(Solution Manual) College Algebra

Blitzer, 7

Math & Statistics

Solution Manual

7th Edition

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Solution Manual for College Algebra, The downloadable file consists of 916 pages, and the total size of the file is 9.69 MB. Purchase now only for highly discounted price.

Author(s): Blitzer

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INSTRUCTOR’ S SOLUTIONS MANUAL DANIEL S. MILLER Niagara County Community College COLLEGE ALGEBRA SEVENTH EDITION Robert Blitzer Miami Dade College iii TABLE OF CONTENTS for INSTRUCTOR SOLUTIONS COLLEGE ALGEBRA 7E Chapter P Fundamental C oncepts of Algebra............................................................................. 1 Chapter 1 Equations and Inequalities ....................................................................................... 71 Chapter 2 Functions and Graphs............................................................................................. 201 Chapter 3 Polynomial and Rational Functions ....................................................................... 325 Chapter 4 Exponential and Logarithmic Functions................................................................ 467 Chapter 5 Systems of Equa tions and Inequalities................................................................... 543 Chapter 6 Matrices and Determinants .................................................................................... 663 Chapter 7 Conic Sections....................................................................................................... . 765 Chapter 8 Sequences, Indu ction, and Probability................................................................... 823 Chapter P Fundamental Concepts of Algebra Solutions Manual. 1 Section P.1 Check Point Exercises 1.2286 ( 3 ) 86 ( 1 33 ) x+−= + − 286 ( 1 0 ) 86 ( 1 0 0 )8 600 608=+=+=+= 2. a. Since 2014 is 14 years after 2000, substitute 14 for x. 2 24 330 3310 4(14) 330(14) 3310 8714Tx x=+ +=+ + = The average cost of tuition and fees at public U.S. colleges for the school year ending in 2014 was $8714. b. The formula underestimates the actual answer by $179. 3. The elements common to {3, 4, 5, 6, 7} and {3, 7, 8, 9} are 3 and 7. { 3 ,4 ,5 ,6 ,7 } { 3 ,7,8 ,9 } { 3 ,7 } ∩= 4. The union is the set containing all the elements ofeither set. { 3 ,4 ,5 ,6 ,7 } { 3 ,7,8 ,9 } { 3 ,4 ,5 ,6 ,7,8 ,9 } ∪= 5. 9, 1.3, 0, 0.3, , 9, 102π −− a. Natu ral numbers: 9 because 93= b. Whole numbers: 0, 9 c. Integers: 9, 0, 9− d. Ratio nal numbers: 9, 1.3, 0, 0.3, 9−− e. Irrational numbers: , 1 02π f. R eal numbers: 9, 1.3, 0, 0.3, , 9, 102π−−6. a. 12− Because 21 . 4 ,≈ the number inside the absolute value bars is negative. The absolute value of x when x < 0 is – x. Thus, () 12 12 2 1−= − − = − b. 3π− Because 3.14,π≈ the number inside th e absolute value bars is positive. The absolu te value of a positive number is the number itself . Thus, 33 .ππ−=− c. x x Because 0, .xx x>= Thus, 1x x xx== 7. 4( 5 ) 9 9−− = − = The distance between –4 and 5 is 9. 8.227(4 3 ) 2(5 )xx x x++ + 22 22 27(4 3 ) 2(5 ) 28 21 10 2 38 23xx x x xx x x xx=+ ++ =+ + +=+ 9. 64 [ 7( 2 ) ] x +− − 64 [ 7 2 ) ] 64[ 9 ] 63 64 42 4x x x x=+ −+=+ −=+ −=− Concept and Vocabulary Check P.1 1. expression 2. b to the nth power; base; exponent 3. formula; modeling; models 4. intersection; AB∩ 5. union; AB∪Chapter P Prerequisites: Fundamental Concepts of Algebra 2 Solutions Manual. 6. natural 7. whole 8. integers 9. rational 10. irrational 11. rational; irrational 12. absolute value; x, x 13. ba+; ba 14. ()ab c++ ; ()ab c 15. ab ac+ 16. 0; inverse; 0; identity 17. inverse; 1; identity 18. simplified 19. a Exercise Set P.1 1. 75 ( 1 0 ) 75 05 7+= + = 2. () 8 6 5 8 30 38+= + = 3. 6(3) 8 18 8 10−= −= 4. ()83 4 2 4 4 2 0 −= −= 5.283 ( 8 ) 6 4 2 4 8 8+= + = 6. ()265 63 6 3 0 6 6+= + = 7.27 6(7) 3 49 42 3 7 3 10− += − +=+= 8. ()28 7 8 4 64 56 4 8 4 12− += − +=+= 9.3345 ( 97 ) 45 ( 2 ) 4 5(8) 4 40 44+−= + =+ =+ =10. () ( ) ()3365 86 65 2 65 8 64 0 4 6+−= + =+ =+ = 11.28 3(8 2) 64 3(6) 64 18 46−− = − =−= 12. () ( )28 4 8 3 64 4 5 64 20 44− − =− =−= 13. 5( 2) 5(10 2) 2 14 2(10) 14 5(12) 6 52 10x x++=−− = =⋅ = 14. 7( 3) 7(9 3) 7(6)73 2 121 62 ( 9 ) 1 6 2x x−−== = ⋅ =−− 15. 23;2 , 41xyxyx+=− =+ () ( )22 3 4 41 2 8821 1 1−+ −+== = = −−+ − − 16. 2; 2 and 42xyxyxy x+=− =− () () ( ) ()22 4 44 0024 2 2 84 4−+ −+== =−− − − + 17. 5(50 32)9C=−5(18) 109== 50°F is equivalent to 10°C. 18. 55 5( 32) (86 32) (54) 3099 9CF=− = − = = 86°F is equivalent to 30°C. 19.224 60 16 4 60(2) 16(2) 4 120 16(4) 4 120 64 124 64 60ht t=+ − =+ −=+ − =+ −=− = Two seconds after it is kicked, the ball’s height is 60 feet. Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers Solutions Manual. 3 20. 2 246 0 1 6 4 60(3) 16(3) 41 8 01 6 ( 9 ) 41 8 01 4 4 184 144 40ht t=+ − =+ − =+ − =+ − =−= Three seconds after it is kicked, the ball’s height is 40 feet. 21. { }{ }{} 1, 2, 3, 4 2, 4, 5 2, 4 =  22. { }{ }{} 1, 3, 7 2, 3, 8 3 =  23. { }{ }{ } ,, ,, ,,set tes set =  24. { }{ }{ } ,,, ,,, ,,,real lear real =  25.{ }{ }{} 1, 3, 5, 7 2, 4, 6, 8,10 =  The empty set is also denoted by .∅ 26. { }{ }{} 1 , 3 , 5 , 7 5 ,3 ,1 −−− = or ∅ 27. { } ,,,abcd ∅=∅ 28. { },,wyz ∅=∅ 29. { }{ }{ } 1, 2, 3, 4 2, 4, 5 1, 2, 3, 4, 5 =  30. { }{ }{ } 1 ,3 ,7,8 2,3 ,8 1 ,2,3 ,7,8 =  31. { }{ } {}1, 3, 5, 7 2, 4, 6, 8,10 1, 2,3, 4,5, 6, 7,8,10= 32. { }{ }{ } 0,1,3,5 2, 4,6 0,1, 2,3, 4,5,6 =  33. { } { } ,, ,, ,, ,,aeiou aeiou ∅= 34. { } { } ,,, , ,,, ,em pt y em pt y ∅= 35. a. 100 b. 0, 100 c. 9, 0, 100− d. 49, , 0, 0.25, 9.2, 1005−− e. 3 f. 49, , 0, 0.25, 3,9.2, 1005−−36. a. 49 b. 0, 49 c. 7, 0, 49− d. 7, 0.6, 0, 49−− e. 50 f. 7, 0.6, 0, 49, 50−− 37. a. 64 b. 0, 64 c. 11, 0, 64− d. 511, , 0, 0.75, 646−− e. 5, π f. 511, , 0, 0.75, 5, , 646π −− 38. a. 4 b. 0, 4 c. 5, 0, 4− d. 5, 0.3, 0, 4−− e. 2 f. 5, 0.3, 0, 2, 4−− 39. 0 40. Answers will vary. An example is1.2 41. Answers will vary. An example is 2. 42. Answers will vary. An example is 2.− 43. true; –13 is to the left of –2 on the number line . 44. false; –6 is to the left of 2 on the number line . 45. true; 4 is to the right of –7 on the number line .Chapter P Prerequisites: Fundamental Concepts of Algebra 4 Solutions Manual. 46. true; –13 is to the left of –5 on the number line. 47. true; −π = −π 48. true; –3 is to the right of –13 on the number line. 49. true; 0 is to the right of –6 on the number line. 50. true; 0 is to the right of –13 on the number line. 51. 300 300 = 52. 203 203−= 53. 12 12−π = −π 54. 77−π = −π 55. 25 5 2−= − 56. 51 3 1 3 5−= − 57. 33133−−== −− 58. 77177−−== −− 59. 37 3 7 4 4−− − = − = −= 60. 51 3 5 1 3 8 8−− − = − = −= 61. 2( 5 ) 3 3 xy+=+ −= − = 62. 2( 5 ) 7 7 xy−=− −= = 63. 25 2 5 7 xy+ = + −=+= 64. 25 2 5 3 xy− = − −=−= − 65. 55155y y−−== = −− 66. 25 251( 1 ) 025 2 5xy xy−+=+ = += + − =−− 67. The distance is 21 7 1 5 1 5−= −= .68. The distance is 41 5 1 1 1 1−= − = . 69. The distance is 25 7 7−− = − = . 70. The distance is 6 8 14 14−− = − = . 71. The distance is 19 ( 4) 19 4 15 15−− −= −+= − = . 72. The distance is 26 ( 3) 26 3 23 23−− −= −+= − = . 73. The distance is 3.6 ( 1.4) 3.6 1.4 2.2 2.2−− − = −+ = − = . 74. The distance is 5.4 ( 1.2) 5.4 1.2 4.2 4.2−− − = −+ = − = . 75. 6 + (–4) = (–4) + 6; commutative property of addition 76. 11 (7 4) 11 7 11 4;⋅+ =⋅ +⋅ distributive property of multiplication over addition 77. 6 + (2 + 7) = (6 + 2) + 7; associative property of addition 78. 6( 23 ) 6( 32 ) ;⋅⋅= ⋅⋅ commutative property of multiplication 79. (2 + 3) + (4 + 5) = (4 + 5) + (2 + 3); commutative property of addition 80. 7( 1 1 8 ) ( 1 1 8 )7 ;⋅⋅ = ⋅ ⋅ commutative property of multiplication 81. 2 (–8 + 6) = –16 + 12; distributive property of multiplication over addition 82. 8(3 11) 24 ( 88)−+ = − + − ; distributive property of multiplication over addition 83. ()131 ; 33xxx+= ≠ −+, inverse property of multiplication 84. ()()44 0 ;xx ++ −+ = inverse property of addition 85. 5(3 4) 4 5 3 5 4 4 15 20 4 15 16xx xx+− = ⋅+ ⋅ − =+ −=+Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers Solutions Manual. 5 86. 2(5 4) 3 2 5 2 4 3 10 8 3 10 5xx xx+− = ⋅+ ⋅ − =+ −=+ 87. 5(3 2) 12 5 3 5 2 12 15 10 12 27 10xx x x xx x−+ = ⋅− ⋅ + =− +=− 88. 2(5 1) 14 2 5 2 1 14 10 2 14 24 2xx x x xx x−+ =⋅ − ⋅ + =− +=− 89. 7(3 5) 2(4 3) 73 75 24 23 21 35 8 629 29yy yy yy y−+ + =⋅ −⋅+⋅ +⋅=− + +=− 90. 4(2 6) 3(5 10) 42 46 35 3 1 0 82 4 1 53 0 23 6yy yy yy y−+ + =⋅ −⋅+⋅ +⋅=−+ +=+ 91. () ()53 2 7 2 1 5 1 0 7 2 81 2yy y y y−− += −− − =− 92. 4(5 3) (6 3) 20 12 6 3 14 15yy y y y−− += −− − =− 93. () [] 74 3 4 5 74 34 5 yy−−− = −− + [] 74 84 73 21 6 16 25y y y=− −=− + =− 94. [] []65 [ 8( 2 4 ) ] 65 82 4 65 1 22 66 01 0 10 54yy y y y−− −= − −+ =− −=− + =− 95. () ()22 22 22 22 22 2218 4 6 2 5 18 4 6 12 5 18 4 6 7 18 4 6 7 18 6 4 7 18 6 11 12 11xx xx xx xx xx xx +− − +  =+ −− +  =+ −− =+ − + =− + + =− + = +96. () ()22 22 22 22 22 2 214 5 7 2 4 14 5 7 14 4 14 5 7 10 14 5 7 10 14 7 5 10 14 7 15 71 5xx xx xx xx xx x x  +− − +    =+ −− +   =+ −− =+ − +=− + + =− + =+ 97. –(–14x) = 14 x 98. ()17 17yy −− = 99. –(2x – 3y – 6) = –2 x + 3y + 6 100. ()51 31 51 31xy xy−−− = − + + 101. []1(3 ) (4 ) ( 4 ) 03xy y xx++ − = + = 102. ()( )127 7 02yx x y y+− + =+= 103. 6 3 6 3 63−− > Since 63 , 6 3 .>− > − 104. 20 50 20 50 20 50−− < Since 20 50, 20 50 .<− < − 105. 3 0.65 0.6 0.6 0.6 0.6 0.6 0.6 − − = Since 30.6 0.6, 0.6 .5== − Chapter P Prerequisites: Fundamental Concepts of Algebra 6 Solutions Manual. 106. 5 2.52 2.5 2.5 .5 2.5 2.5 2.5 2− − = Since 52.5 2.5, 2.5 .2== − 107. 30 3 14 15 40 4 15 14 30 30 14 40 40−⋅ − 1515⋅ 14 0 1 0 1< Since 30 3 14 1501 ,40 4 15 14<− < ⋅ . 108. 17 18 50 5 18 17 60 6 17⋅− 1818⋅ 1750 50 60 60 1 0 1 0− > Since 17 18 50 510 , > 18 17 60 6>⋅ − . 109. 88 113 13 81 3 1 13 8 1 1 1 1 ÷− ⋅ = Since 8811 , =1 .13 13=÷ − 110. 44217 17 41 7 2 17 4 2 1 21 −÷ ⋅ > Since 4421 , 2 > .17 17>− ÷111.228 16 2 4 3 64 16 4 4 3 64 4 4 3 64 16 3 48 345−÷⋅ − = −÷ ⋅ − =− ⋅ −=− −=−= 112.2210 100 5 2 3 100 100 25 2 3 100 4 2 3100 8 3 92 3 89−÷ ⋅ − =−÷ ⋅ − =− ⋅ −=− −=−= 113. 2 22 2 2 252 3 52 9 [3 ( 2)] [9 ( 2)] 10 9 [9 2] 10 9 11 1 121⋅− ⋅−= −− −− −= + −= = 114. 22 210 2 3 4 5 12 (12 3 2) (12 6) 17 6 17 36÷+⋅ += −⋅ − = = 115. 83 [ 2 ( 25 )4 ( 86 ) ]83 [ 2 ( 3 )4 ( 2 ) ] 83 [ 68 ] 83 [ 2 ]86 14−− − − − =−−−− =− −=−−=+= 116. 8 3 [2 ( 5 7 ) 5 ( 4 2 ) ] 8 3 [2 (2 ) 5 ( 2 ) ] 83 [ 41 0 ] 83 [ 6 ]81 8 26−− − − − =−−−− =− −=−−=+= 117. 2( 2) 4( 3) 4 12 58 3 8 3 8 3−−− − +=−− =− =−Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers Solutions Manual. 7 118. 6( 4) 5( 3) 24 15 91 0 1 9 1 9−−− − +=−− −=− = 119. 22 2(5 6) 2 3 7 ( 1) 2 4 89 3 25 89 3 5 12 ( 4 ) 89 75 18 14 7 14 1 2−−− − − −=−⋅ −⋅ −=− −= −= =− 120. 22 212 3 5 2 3 12 3 5 4 9 733 6 736 45 1 3 10 36 20(13) 26 260 26 10÷⋅ + ÷⋅ +=+− +− ⋅=− =− =− =− 121. ()44 4 xx x x−+= − − = − 122. ()88 2 8xx x x x−− = − + = − 123. ()65 3 0 xx−= − 124. ()10 4 40 xx−= − 125. 52 3xxx−= 126. () 62 6 2 8xx x x x−− = + = 127. ()8 36836 56xx x x x−+ = − − = − 128. () 83 6 83 1 8 3 1 0 xx x−+ = − − = − −129. a. () () ()722010 7220 2010 720010 140Ha H=−=−= = The lower limit of the heart rate for a 20-year- old with this exercise goal is 140 beats per minute. b. () () ()42205 4220 205 42005 160Ha H=−=−= = The upper limit of the heart rate for a 20-year- old with this exercise goal is 160 beats per minute. 130. a. () () ()12202 1220 302 11902 95Ha H=−=−= = The lower limit of the heart rate for a 30-year- old with this exercise goal is 95 beats per minute. b. () () ()32205 3220 305 31905 114Ha H=−=−= = The upper limit of the heart rate for a 30-year- old with this exercise goal is 114 beats per minute. 131. a.2 221 862 15,552 21(14) 862(14) 15,552 31,736Tx x=++=++ = The formula estimates the cost to have been $31,736 in 2014. Chapter P Prerequisites: Fundamental Concepts of Algebra 8 Solutions Manual. b.This overestimates the value in the graph by $35. c.2 221 862 15,552 21(20) 862(20) 15,552 41,192Tx x=++ =++ = The formula projects the cost to be $41,192 in 2020. 132. a.2 221 862 15,552 21(12) 862(12) 15,552 28,920Tx x=++ =++ = The formula estimates the cost to have been $28,920 in 2012. b.This underestimates the value in the graph by$136. c. 2 221 862 15,552 21(22) 862(22) 15,552 44,680Tx x=++ =++ = The formula projects the cost to be $44,680 in 2022. 133. a. () 0.05 0.12 10,000 0.05 1200 0.12 1200 0.07xx xx x+− =+ − =− b.1200 0.07 1200 0.07(6000) $780x −= − = 134. a. 0.06 0.5(50 ) 0.06 25 0.5 25 0.44tt t t t+− = + − =− b. 0.06(20) 0.5(50 20) 1.2 0.5(30)1.2 1516.2 miles+− =+=+= 135. – 143. Answers will vary. 144. does not make sense; Explanations will vary. Sample explanation: Models do not always accurately predict future values. 145. does not make sense; Explanations will vary. Sample explanation: To use the model, substitute 0 for x.146. makes sense 147. does not make sense; Explanations will vary. Sample explanation: The commutative propertychanges order and the associative property changes groupings. 148. false; Changes to make the statement true will vary. A sample change is: Some rational numbers are notintegers. 149. false; Changes to make the statement true will vary.A sample change is: All whole numbers are integers. 150. true 151. false; Changes to make the statement true will vary.A sample change is: Some irrational numbers arenegative. 152. false; Changes to make the statement true will vary. A sample change is: The term x has a coefficient of 1. 153. false; Changes to make the statement true will vary. A sample change is: 53 ( 4 ) 53 1 2 3 7 . xx x+− = + − = − 154. false; Changes to make the statement true will vary. A sample change is: 2. xx x−− = − 155. true 156. 21 . 4 1.4 1.5 21 . 5≈ < < 157. 3.5 π−> − 158. 3.141.572 1.5712 1.57 1.571 3.14 22−= − π−≈ − −> − π−> − 159. a. 43 7() ( ) b b bbbb bbb b⋅=⋅ ⋅ ⋅ ⋅ ⋅= b.55 1 0() () b b bbbbb bbbbb b⋅ = ⋅⋅⋅⋅ ⋅⋅⋅⋅ = c.add the exponentsSection P.2 Exponents and Scientific Notation Solutions Manual. 9 160. a.7 4 3b bbbbbbbbbbb b⋅⋅⋅⋅⋅⋅==⋅⋅ b. 8 6 2b bbbbbbbbbbb b⋅⋅⋅⋅⋅⋅⋅==⋅ c.subtract the exponents 161.36.2 10 6.2 10 10 10 6200×= × × × = It moves the decimal point 3 places to the righ t. Section P.2 Chec k Point Exercises 1. a. 32 3 2 53 3 3 3 or 243+⋅= = b.() ( )34 26 3 2 4 641 0 4 1 0xy xy x x y y =⋅ ⋅ ⋅ ⋅ ⋅ 32 46 51 040 40xyxy++=⋅= 2. a.6 3 3(3 )(3 ) 2 7 (3 )−=− = − − b. 14 8 14 8 14 3 8 5 11 3 35 3 527 27993 3xy x yxy x y xy x y−−=⋅ ⋅= = 3. a.2 211525 5−== b.3 3111(3 )27 27 (3 )−−= = = −− − c. 2 2 2 211 414 1 61 1 4 4−== ⋅ = = d. 4 64 4 661333yxy y xx−=⋅ ⋅ = 4. a. ()233 2 633 3 o r 7 2 9⋅== b.()277 ( 2 ) 1 4 141yyy y−−−== =c.()433 ( 4 ) 1 2bbb−−− −== 5.33 3 3( 4 ) ( 4) ( ) 64xx x−= − = − 6. a.()5 5 552 23 2 y yy−  −−= =  b. ()33 551 5 332 7 3xxx == 7. a. () ()() ()44 44 3 6 3 6 12 2422 1 6xy x y x y == b.() ( ) ()25 3 2 5 3 3863 6 3 18xy x y x xy y xy−= − ⋅ ⋅ ⋅ ⋅ ⋅ =− c. 12 2 12 2 16 4 16 4100 100 20 20xy x y xy x y−− =    12 16 2 ( 4) 46 6 45 5 5xyxy y x−− − −== = d. () () ()2 22 4245 5 x x yy− −− −= () () ()22 24 22 8 8 22 8 25 5 5 25x y x y y x y x−− − −− −= = = = 8. a. 92.6 10 2,600,000,000−× = − b.63.017 10 0.000003017−×= 9. a. 95, 210,000,000 5.21 10 =× b.80.00000006893 6.893 10−−= − ×Chapter P Prerequisites: Fundamental Concepts of Algebra 10 Solutions Manual. 10. ()72 7410 10 4.1 10 10×= × × ()27 94.1 10 10 4.1 10=× ×=× 11 a. () ( )577.1 10 5 10−×× () ()57 2 12 12 17.1 5 10 10 35.5 10 3.55 10 10 3.55 10 10 3.55 10− − − − −=⋅ ×⋅ =× =× × =× × =× b. 66 331.2 10 1.2 10 3 31 0 1 0−−×=⋅ × 6(3 ) 9 80.4 10 0.4 10 41 0−−=×=×=× 12. 10 10 10 55 54.08 10 4.08 10 4.08 10 680,000 6.8 6.8 10 10 0.6 10 60,000××== ⋅ × =×= The average salary was $60,000 per U.S. police officer. Concept and Vocabulary Check P.2 1.mnb+; add 2.mnb−; subtract 3. 1 4. 1 nb 5. false 6.nb 7. true 8. a number greater than or equal to 1 and less than 10; integer 9. true10. false Exercise Set P.2 1.252( 5 5 ) 22 5 25 0⋅= ⋅ ⋅= ⋅= 2. 262( 6 6 ) 23 6 27 2⋅= ⋅ ⋅= ⋅= 3.6(2 ) (2 ) (2 ) (2 ) (2 ) (2 ) (2 ) 6 4− = −−−−−−= 4.4(2 ) (2 ) (2 ) (2 ) (2 ) 1 6− = −−−−= 5.62 222222 6 4− = − ⋅⋅⋅⋅⋅= − 6.42 2222 1 6−= − ⋅ ⋅ ⋅ = − 7.0(3 ) 1−= 8.0(9 ) 1−= 9.031−= − 10.091−= − 11.3 31114444 6 4 4−== =⋅⋅ 12.6 61112222222 6 4 2−== =⋅⋅⋅⋅⋅ 13.23 2 3 52 2 2 2 22222 3 2+⋅ = = =⋅⋅⋅⋅= 14.32 3 2 53 3 3 3 33333 2 4 3+⋅ = = =⋅⋅⋅⋅= 15.23 2 3 6( 2) 2 2 222222 6 4⋅= = =⋅⋅⋅⋅⋅= 16.32 3 2 6( 3) 3 3 333333 7 2 9⋅= = =⋅⋅⋅⋅⋅= 17. 8 84 4 422 2 2222 1 6 2−== = ⋅ ⋅ ⋅ = 18. 8 84 4 433 3 3333 8 1 3−= = =⋅⋅⋅= 19.33 1 2 211 133 3 333 9 3−− + −⋅= = = = =⋅ 20.33 1 2 211 122 2 222 4 2−− + −⋅= = = = =⋅ 21. 3 37 4 7421 1 1222222 1 6 22−−== = = =⋅⋅⋅Section P.2 Exponents and Scientific Notation Solutions Manual. 11 22. 4 47 3 7331 1 133333 2 7 33−−== = = =⋅⋅ 23.2 221 yxy y xx−=⋅ = 24.3 331 xxy x yy−=⋅ = 25.05 5 51 xy y y =⋅ = 26.70 7 71 xy x x⋅=⋅ = 27.37 3 7 1 0xx x x+⋅= = 28.11 5 11 5 16xx x x+⋅= = 29.51 0 5 1 0 5xx x x−− +⋅= = 30.61 2 6 1 2 6xx x x−− +⋅= = 31.37 3 7 2 1()xx x⋅== 32.11 5 11 5 55()xx x⋅== 33.53 5 3 1 5 151()xx x x−− ⋅ −== = 34.64 6 4 2 4 241()xx x x−− ⋅ −== = 35. 14 14 7 7 7xxx x−== 36. 30 30 10 20 10xxx x−== 37. 14 14 ( 7) 14 7 21 7xxx x x−− + −== = 38. 30 30 ( 10) 30 10 40 10xxx x x−− + −== = 39.32 2 32 2 3 2 6(8 ) 8 ( ) 8 64xx x x⋅== = 40.42 2 42 2 4 2 8(6 ) (6) ( ) 6 36xx x x⋅== = 41. 3 3 334 ( 4) 64 x xx− −= = −42. 33 336( 6 ) 2 1 6 y yy −−= = −  43.25 2 2 2 2 5 2 22 52 41 0(3 ) (3 )( ) ( ) 9 9xy x y xy xy⋅⋅−= −⋅ = = 44.4 63 3 43 63 43 63 12 18(3 ) (3 )( )( ) 2727xy x y xy xy⋅⋅−= − =− =− 45.47 4 7 4 7 1 1(3 )(2 ) 3 2 6 6xx x x x x+=⋅ ⋅ = = 46.51 2 5 1 2 5 1 2 1 7(11 )(9 ) 11 9 99 99xx x x x x+=⋅ = = 47.36 4 3 6 4 36 14 95(9 ) (2 ) (9 ) (2 ) 18 18xy xy xxy y xy xy++−− = − − = = 48.4 7 11 4 7 11 47 1 1 1 11 12(5 ) (6 ) (5 ) (6 ) 3030xy xy xxy y xy xy++−− = − − = = 49. 20 20 20 4 16 4488442 2xxxx xx−== = 50. 24 24 24 6 18 6620 202210 10xxxx xx−== = 51. 13 4 13 4 23 2 3 13 2 4 3 1125 25 5 5 5 5ab a b ab a b ab ab−−   ⋅=    − −⋅    =− =− 52. 14 6 14 6 73 7 3 14 7 6 3 7335 35 7 7 55ab a b ab a b ab ab−−  =    − −    =− =− 53. 77 71 4 7 14 14 714 14 2227 7bbbb bb b−−== ⋅ = = Chapter P Prerequisites: Fundamental Concepts of Algebra 12 Solutions Manual. 54. 10 10 20 20 10 20 10 1020 20 10 10 2 2 2bb bb b b b− −= = = = 55.32 2 32 26 26 6(4 ) (4 )( ) 4 1 4 1 16xx x x x−− − −−= = = = 56.23 3 2 ( 3 ) 36 36 6(10 ) 10 10 1 10 1 1000xx x x x−− ⋅ − −−= = = = 57. 35 37 5(9 ) 79 41 4 14 424 3 4 32 3 4 3 4xyxy xy xy y x−− − − −⋅= = = 58. 49 41 2 9(3 ) 12 3 81 2 12 810 1 3 30 1 3 3xyxy xy xy y x−− − − −== = 59. 232 6 2 655 2 25xx y y x y−−−  ==  − 60. 33 4 4 3 34 3 3 123 3 3 27xy y x y x y x− ⋅ =     = = 61. ()3 323 42 10 3 10 4 35 6 15 1815 3 5 3 27ab b ab a b a b a−− −− −−=    −= −= 62. ()3 382 14 8 17 2 17 14 310 30 930 3 10 3 3 27ab b ab a b a b a−− −− −−=    −= −= 63. 052 3431 12ab ab− −=  64. 053 3541 12ab ab− −=  65.23.8 10 380×= 66.29.2 10 920×= 67.461 0 0 . 0 0 0 6−×= 68.57 10 0.00007−×= 69.67.16 10 7,160,000−×= − 70.68.17 10 8,170,000−×= − 71.17.9 10 0.79−×= 72.16.8 10 0.68−×=Section P.2 Exponents and Scientific Notation Solutions Manual. 13 73.34.15 10 0.00415−−× = − 74.33.14 10 0.00314−−× = − 75.106.00001 10 60,000,100,000−× = − 76.107.00001 10 70,000,100,000−× = − 77.432,000 3.2 10 =× 78.464,000 6.4 10 =× 79. 17638,000,000,000,000,000 6.38 10=× 80.17579,000,000,000,000,000 5.79 10 =× 81.35716 5.716 10−= −× 82.33829 3.829 10−= −× 83.30.0027 2.7 10−=× 84.30.0083 8.3 10−=× 85.90.00000000504 5.04 10−−= − × 86.90.00000000405 4.05 10−−= − × 87.() ( ) () ()43 4 3 43 73 10 2.1 10 3 2.1 10 10 6.3 10 6.3 10+×× = ×× =× =× 88.() ( )43 721 0 4 . 11 0 8 . 21 0×× = × 89.() ( ) () ()15 11 15 11 15 ( 11) 41.6 10 4 10 1.6 4 10 10 6.4 10 6.4 10−− +−×× = ×× =×=× 90.() ( ) () ()15 11 15 11 15 ( 11) 41.4 10 3 10 1.4 3 10 10 4.2 104.2 10−− +−×× = ×× =×=× 91.() ( ) () ()84 8 4 8(4 ) 12 116.1 10 2 10 6.1 2 10 10 12.2 10 12.2 101.22 10−− − − −+− − −×× = ×× =× =×=×92.( )( )84 1 2 115.1 10 3 10 15.3 10 1.53 10−− − −×× = × =× 93. ( )( ) () ()84 84 84 12 13 134.3 10 6.2 10 4.3 6.2 10 10 26.66 10 26.66 102.666 10 2.67 10+×× =× × =× =×=× ≈ × 94. ( )( )84 84 1 2 13 138.2 10 4.6 10 37.72 10 37.72 10 3.772 10 3.77 10+×× =×=×=× ≈ × 95. 88 55 85 38.4 10 8.4 10 4 41 0 1 0 2.1 10 2.1 10−×=× × =× =× 96. 8 85 3 56.9 102.3 10 2.3 10 31 0− ×=× =× × 97. 44 22 4(2 ) 653.6 10 3.6 10 9 91 0 1 0 0.4 10 0.4 10 4 10−− −−×=× × =×=× = × 98. ()4 4(2 ) 6 2 16 51.2 100.6 10 0.6 10 21 0 61 0 1 0 61 0−− − −×=× =× × =× × = × 99. 22 66 26 84.8 10 4.8 10 2.4 2.4 10 10 21 0 21 0−− −− −×=× × =× =× 100. 2 26 8 67.5 1031 0 31 0 2.5 10− −− − ×=× =× × 101. 22 66 2(6 ) 432.4 10 2.4 10 4.8 4.8 10 10 0.5 10 0.5 10 5 10−− −− −−−×=× × =×=× = ×Chapter P Prerequisites: Fundamental Concepts of Algebra 14 Solutions Manual. 102. 2 2(6 ) 6 431.5 100.5 10 51 0 0.5 10 5 10− −−− −×=× × =× = × 103. 11 4 11 4 11 ( 4) 15480,000,000,000 4.8 10 0.00012 1.2 10 4.8 10 1.2 10 41 0 41 0− − −−×= × =× =× =× 104. 11 3 11 ( 3) 14282,000,000,000 2.82 10 0.00141 1.41 10 21 0 21 0− −−×= × =×=× 105. () ( )43 4 43 3 40.00072 0.003 0.00024 7.2 10 3 10 2.4 10 7.2 3 10 1091 02.4 10−− − −− − −× ×× = × ×⋅=× = × 106. () ( ) () () ()43 33 116 6 76.6 10 1 1066000 0.001 0.003 0.002 31 0 21 0 6.6 101.1 10 61 0 1.1 10− −− −− −×××=× ×× ×== × × =× 107. () () ()3263 36 321 33 66 0 01xyxy xyxy xy x y−−− −− −−− −= == = 108. () () ()2224 36 32 7 43 26 8 7 8xyxy xyxy yxy x y x−−− −−− −− −− −= == = 109.() ()5 36 36 55 48 6 48 6 8 622 2 2 2 21 6xy z x xy z x yyxy z xz xz− −− −− −− −− −=⋅ ⋅ == =110.() ()3 47 4733 27 7 27 7 7 733 3 3 3 39xy z x xy z x yyxy z xz xz− −− −−−− −− −=⋅ ⋅ == = 111. ()234 5268 1 0 345 12 16 20 12 16 201xyzxyz xyz xyz xyz− − −−− −−−=  == 112. ()445 6481 01 2 45 6 32 40 48 32 40 481xyzxy z xyz xyz xyz− − −− − −− − =  == 113. () ( ) ( ) () () ( ) () ()220121 4 3 3 3 235 222 28 6 261 0 18 622 1 6 2 22 1 2 4xy xy xy xy xy xy xy xy−−−−− − − −− −− −−= = 114. () ( ) ( ) () () ( ) () ()22 0131 6 4 33 246 262 21 2 8 281 2 26 622 9 2 22 1 2 4xy xy x y xy xy x y xy xy−−−−− − − −− −− −−= = 115. a. 123.18 10× b.83.20 10× c. 12 12 88 43.18 10 3.18 10 3.20 3.20 10 10 0.9938 10 9938×=× × ≈× ≈ $993 8 per American 116. a. 123.02 10× b.83.19 10×Section P.2 Exponents and Scientific Notation Solutions Manual. 15 c. 12 12 88 43.02 10 3.02 10 3.19 3.19 10 10 0.9467 10 9467×=× × ≈× ≈ $946 7 per American 117. a. 131.89 10× b.461 0× c. 13 13 44 9 81.89 10 1.89 10 6 61 0 1 0 0.315 10 3.15 10315,000,000×=× × =× =× = 315,0 0,000 Americans 118. a. 131.89 10× b.112.54 10× c. 13 13 11 11 21.89 10 1.89 10 2.54 2.54 10 10 0.74 1074×=× × ≈×≈ approx imately 74 years 119. a. 121.09 10× b.73.2 10× c. 12 12 77 51.09 10 1.09 10 3.2 3.2 10 10 0.340625 10 34,062.5×=× × =×= 34,06 2.5 years 120. – 128. Answers will vary . 129. does not make sense; Explanations will vary . Sample explanation: 3 9 27 1236( ) 36 not 36 .xx x = 130. makes sense 131. does not make sense; Explanations will vary . Sample explanation: 124.6 10× represents ov er 4 trillion . The entire world population is measured in billions (910).132. makes sense 133. false; Changes to make the statement true will vary . A sample change is: 2344 .−−> 134. true 135. false; Changes to make the statement true will vary . A sample change is: 44 1( 2) 2 because 16 .16−−≠ ≠ 136. false; Changes to make the statement true will vary . A sample change is: 22 5555 22 .−−⋅= ⋅ 137. false; Changes to make the statement true will vary . A sample change is: 534.7 5347. ≠ 138. false; Changes to make the statement true will vary . A sample change is: 30 30 ( 5) 35 581 041 0 41 0. 21 0−− −×=× =× × 139. false; Changes to make the statement true will vary . A sample change is: 53(7 10 ) (2 10 ) 700,000.002.−×+ × = 140. true 141. The doctor has gath ered: 12 11 2 1 3222 24 4 42−−+= += + = So, 31144−= is remaining. 142. ,,AC DbM N bM bN== = AC Dbb b ACD= =+ 143. 70 bts min60 min⋅ hr24 hrs⋅ day365 days⋅ yr80 yrs⋅ 9 970 60 24 365 80 beats 2943360000 beats 2.94336 10 beats 2.94 10 beats=⋅ ⋅⋅ ⋅ = =× ≈× The heartbeats approximately 92.94 10× times over a lifetime of 80 years. 144. Answers will vary.Chapter P Prerequisites: Fundamental Concepts of Algebra 16 Solutions Manual. 145. a. 16 4 4 2 8⋅= ⋅ = b. 16 4 64 8⋅= = c. 16 4 16 4⋅= ⋅ 146. a. 300 17.32≈ b.10 3 17.32 ≈ c. 300 10 3= 147. a. 21 10 31xx x+= b. 21 2 10 2 31 2 += Section P.3 Check Point Exercises 1. a. 81 9= b. 93−= − c. 11 25 5= d. 36 64 100 10+= = e. 36 64 6 8 14+= + = 2. a. 75 25 3 25 3 5 3=⋅ = = b. 2 2 251 0 5 1 0 50 25 2 25 2xxx x x x x⋅= ⋅ = =⋅ =⋅ 52x= 3. a.25 25 5 16 4 16== b. 33150 150 2 2xx x x= 2 275 25 3 53x x x= =⋅ =4. a. 81 3 91 3 ( 8 9 )3 17 13+= + = b. 17 20 17 11 7 2 01 7 (1 20) 17 19 17xx xx x x− =−=−=− 5. a. 52 7 1 2 + 59 3 4 353 3 2 3 15 3 2 3 (15 2) 3 17 3=⋅ + ⋅=⋅ +=+ =+ = b. 61 8 48 xx− 69 2 44 263 2 42 2 18 2 8 2 (18 8) 2 10 2xx xx xx x x=⋅ −⋅=⋅ −⋅ =− =−= 6. a. If we multiply numerator and denominator by 3, the denominator becomes 33 93 .⋅= = Therefore, multiply by 1, choosing 3 3 for 1. 5 5 35 35 3 3 33 3 9=⋅= = b. The smallest number that will produce a perf ect square in the denominator of 6 12 is 3 becau se 12 3⋅= 36 6.= So multiply by 1, choosing 33 for 1. 66 3 6 3 6 336 12 12 3 36=⋅ = = =Section P.3 Radicals and Rational Exponents Solutions Manual. 17 7. Multiply by45 45−−. 88 4 5 45 45 45 8(4 5) 224( 5 ) 8(4 5) 16 5 8(4 5) 32 8 5 or 11 11−=⋅ ++ − −= − −=− −−= 8. a.33 3 3340 8 5 8 5 2 5=⋅ =⋅= b.55 55 5 58 8 64 32 2 2 2⋅= = ⋅= c. 3 3 3125 125 5 27 3 27== 9.3338 1 43 − 33 33 33 3 332 7 3 43 33 3 4 3 93 43 (9 4) 3 53=⋅ − =⋅ − =− =−= 10. a.1 225 25 5== b. 1 3 388 2== c. 1 4 481 81 3−= − = − d. ()1 3388 2−= − = − e. 1 3 1 3 311 1273 27 27− == = 11. a. ()444 3 327 27 (3) 81 == = b. ()333 2 244 ( 2 ) 8== = c. ()2 5 22 25 511 1 1324 23232− == = = 12. a.() ()4/3 8/325xx ()4/3 8/3 4/3 (8/3) 12/3 425 10 1010xx xxx +=⋅ ⋅ = == b. 33224420 20 5 5xx xx= () () ()32 8322 5244 4 4x x x− −= = = 13. 3 16 263xx x x=== Concept and Vocabulary Check P.3 1. principal 2.28 3. a 4. ab⋅ 5. a b 6. 18 3 7. 5; 63 8. 73− 9. 10 2+ 10. index; radican d 11.5(2 )− 12. a;a 13.na 14. 2; 8Chapter P Prerequisites: Fundamental Concepts of Algebra 18 Solutions Manual. Exercise Set P.3 1.236 6 6== 2.225 5 5== 3.236 6 6−= −= − 4.225 5 5−= −= − 5. 36−, The square root of a negative number is not real. 6. 25−, The square root of a negative number is not real. 7. 25 16 9 3−= = 8. 144 25 169 13+= = 9. 25 16 5 4 1−= − = 10. 144 25 12 5 17+= + = 11.2(1 3 ) 1 6 9 1 3−= = 12.2( 17) 289 17−= = 13. 50 25 2 25 2 5 2=⋅ = = 14. 27 9 3 9 3 3 3=⋅ = = 15.22 2 245 9 5 95 95 35xx x x x=⋅ = = = 16.22 2 2125 25 5 25 5 25 5 55xx x x x=⋅ = = =17. 26 2 6xx x x⋅=⋅ 2 212 43 23x x x= =⋅ = 18. 210 8 10 8 80xx x x x⋅= ⋅ = 216 5 45x x=⋅= 19.32xx x x x=⋅ = 20.32yy y y y=⋅ = 21.22262 6xxx x⋅= ⋅ 3 212 43 23x xx xx= =⋅ = 22.2263 6 3xx x x⋅= ⋅ 2318 92 32x xx xx= =⋅ = 23. 11 1 81 9 81== 24. 11 1 49 7 49== 25. 49 49 7 16 4 16== 26. 121 121 11 93 9== 27. 33 2 48 4816 43 3xxxxx x== = 28. 33 2 72 72938 8xxxxx x== = Section P.3 Radicals and Rational Exponents Solutions Manual. 19 29. 44 3 2150 150 3 3 50 25 2 52xx x x x xx xx= = =⋅ = 30. 44 3 224 24 3 3 8 42 22xx x x x xx xx= = =⋅ = 31. ()33 11 31 4 4200 200 10 10 20 20 45 225xx x x x x x x−− −−= = = =⋅ = 32. ()33 31 11 44 2500 50050 10 10 50 25 2 5 2xxx x x xx x−− −−== == ⋅ = 33. 73 63 ( 7 6 )3 1 33 += + = 34. 85 1 15 ( 81 1 )5 1 95 += + = 35. 6 17 8 17 (6 8) 17 2 17 xx x x−= − = − 36. 4 13 6 13 (4 6) 13 2 13 xx x x−= − = − 37. 83 2 4 23 2 22 32 (2 3) 2 52+= ⋅ + =+ =+= 38. 2 0 65 4 5 65 25 65 (2 6) 5 85+= ⋅ + =+=+=39. 50 8 25 2 4 2 52 22 (5 2) 2 32xx x x xx x x−=⋅ − ⋅ =−=−= 40. 6 3 2 8 97 47 37 27 (3 2) 7 7xx x x xx x x−= ⋅ − ⋅ =−=−= 41. 31 8 55 0 392 52 52 33 2 55 292 2 52 (9 25) 2 34 2+= ⋅ + ⋅ =⋅ +⋅=+ =+ = 42. 4 12 2 75 4 4 3 2 25 3 42 3 25 3 83 1 03 (8 10) 3 23−= ⋅ − ⋅ =⋅ −⋅=− =− =− 43. 38 3 2 37 2 7 5−+ − 3 42 1 62 3 3 62 2 53 32 2 4 2 36 2 5 3 62 42 1 82 53 20 2 5 3=⋅ − ⋅ + ⋅ −⋅=⋅ − +⋅ − =−+ −=− 44. 3 54 2 24 96 4 63 −− + 39 6 24 6 1 6 6 49 7 3 36 2 26 46 4 37 96 46 46 1 27 61 2 7=⋅ −⋅ −⋅ +⋅ =⋅ −⋅ − +⋅ =−−+=+ 45. 11 77 7 77 7=⋅= 46. 22 1 0 2 1 0 1 0 10 5 10 10 10=⋅= = 47. 22 5 1 0 5 55 5=⋅=Chapter P Prerequisites: Fundamental Concepts of Algebra 20 Solutions Manual. 48. 77 32 1 3 33 3=⋅= 49. 2213 13 3 11 31 1 31 1 31 1 13(3 11) 3( 1 1 ) 13(3 11) 91 1 13(3 11) 2−=⋅ ++ − −= − −=− −=− 50. 2233 3 7 37 37 37 3(3 7 ) 3( 7 ) 3(3 7 ) 97 3(3 7 ) 2−=⋅ ++ − −= − −=− −= 51. 2277 5 2 52 52 52 7( 5 2) (5 ) 27( 5 2) 54 7( 5 2)+=⋅ −− + += − +=− =+ 52. 2255 3 1 31 31 31 5( 3 1) (3 ) 1 5( 3 1) 31 5( 3 1) 2+=⋅ −− + += − +=− +=53. 2266 5 3 53 53 53 6( 5 3) (5 ) (3 )6( 5 3) 53 6( 5 3) 2 3( 5 3)−=⋅ ++ − −= − −=− −= =− 54. 2211 11 7 3 73 73 73 11( 7 3) (7 ) (3 ) 11( 7 3) 73 11( 7 3) 4+=⋅ −− + += − +=− += 55.33 3125 5 5== 56.33 382 2== 57.3 3 38( 2 ) 2−= − = − 58.3 3 3125 ( 5) 5−= − = − 59.416− is not a real numb er. 60.481− is not a real numb er. 61.4 4(3 ) 3 3−= − = 62.4 4(2 ) 2 2−= − = 63.5 5(3 ) 3−= − 64.5 5(2 ) 2−= − 65. 5 5 511 1 32 2 2−= −= −Section P.3 Radicals and Rational Exponents Solutions Manual. 21 66. 6 6 6611 1 64 22== 67.33 33 332 8 4 8 4 2 4=⋅ = = ⋅ 68.3150 cannot be simplified further. 69.3343 3xx x x x=⋅ = ⋅ 70.33 353 2 2xx x x x== 71.33 33 3 3 396 5 4 2 7 2 2 7 2 3 2⋅= = ⋅ = = 72.33 33 312 4 48 8 6 2 6⋅= = ⋅ = 73. 5 66 5 5 5 564 6432 22 2xxxxx x== = 74. 4 55 4 4 4 4162 16281 32 2xxxxx x== = 75.55 542 32 72 += 76.55563 23 83 += 77.33 3 351 6 5 4 582 2 72 += ⋅ + ⋅ 33 33 352 2 3 2 10 2 3 2 13 2=⋅ + =+= 78.3 33 332 4 8 1 8 3 2 7 3 += ⋅ +⋅ 33 33 332 3 3 3 63 33 93=⋅ +=+= 79.3 3 354 128xy y x− 3 3 3 33 327 2 64 2 32 42 2xy y x yx yx yx=⋅ − ⋅ =− =−80.3 3 324 81xy y x− 3 3 3 33 383 2 73 23 33 3xy y x yx yx yx=⋅ − ⋅ =− =− 81.328 2 2+=+ 82.331 5+ will not simplify . 83.1/236 36 6 == 84.1/2121 121 11 == 85.1/3 388 2== 86.1/3 327 27 3 == 87. ()22/3 2 3125 125 5 25 == = 88. ()22/3 388 4== 89.4/5 4/5 411 13216 32 2−== = 90.5/2 5/2 5 511 1 1161024 16 ( 16) 4−== = = 91. () ()1/3 1/4 1/3 1/4 1/3 1/4 7/1272 7 2 14 14xx x x x x+=⋅ ⋅ =⋅ = 92.2/3 3/4 2/3 3/4 2/3 3/4 17/12(3 )(4 ) 3 4 12 12xx x x x x+=⋅ ⋅ =⋅= 93. 1/2 1/2 1/4 1/4 1/2 1/4 1/420 20 5 5 4 4xx xx x x−= =⋅ = 94. 3/4 3/4 3/4 1/3 5/12 1/3 1/372 72889 9xxxx xx−== ⋅ =Chapter P Prerequisites: Fundamental Concepts of Algebra 22 Solutions Manual. 95.()32/3 2/3 3 2xx x⋅== 96.4/5 5 4/5 5 4()xx x⋅== 97.3 4 6 1 /2 1 /2 41 /2 61 /2 2(25 ) 25 5 xy x y x y⋅⋅== 98.9 6 1/3 1/3 9/3 6/3 3 2(125 ) 125 5 xy x y xy == 99. 31 4 3 314 41 2 11 12 123 2727y yy yy− == 2 8 3 1227 27yy == 100. () ()441/5 4 1/5 3/10 3/1022yy yy= 4/5 4/5 3/10 1/2 3/101616 16yyy y−== = 101.422 / 4 1 / 255 5 5=== 102.422 / 4 1 / 277 7 7== = 103.366 / 3 2xx x== 104.3 412 12/4xx x== 105.66 / 2 344 / 2 2xxx== 106.99 / 3 366 / 3 2xxx== 107. 63 21 63 2 93 99 33 xy x y x y xy === 108. 48 1 2 48 2 12 312 12 3 3 xy x y x y x y == = 109.3 33 416 625 2 25 27 3+= + = =110.3 33 3 3 3 3169 9 1000 216 13 3 10 6 16 16 44 8 2++ + =+ + +=+ =+ = = 111.() ( ) () ()( )( ) ( ) () ( ) ()() ()1/224 1 / 2 1/2 1/21/2 24 1 / 2 21 / 2 41 / 2 1/2 1/2 21 / 2 12 1 / 2 1 1 2 23 / 2 3/249 49 1 49 11 77 1 7 7xy x y xy x y xy x y xy x y x y xxy y−− −−− − −− − −+ −+ −= = =⋅ = == 112.() ( ) ( ) () ( ) () ( ) ( ) ( ) ( ) ()1/3 663 5 / 6 1 / 3 61 / 3 31 / 3 5 / 66 1 / 36 1/3 12 215 2 2 5 3 318 8 22 22xy x y xy x y xy x y x y xxyy−− −− +− −− − + −= == == 113.() ()( ) ()( ) () ( ) ()65/4 1/3 65/4 3/4 1/3 3/4 62/4 6 1/3 6 2/4 1/3 3 32 2xyxy x xy x y xxy y−− −−− − − −−− − − −=  == == 114.() ()( ) ()() ( ) ( ) ( )41/2 7/4 47/4 5/4 1/2 5/4 41 / 24 2 / 44 1/2 2/4 2 22 2xyxy y xy x y yxy x−− −−− − − −−−− − −=  == == 115. The message is “Paige Fox is bad at math .”Section P.3 Radicals and Rational Exponents Solutions Manual. 23 116. a. For 2030: 5.8 56.4 5.8 10 56.4Ex=+=+ For 2060: 5.8 56.45.8 40 56.45.8 2 10 56.4 11.6 10 56.4Ex=+=+=⋅ +=+ Difference: (11.6 10 56.4) (5.8 10 56.4) 11.6 10 56.4 5.8 10 56.411.6 10 5.8 10 56.4 56.4 5.8 10+− + =+ − −=− + −= The difference is 5.8 10. b. 5.8 10 18.3 ≈ This underestimates the difference projected by the graph of 98.2 74.1 24.1 −= by 5.8. This represents a difference of 5.8 million peopl e. 117. 25 1 2 ( 5 1 ) 51 51 51 2( 5 1) 4 51 2 1.62++⋅=− −+ += += ≈ About 1.62 to 1. 118. ()2 2 21 0.91 10 . 9 0.19 0.44af f f f fvRRc cRc R R R=−  =−  =− = ≈ 0.44 44 0.44 0.44 44 0.44 0.44 100af f f fRR R R R= = = = If you are gone for 44 weeks, then 100 weeks will have passed for your friend. 119. Perimeter: 22 2 125 2 2 20 22 5 5 4 4 525 5 42 5 10 5 8 518 5 feetPlw=+ =⋅ +⋅ =⋅ ⋅+ ⋅=⋅ +⋅ =+ = Area: 125 2 20 2 125 202 2500 25 0 100 square feetAl w= =⋅ =⋅= =⋅= 120. Perimeter: 22 24 2 0 2 8 0 84 5 21 6 5 82 5 24 5 16 5 8 5 24 5 feetPlw=+ =⋅ +⋅ =⋅ + ⋅=⋅ +⋅ =+ = Area: 42 0 8 042 0 8 04 1600 44 0 160 square feetAl w= =⋅ =⋅= =⋅= 121. – 128. Answers will vary. 129. does not make sense; Explanations will vary . Sample explanation: The denominator is rationalized correctly . 130. makes sense 131. does not make sense; Explanations will vary . Sample explanation: 22 0 47 5 + simplifies to 45 2 03 + and thus the radical terms are no t common. 132. does not make sense; Explanations will vary . Sample explanation: Finding the nth root first often gives smaller numbers on the middle step.