computer after 3 yr.c. In how many years will the value of the computer be less than $500?

145. a. Graph the functions y = x and y = 2X on the same axes.• b. What do you notice about the graphs for the values of x between 1 and 3?

c. Reasoning How do you think the graph of y = 8*would compare to the graphsofy = x2andy = 2V?

46. Writing Find the range of the function fix) = 500 • lx using the domain{1, 2, 3, 4, 5}. Explain why the definition of exponentialfunction states that b =£ 1.

t PowerAlgebra.com Lesson 7-6 Exponential Functions 451

/^Challenge Solve each equation.

47. 2V = 64 48. 3- = i 49. 3 • 2X = 24 x _50. 5 • 2

51. Suppose (0, 4) and (2, 36) are on the graph of an exponential function.a. Use (0, 4) in the general form ofan exponential function, y = a • bx, to find the

value of the constant a.

b. Use your answer from part (a) and (2, 36) to find the value of the constant b.c. Write a rule for the function.

d. Evaluate the function for x = —2 and x = A.

152 = 8


MA.912.A.2.2

MA.912.A.3.3

MA.912.A.3.6

52. A population of 30 swans doubles every 10yr. Which graph represents thepopulation growth?

120

60

0 I;

*, 30\

1 15

n

120

I 60

//

/

120

60

0

M0 20 40

Years

0 20 40

Years

0 20 40

Years

53. Which equation do you get when you solve y = 2x —12 for x?

CD * = y ~ 6 dD x = y + 6 GD x = 0.5y - 6 CD *

54. Short Response What are the solutions of the inequality 11 < \x - 2| +4?Graph the solutions on a number line. Show your work.

0 20 40

Years

0.5y + 6

rMixed Review


55.2\-4

56.4\-7?>f

57.t-3

Write an equation for the line that is parallel to the given line and passesthrough the given point.

59. y = 5x + 1; (0, 0) 60. y = 3x - 2; (0, 1)


Tell whether each percent change is an increase or decrease. Then findthe percent change. If necessary, round to the nearest percent.

4p See Lesson 7-5.pq°\5

58.

4^ See Lesson 5-6.

61. y- 0.4.Y + 5; (2, -3)

4(d See Lesson 2-10.

62. original price: $25; sale price: $22

64. price this year: $999; price last year: $1450

63. height last week: 15 cm; height this week: 18 cm

65. birth weight: 220 lb; weight at 1 month: 300 lb


Will the pressure everbe negative?No. The range of anexponential decayfunction isall positivereal numbers. Thegraphof an exponential decayfunction approaches butdoes not cross the x-axis.

Problem 3 Modeling Exponential Decay

Physics The kilopascal is a unit of measure for atmospheric pressure. Theatmospheric pressure at sea level is about 101 kilopascals. For every 1000-m increasein altitude, the pressure decreases about 11.5%. What is the approximate pressure atan altitude of 3000 m?

Relate y = a • bx Use an exponential function.

Define Let.v = the altitude (in thousands of meters).Lety = the atmospheric pressure (in kilopascals).

Let a = the initial pressure (in kilopascals), 101.Let b = the decay factor, which is 1 - 0.115 • 0.885.

Write y = 101 • 0.885*

Use the equation to estimate the pressure at an altitude of 3000 m.

y = 101 • 0.885-v

= 101 • 0.8853 Substitute 3 forx.

~ 70 Round to the nearest kilopascal.

The pressure at an altitude of 3000 m is about 70 kilopascals.

Got It? 3. a. Whatis the atmospheric pressure at an altitude of5000 m?b. Reasoning Why do you subtract the percent decrease from 1 to find the

decay factor?

N*

*F^Do

1.

2.

3.

4.

5.

Lesson Check

you know HOW?What is the growth factor in the equation y = 34 • 4*?

What is the initial amount in the function y = 15 • 3V?

What is the decay factor in the function y = 17 • 0.2*?

A population offish in a lake decreases 6% annually.What is the decay factor?

Suppose your friend's parents invest $20,000 in anaccount paying 5% interest compounded annually.What will the balance be after 10 yr?

Do you UNDERSTAND?

6. Vocabulary How can you tell if an exponentialfunction models growth or decay?

7. Reasoning How can you simplify the compoundinterest formula when the interest is compounded

annually? Explain.

8. Error Analysis A student deposits $500 into anaccount that earns 3.5% interest compoundedquarterly. Describe and correct the student's errorin calculating the account balance after 2 yr.


">

^jj$r Practice and Problem-Solving Exercises

Wff Practice'

©Apply

Identify the initial amount a and the growth factor b in eachexponential function.

9. g{x) = 14 • 2* 10. y = 150 • 1.0894*

11. y = 25,600 • 1.01* 12. /(/) = 1.4'

13. College Enrollment The number of students enrolled at a college is 15,000 andgrows 4% each year.a. The initial amount a is •.

b. The percent rate of change is 4%, so the growth factor b is 1 + • = •.c. To find the number of students enrolled after one year, you calculate

15,000 • •.

d. Complete the equation y = • • • to find the number of students enrolledafter x years.

e. Use your equation to predict the number of students enrolled after 25 yr.

14. Ecology A town has 10 acres of conservation land. The town plans to increase theamount of conservation land about 5% every 10 yr. If the town continues to followtheir plan, how much conservation land will there be after 50 yr?

Find the balance in each account after the given period.

15. $4000 principal earning 6% compounded annually, after 5 yr

16. $12,000 principal earning 4.8% compounded annually, after 7 yr

17. $500 principal earning 4% compounded quarterly, after 6 yr

18. $20,000 deposit earning 3.5% compounded monthly, after 10 yr

19. $5000 deposit earning 1.5% compounded quarterly, after 3 yr

20. $13,500 deposit earning 3.3% compounded monthly, after 1 yr

21. $775 deposit earning 4.25% compounded annually, after 12 yr

22. $3500 deposit earning 6.75% compounded monthly, after 6 months

4^.< See Problem 1.

4fj See Problem 2.

Identify the initial amount a and the decay factor b in eachexponential function.

23. y = 5 • 0.5* 24. f{x) = 10 • 0.1* 25. g(.r) = 100(fVX

4& See Problem 3.

26. y = 0.1 • 0.9*

27. Population The population of a city is 45,000 and decreases 2%each year. If thetrend continues, what will the population be after 15 yr?

State whether the equation represents exponential growth, exponential decay,or neither.

28. y = 0.93 • 2* 29. y = 2 • 0.68* 30. y = 68 • x2 31. y = 68 • 0.2-1

Lesson 7-7 Exponential Growth and Decay 459

32. Sports In a single-elimination tournament, half of the remaining teams areeliminated in each round. If the tournament starts with 128 teams, how manyteams will be left after 5 rounds?

33. Car Value A family buys a car for $20,000. The value of the car decreases about20% each year. After 6 yr, the family decides to sell the car. Should they sell it for$4000? Explain.

34. Think About a Plan You invest $100 and expectyour money to grow 8% each year.About how many years will it take for your investment to double?• What function models the growth of your investment?• How can you use a table to find the approximate amount of time it takes for your

investment to double?

35. Reasoning Givean example of an exponential function in the form y = a • bxthat is neither an exponential growth function nor an exponential decay function.Explain your reasoning.

State whether each graph shows an exponentialgrowthfunction, anexponential decayfunction, or neither.

36.

J.

\ 1

40. Writing Wouldyou rather have $500 in an account paying 6% interestcompounded quarterly or $600 in an account paying 5% interest compoundedannually? Explain your reasoning.

41. Medicine Doctors can use radioactive iodine to treat some forms of cancer. The

half-life of iodine-131 is 8 days. A patient receives a treatment of 12 millicuries ofiodine-131. (A millicurie is a unit of radioactivity.) How much iodine-131 remainsin the patient 16 days later?

42. Business Suppose you start a lawn-mowingbusiness and make a profit of $400inthe first year. Each year, your profit increases 5%.a. Write a function that models your annual profit.b. Ifyou continue your business for 10yr,what will your total profit be?


«

r **±

43. Medicine Cesium-137 is a radioisotope used in radiology wherelevels are measured in millicuries (mci). Use the graph at the right.What is a reasonable estimate of the half-life of cesium-137?

ym Challenge 44. Credit Suppose you use a credit card tobuya newsuitfor $250. Ifyou do not pay the entire balance after one month, you are charged1.8%monthly interest on your account balance. Suppose you canmake a $30 payment each month.a. What is your balance after your first monthly payment?b. How much interest are you charged on the remaining balance

after your first payment?c. What is your balance just before you make your second payment?d. What is your balance after your second payment?e. How many months will it take for you to pay offthe entire bill?f. How much interest will you have paid in all?

Cesium-137 Decay

- 500

E 400

5 300I 200-1 100

0

45. Open-Ended Write two exponential growth functions /(.v) and g(x) such thatf(x) < g{x) for x < 3 and f{x) > g(x) for x > 3.

20 40 60

Time (yr)

"SSunshine State Standards Practice

MA.912.A.1.4 46. A new fitness center opens with 120 members. Every'month the fitness centerincreases the number of members by 40 members. How many members will thefitness center have after being open for 3 months?

MA.912.A.3.9 47. What is the slope of the line at the right?

MA.912.A.1.1 48. Whatis the simplified formof 8°?

MA.912.A.1.1 49. What is the simplified form of5~2?

MA.912.A.1.1 50. A manufacturing company is making metal sheets with a thicknessof 5.4 X 10-2 mm. What is 5.4 X 10~2 written in standard form?

GRllDDIED RESPONSE

\ y

X

0 L

t—Mixed Review

Graph each function. 4(J See Lesson 7-6.

51. y= 2• 10* 52. f(x) = 100 •0.9* 53. g(x) =^ •2*

Solve each inequality. 4& See Lesson 3-4.

54. 7x + 2< 16 55. \t - 4> 5 56. -0.08 > 0.35fc - 0.15


Simplify each expression. 4fr See Lesson 1-7.

57. 6f + 13* 58. Ik - ISk 59. 2b - 6 + 9/? 60. 4/?2 - 7/?2 61. Ox2 + x2

c PowerAlqebra.com Lesson 7-7 Exponential Growth and Decay 461

Chapter Review

Connecting BIG ideas and Answering the Essential Questions^v

1 EquivalenceOne way to representnumbers is in scientific

notation. This form uses

powers of ten to write verylarge or verysmall numbers.

2 PropertiesJust as there are propertiesthat describe how to rewrite

expressions involvingaddition and multiplication,there are properties thatdescribe how to rewrite

and simplify exponentialexpressions.

3 Function

Thefamily of exponentialfunctions has equationsof the form y = a • bx.They can be used to modelexponential growth or decay.

Zero and Negative Exponents(Lesson 7-1)

10° = 1

10~3 =103

Properties of Exponents(Lessons 7-3, 7-4, and 7-5)c2 . c4 _ c2 + 4 _ c6

(37)4 = 37-4 = 328

(6x)4 = 6V7f7-

= 78~5 = 73

(f)5 =

Exponential Functions (Lesson 7-6)

y = 2

y = 3

(ir4.

Chapter Vocabularycompound interest (p. 456)decay factor (p. 457)exponential decay (p. 457)

• exponential function (p. 447)• exponential growth (p. 455)

Choose the correct term to complete each sentence.

1. ? is a shorthand way to write very large and very small numbers.

2. For a function y = a • bx,where a > 0 and b > I, bis the ? .

3. For a function y = a • bx, where a > 0 and 0 < b < 1, bis the '? .

4. The function y —a • bx models ? for a > 0 and b > 1.

5. The function y = a • bx models _?_ for a > 0 and 0 < b < 1.

Scientific Notation (Lesson 7-2)

175,000,000,000,000

= 1.75 X 1014

0.0000568 = 5.68 X 10~5

Exponential Growth andDecay (Lesson 7-7)

Exponential growth is modeled bythe function y = a • o*, wherea > 0 and b > 1. Exponentialdecay is modeled bythe functiony = a • bx, where a > 0 and0<6 < 1.

growth factor (p. 455)scientific notation (p. 420)

Chapter 7 Chapter Review 463

7-1 Zero and Negative Exponents

Quick ReviewYou can use zero and negative integers as exponents.For ever)7 nonzeronumber a, a° = 1. For every nonzeronumber a and any integer n, a~" — —,. When you evaluatean exponential expression, you can simplify the expressionbefore substituting values for the variables.

Exampl

What is the value of a2b 4c° for a = 3, b = 2, andc= -5?

2 09,-4 0 tf~C—j- Use thedefinition ofnegative exponents.

«2(D-4— Use thedefinition ofzero exponent.

= ^r Substitute.

= Ye Simplify.

7-2 Scientific Notation^

Quick ReviewYoucan use scientific notation to write very large or verysmall numbers. A number is written in scientific notation

if it has the forma x 10", where 1 < a < 10 and n is

an integer.

ExampleWhat is each number written in scientific notation?

a. 510,000,000,000

510,000,000,000 = 5.1 X 1011

b. 0.0000087

0.0000087

Move the decimal

point 11 places tothe left.

Move the decimal

- 8.7 X 10-6 point 6 places tothe right.

464 Chapter 7 Chapter Review

*Exercises


6.5°

8.•l.v

--2

y

-27. 7

9. ?a-4jDp-q

Evaluate each expression for x = 2, y = —3, and z = -5.

io. xV n.(-x)-y

12. x°z°

-2,214. y

13.y

15.2.v

fz -1

16. Reasoning Is it true that (-3b)4 = -12b4?Explainwhy or why not.

Exercises

Is the number written in scientific notation? If not,

explain why not.

17. 950 X 105 18. 7.23 X 1008

19. 1.6 X 10~6 20. 0.84 X 10~5


21. 2,793,000 22. 189,000,000

23. 0.000043 24. 0.0000000027

25. 3,860,000,000,000 26. 0.00000478

7-3 and 7-4 Multiplication Properties of Exponents

Quick ReviewTo multiply powers with the same base, add the exponents.

a'" • a" = am+", where a =r= 0 and m and n are integers

To raise a power to a power, multiply the exponents.

[a"1)" = a"1", where a =£ 0 and m and n are integers

To raise a product to a power, raise each factor in theproduct to the power.

(ab)n = anbn, where a =£ 0, b =£ 0, and n is an integer

ExampleWhat is the simplified form of each expression?

a. 31() • 31 = 3104"4 = 314

b. (.v5)7 = x5*7 = x35

c. [pq? = ptq8

7-5 Division Properties of Exponents

Quick ReviewTo divide powers with the same base, subtract theexponents.

a"1 —*V = a'" ", where a # 0 and m and n are integers

To raise a quotient to a power, raise the numerator and thedenominator to the power.(a \" a"j) = 777< where a =£ 0, b ± 0, and n is aninteger

ExampleWhat is the simplified form of I ^-)

'5x"\3 (5a.-1)3 53.V'-3 125x12m- 2->3

(« z2-

A \35x

•£

GPowerAlgebra.com

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Exercises

Complete each equation.

27. 32 • 3 = 310

29. a-2}/5 • x y =.rV1

31. (if4) = b2Q


33. 2d2 • d3

35. (5c-4)(-4m2c8)

37. {I2xiy-2)5{4xy~3)~7

28. a6 -a - a8

30. (55) = 515

32. {4x3y5) = 16x63'10

34. (^r)4

36. (1.342)5(1.34)-8

38. (-2;--4)2(-3^z8)-'

39. Estimation Each square inch of your body has about6.5 X 102 pores.Suppose the back ofyour hand hasan area ofabout 0.12 X 102 in.2. Abouthowmanypores are on the back of your hand? Write youranswer in scientific notation.

Exercises


40- 4

42.

41.21.V3

-l3x

if -4

Simplify each quotient. Write your answer in

scientific notation.

44.

46.

4.2 X 1082.1 X 1011

4.5 X 103

9 X 107

45.

47.

3.1 X 1041.24 X 102

5.1 X 1051.7 X 102

48. Writing List the steps that you would use to

\i0fl6-3

simplify

Chapter 7 Chapter Review 465

7-6 Exponential Functions

Quick ReviewAn exponential function involves repeated multiplicationof an initial amount a by the same positive number b. Thegeneral form of an exponential function is y = a • bx,where a =£ 0, b > 0, and b =£ 1.

ExampleWhat is the graph of y = \ •5*?Make a table of values. Graph the ordered pairs.

f y—

X

7 0'i

a-2

D150

-1110

012

152

225

2

VI i^H V

7-7 Exponential Growth and Decay

Quick ReviewWhen a > 0 and b > 1, the function y = a • bxmodelsexponential growth. The base b is called the growthfactor. When a > 0 and 0 < b < 1, the function

y = a • b*models exponential decay. In this case thebase b is called the decay factor.

ExampleThe population of a city is 25,000 and decreases 1% eachyear. Predict the population after 6 yr.

y = 25,000 • 0.99* Exponential decay function

= 25,000 • 0.996 Substitute 6 for x.

* 23,537 Simplify.

The population will be about 23,537 after 6 yr.

466 Chapter 7 Chapter Review

T

Exercises

Evaluate each function for the domain {1, 2, 3}.

49. f{x) = 4*

51. 7=40(1)*Graph each function.

53. J{x) = 2.5*

55. X*) =|-3*

50. y = 0.01*

52. /(*) = 3 • 2*

54. y = 0.5(0.5)*

56. y - 0.1*

57. Biology A population of 50 bacteria in a laboratoryculture doubles every 30 min. The functionp[x) = 50 • 2*models the population, where .vis thenumber of 30-min periods.

a. How many bacteria will there be after 2 h?

b. How many bacteria will there be after 1 day?

Exercises iTell whether the function represents exponential growthor exponential decay. Identify the growth or decay factor.

58. y = 5.2 • 3*

60. y= 0.15(|)'V59. f{x) = 7 • 0.32

61. gW 'LSI J*

62. Finance Suppose $2000 is deposited in an accountpaying 2.5% interest compounded quarterly. Whatwill the account balance be after 12 yr?

63. Music A band performs a free concert in a localpark. There are 200 people in the crowd at the startof the concert. The number of people in the crowdgrows 15% every half hour. How many people are inthe crowd after 3 h? Round to the nearest person.

7 Chapter Test

Do you know HOW?Simplify each expression.

1. r3(",3,-7

3\-4

2. (ft

3. <?v*c-l<P

4. 2j-9/z2(2/h-4)-6

5. (1.2)5(1.2)~2

6- (-3^-W7. If n = -3, which expression has the least value?

CE) n2n° <&n8n-5

C3D nn -nnn~4


8. 79,500,000,000

9. 0.0000000405

10. Pets More than 450,000 households in the United

States have reptiles as pets.


11. 8.4 X 10~6

12. 9.52 X 1011

13. 5.1807 X 104

Simplify each expression. Write each answer inscientific notation.

14. (6 X 104)(4.8 X 102;15 1.5 X107

.

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i Go to PowerAlgebra.com

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16. Medicine The human body normally producesabout 2 X 106 red blood cells per second.

a. Use scientific notation to express how many redblood cells your body produces in one day.

b. Onepintofbloodcontainsabout 2.4 X 1012red blood cells. How many seconds will it takeyour body to replace the red blood cells lost bydonating one pint of blood? How many days?


17. 3/3/4

18. (m4/?)5

19. (8rt2fc-3)4(3fl/?-2r3

20. (83) • 8-521 /35m4,

22.

V7m2

Do you UNDERSTAND?23. Error Analysis Find and correct the error in the

work shown below.

3*^=z$L

24. Open-Ended Write two equivalent expressions. Usea negative exponent in one of the expressions.

25. Compare and Contrast How is dividing powers withthe same base different from multiplying powers

with the same base?

26. Simplifythe expression f-jl in two different ways.Justify each step.

27. Reasoning Explain how you can use the propertyfor dividing powers with the same base to justify thedefinition of a zero exponent.

Chapter 7 Chapter Test

J

467

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