Chapter R Review
Section R.1
1. rational
2. 4563430331
3. Distributive
4. c
5. a
6. b
7. True
8. False;TheZero-ProductPropertystatesthatifa productequals0,thenatleastoneofthefactors mustequal0.
9. False;6istheGreatestCommonFactorof12 and18.TheLeastCommonMultipleisthe smallestvaluethatbothnumberswilldivide evenly.TheLCMfor12and18is36.
10. True
AB
11. 1,3,4,5,92,4,6,7,8 1,2,3,4,5,6,7,8,9
12. 1,3,4,5,91,3,4,6 1,3,4,5,6,9 AC
13. 1,3,4,5,92,4,6,7,84 AB
14. 1,3,4,5,91,3,4,61,3,4 AC
15.
ABC
16.
ABC
()
1,3,4,5,92,4,6,7,81,3,4,6 41,3,4,6 1,3,4,6
17. 0,2,6,7,8 A
18. 0,2,5,7,8,9 C
AB
19. 1,3,4,5,92,4,6,7,8 40,1,2,3,5,6,7,8,9
BC
20. 2,4,6,7,81,3,4,6 1,2,3,4,6,7,80,5,9
AB
21. 0,2,6,7,80,1,3,5,9 0,1,2,3,5,6,7,8,9
BC
22. 0,1,3,5,90,2,5,7,8,9 0,5,9
23. a. 2,5
b. 6,2,5
c. 1 6,,1.333...1.3,2,5 2
d.
e. 1 6,,1.333...1.3,,2,5 2
()
1,3,4,5,92,4,6,7,81,3,4,6 1,2,3,4,5,6,7,8,91,3,4,6 1,3,4,6
24. a. 1
b. 0,1
c. 5,2.060606...2.06,1.25,0,1 3
d. 5
e. 5,2.060606...2.06,1.25,0,1,5 3
25. a. 1
b. 0,1
c. 1110,1,,,234
d. None
e. 1110,1,,,234
26. a. None
b. 1
c. 1.3,1.2,1.1,1
d. None
e. 1.3,1.2,1.1,1
27. a. None
b. None
c. None
d. 2,,21,1 2
e. 2,,21,1 2
28. a. None
b. None
c. 110.3 2
d. 2,2 e. 1 2,2,10.3 2
29. a. 18.953 b. 18.952
30. a. 25.861 b. 25.861
31. a. 28.653 b. 28.653
32. a. 99.052 b. 99.052
33. a. 0.063 b. 0.062
34. a. 0.054 b. 0.053
35. a. 9.999 b. 9.998
36. a. 1.001 b. 1.000
37. a. 0.429 b. 0.428
38. a. 0.556 b. 0.555
39. a. 34.733 b. 34.733
40. a. 16.200 b. 16.200
41. 325
42. 5210
43. 234 x
44. 322 y
45. 312 y
46. 246 x
47. 26 x
48. 26 y
49. 6 2 x
50. 26 x
51. 942527
52. 643235
53. 6436126
54. 842880
55. 185218108
56. 1001021002080
57. 4112113 333
58. 21413 222
21100374253 253253
73. 3215823 452020
74. 418311 3266
75. 74493281 875656
76. 81516135151 921818
77. 5110313 18123636
2864046 1594545
79. 5825643913 241512012040
80. 32945 14214242
81. 32981 20156060
82. 6312153 35147070
83. 5 1852759359 1118119211 27
5 2153557557 2212732
85. 141741721 1 3721212121
86. 24122222222 35635323532315 25210210212 351515151515 43434 53535
87. 3323363623 2 4814848428 12312315 8888
88. 51351351351 3 62162322322 51514 2 2222
64624 xx
2 44 xxxx
2 43412 xxxx
2121323 333 3636332 3231 2 3322 x
2 2 24428 68 xxxxx
2 2 5155 65
xxxxx
2 2 927271863 21163
98. 2 2 3153155 3145 xxxxx xx
99. 2 2 822816 1016 xxxxx xx
100. 2 2 42248 68 xxxxx xx
101. 2 22 3(5)360 315360 1560 4 xxkxx xxkxx xkx k
102. 2 222 222 222 222 ()(3)412 33412 (3)3412 (3)3412 (2)3412 24 2
xkxkxx xkxkxkxx xxkkkxx xxkkkxx xxkkxx k k
103. 2323 23 5 5 xxxx x x x
104. 23421214 sincemultiplicationcomesbeforeadditioninthe orderofoperationsforrealnumbers.
2345420 sinceoperationsinsideparenthesescomebefore multiplicationintheorderofoperationsforreal numbers.
105. 23421224 23246848
106. 4371 257 ,but 43453220626132.6 251010105
107. Subtractionisnotcommutative;for example:231132
108. Subtractionisnotassociative;for example: 52124521 .
109. Divisionisnotcommutative;forexample: 23 32 .
110. Divisionisnotassociative;for example: 1222623 ,but
122212112
111. TheSymmetricPropertyimpliesthatif2= x, then x =2.
112. Fromthe principleofsubstitution, if5 x ,then
113. Therearenorealnumbersthatarebothrational andirrational,sinceanirrationalnumber,by definition,isanumberthatcannotbeexpressed astheratiooftwointegers;thatis,notarational number
Everyrealnumberiseitherarationalnumberor anirrationalnumber,sincethedecimalformofa realnumbereitherinvolvesaninfinitely repeatingpatternofdigitsoraninfinite,nonrepeatingstringofdigits.
114. Thesumofanirrationalnumberandarational numbermustbeirrational.Otherwise,the irrationalnumberwouldthenbethedifferenceof tworationalnumbers,andthereforewouldhave toberational.
115. Answerswillvary.
116. Since1day=24hours,wecompute 12997541.5416 24
Nowweonlyneedtoconsiderthedecimalpart oftheanswerintermsofa24hourday.Thatis,
0.54162413 hours.Soitmustbe13hours laterthan12noon,whichmakesthetime1AM CST.
117. Answerswillvary.
Section R.2
1. variable
2. origin
3. strict
4. base;exponent(orpower)
5. 31.234567810
6. d
7. a
8. b
9. True
10. False;theabsolutevalueofarealnumberis nonnegative.00 whichisnotapositive number.
11. False;anumberinscientificnotationis expressedastheproductofanumber,x, 110 x or101 x ,andapowerof10.
12. True 13.
15. 10 2 16. 56 17. 12 18. 35 2 19. 3.14 20. 21.41
21. 10.5 2 22. 10.33 3 23. 20.67 3 24. 10.25 4 25. 0 x
26. 0 z 27. 2 x 28. 5 y
29. 1 x
30. 2 x
31. Graphonthenumberline:2 x
32. Graphonthenumberline:4 x
33. Graphonthenumberline:1 x
34. Graphonthenumberline:7 x
35. (,)(0,1)1011dCDd
36. (,)(0,3)3033dCAd
37. (,)(1,3)3122dDEd
38. (,)(0,3)3033dCEd
39. (,)(3,3)3(3)66dAEd
40. (,)(1,1)1122dDBd
41. 2223264xy
42. 33(2)3633 xy
43. 525(2)(3)230228 xy
44. 22(2)(2)(3)462 xxy
45. 2(2)4 24 2355 x xy
46. 2311 2355 xy xy
47. 323(2)2(3)6600 22355 xy y
48. 2(2)343 237 333 x y
49. 3(2)11 xy
50. 3(2)55 xy
51. 32325 xy
52. 32321 xy
53. 331 33 x x
54. 221 22 y y
55. 454(3)5(2) 1210 22 22 xy
56. 323(3)2(2)9455 xy
57. 454(3)5(2) 1210 1210 2 2 xy
58. 323322 3322 94 13 xy
59. 21 x x
Part(c)mustbeexcluded.Thevalue0 x must beexcludedfromthedomainbecauseitcauses divisionby0.
60. 21 x x
Part(c)mustbeexcluded.Thevalue0 x must beexcludedfromthedomainbecauseitcauses divisionby0.
61. 2(3)(3) 9 xx xxx
Part(a),3 x ,mustbeexcludedbecauseit causesthedenominatortobe0.
62. 29 x x
Noneofthegivenvaluesareexcluded.The domainisallrealnumbers.
63. 2 21 x x
Noneofthegivenvaluesareexcluded.The domainisallrealnumbers.
64. 33 2(1)(1) 1 xx xxx
Parts(b)and(d)mustbeexcluded.Thevalues 1,and1xx mustbeexcludedfromthe domainbecausetheycausedivisionby0.
65. 22 3 510510 (1)(1) xxxx xxxxx
Parts(b),(c),and(d)mustbeexcluded.The values0,1,and1 xxx mustbeexcluded fromthedomainbecausetheycausedivisionby 0.
66. 22 32 9191 (1) xxxx xxxx
Part(c)mustbeexcluded.Thevalue0 x must
beexcludedfromthedomainbecauseitcauses divisionby0.
67. 4 5 x 5 x mustbeexludedbecauseitmakesthe denominatorequal0.
Domain5 xx
68. 6 4 x 4 x mustbeexcludedsineitmakesthe denominatorequal0.
Domain4 xx
69. 4 x x 4 x mustbeexcludedsineitmakesthe denominatorequal0.
Domain4 xx
70. 2 6 x x 6 x mustbeexcludedsineitmakesthe denominatorequal0.
Domain6 xx
71. 555 (32)(3232)(0)0C 999 CF
72. 555 (32)(21232)(180)100C 999 CF
73. 555 (32)(7732)(45)25C 999 CF
74. 55(32)(432)99 5(36) 9 20C CF
75. 2 (9)(9)(9)81
76. 22 4(4)16
77. 2 2 411 416
78. 2 2 411 416
79. 64642 2 333311 39
80. 23231 44444
131 33 44464
90.
33311333 3 y xyxyxy x
91. 25 235411 34 xyy xyxy xyx
92. 2 211231 23 1 xy xyxy xyxy
93. 253533 37272 315732 221 2 2 (4)()16 (3)27 16 27 16 27 16 27
95. 22 2 33266 13222 3 2339 2 3224
122 2 24 1 x xy y
yxzyxz xyzxyz xyz xyz xz y 94. 21211 344 24111 621 62 4()4 28 4 8 1 2 1 2 xyzxyz xyxy xyz xyz xyz
222221415 xy
222221414 xy
2 222124 xy
2 222111 xy 103. 222xx 104.
2 2 xx 105.
222221415 xy 106. 2221213 xyxy 107. 211 2 y x
114. 5 (3.7)693.440 115. 3 (6.1)0.004
116. 5 (2.2)0.019 117. 6 (2.8)481.890
118. 6 (2.8)481.890
119. 4 (8.11)0.000
120. 4 (8.11)0.000
121. 2 454.24.54210
122. 132.143.21410
123. 0.0131.3102 124. 0.004214.21103
125. 432,1553.215510
126. 4 21,2102.12110
127. 0.0004234.23104
128. 0.05145.14102
129. 4 6.151061,500
130. 3 9.7109700
131. 3 1.214100.001214
132. 4 9.88100.000988
133. 8 1.110110,000,000
134. 2 4.11210411.2
135. 2 8.1100.081
136. 1 6.453100.6453
137. Alw
138. 2 Plw
139. Cd
140. 1 2 Abh
141. 32 4 Ax
142. 3 Px
143. 43 3 Vr
144. 42Sr
145. 3 Vx
146. 62Sx
147. a. If1000, x 40002 40002(1000) 40002000 $6000 Cx
Thecostofproducing1000watchesis $6000.
b. If2000, x 40002 40002(2000) 40004000 $8000 Cx
Thecostofproducing2000watchesis $8000.
148. 210801202560325$98 Hisbalanceattheendofthemonthwas$98.
149. Wewantthedifferencebetween x and4tobeat least6units.Sincewedon’tcarewhetherthe valuefor x islargerorsmallerthan4,wetake theabsolutevalueofthedifference.Wewantthe inequalitytobenon-strictsincewearedealing withan‘atleast’situation.Thus,wehave 46 x
150. Wewantthedifferencebetween x and2tobe morethan5units.Sincewedon’tcarewhether thevaluefor x islargerorsmallerthan2,we taketheabsolutevalueofthedifference.We wanttheinequalitytobestrictsinceweare dealingwitha‘morethan’situation.Thus,we have 25 x
151. a. 110108110225 x 108voltsisacceptable.
b. 110104110665 x 104voltsis not acceptable.
152. a. 220214220668 x 214voltsisacceptable.
b. 22020922011118 x 209voltsis not acceptable.
153. a. 32.9993 0.001 0.0010.01 x Aradiusof2.999centimetersisacceptable.
b. 32.893 0.11 0.110.01 x Aradiusof2.89centimetersis not acceptable.
154. a. 98.69798.6 1.6 1.61.5 x 97˚Fisunhealthy.
b. 98.610098.6 1.4 1.41.5 x 100˚Fis not unhealthy.
155. ThedistancefromEarthtotheMoonisabout 8 410400,000,000 meters.
156. TheheightofMt.Everestisabout 388488.84810 meters.
157. Thewavelengthofvisiblelightisabout 7 5100.0000005 meters.
158. Thediameterofanatomisabout 10 1100.0000000001 meters.
159. Thediameterisabout2 0.04034.0310 inches.
160. Thetiniestmotorislessthan5 0.00004410 millimeterstall.
161. 5112 2 1.86106102.4103.6510
186,000606024365 1012 586.5696105.86569610 Thereareabout12 5.910 milesinonelightyear.
162. 7 2 5 93,000,0009.310510 186,0001.8610 500seconds8min.20sec.
Ittakesabout8minutes20secondsforabeam oflighttoreachEarthfromtheSun.
163. 10.333333...0.333 3 1 3islargerbyapproximately0.0003333...
164. 2 30.666666...0.666 2 3islargerbyapproximately0.000666...
165. 61319 20 5.24106.51034.0610 3.40610
166. 44 6 1010 5 1.62101.62100.3610 4.5 4.51010 3.610
167. No.Foranypositivenumber a,thevalue2 a is smallerandthereforecloserto0.
168. Wearegiventhat2110 x .Thisimpliesthat 110 x .Since103.162 x and 3.142 x ,thenumbercouldbe3.15or3.16 (whicharebetween1and10asrequired).The numbercouldalsobe3.14sincenumberssuchas 3.146whichliebetween and10would equal3.14whentruncatedtotwodecimalplaces.
169. Answerswillvary.
170. Answerswillvary. 5<8isatruestatementbecause5isfurtherto theleftthan8onarealnumberline.
Section R.3
1. right;hypotenuse
2. 1 2 Abh 3. 2 Cr 4. similar 5. c 6. b 7. True.
8. True.222 68366410010
9. False;thesurfaceareaofasphereofradius r is givenby2 4 Vr
10. True.Thelengthsofthecorrespondingsidesare equal.
11. True.Twocorrespondinganglesareequal.
12. False.Thesidesarenotproportional.
13. 222 22 5,12, 512 25144 16913 ab cab c
14. 222 22 6,8, 68 3664 10010 ab cab c
15. 222 22 10,24, 1024 100576 67626 ab cab c
16. 222 22 4,3, 43 169 255 ab cab c
17. 222 22 7,24, 724 49576 62525 ab cab c
18. 222 22 14,48, 1448 1962304 250050 ab cab c
19. 222 534 25916 2525
Thegiventriangleisarighttriangle.The hypotenuseis5.
20. 222 1068 1003664 100100
Thegiventriangleisarighttriangle.The hypotenuseis10.
21. 222 645 361625 3641false
Thegiventriangleisnotarighttriangle.
22. 222 322 944 98false
Thegiventriangleisnotarighttriangle.
23. 222 25724 62549576 625625
Thegiventriangleisarighttriangle.The hypotenuseis25.
24. 222 261024 676100576 676676
Thegiventriangleisarighttriangle.The hypotenuseis26.
25. 222 634 36916 3625false
Thegiventriangleisnotarighttriangle.
26. 222 754 492516 4941false Thegiventriangleisnotarighttriangle.
27. 67422 in Alw
28. 94362 cm Alw
29. 112 22(14)(4)28in
Abh
30. 112 22(4)(9)18cmAbh
31. 222 (5)25m 22(5)10m Ar Cr
32. 222 (2)4ft 22(2)4ft Ar Cr
33. 6852403 ft Vlwh
Slwlhwh
2 222 268265285 966080
236ft
34. 9482883 in Vlwh
Slwlhwh
2 222 294298248 7214464 280in
35. 333 222 445005cm 333 445100cm
Vr Sr
36. 333 222 44336f 33 44336ft
Vrt Sr
37. 223 (9)(8)648in Vrh
2 2 2 22 29298 162144
38. 223 (8)(9)576in Vrh
2 2 2 22 28289 128144
39. Thediameterofthecircleis2,soitsradiusis1. 22(1)squareunits Ar
40. Thediameterofthecircleis2,soitsradiusis1. 22 2(1)4squareunits A
41. Thediameterofthecircleisthelengthofthe diagonalofthesquare.
222 22 44 8 822 222 22 d d d r
Theareaofthecircleis:
2222squareunits Ar
42. Thediameterofthecircleisthelengthofthe diagonalofthesquare. 222 22 44 8 822 222 22 d d d r
Theareais:
22 2224squareunits A
43. Sincethetrianglesaresimilar,thelengthsof correspondingsidesareproportional.Therefore, weget 8 42 82 4 4 x x x
Inaddition,correspondinganglesmusthavethe sameanglemeasure.Therefore,wehave 90 A ,60 B ,and30 C
44. Sincethetrianglesaresimilar,thelengthsof correspondingsidesareproportional.Therefore, weget 6 1216 616 12 8 x x x
Inaddition,correspondinganglesmusthavethe sameanglemeasure.Therefore,wehave 30 A ,75 B ,and75 C
45. Sincethetrianglesaresimilar,thelengthsof correspondingsidesareproportional.Therefore, weget 30 2045 3045 20 135 or67.5 2 x x xx
Inaddition,correspondinganglesmusthavethe sameanglemeasure.Therefore,wehave 60 A ,95 B ,and25 C
46. Sincethetrianglesaresimilar,thelengthsof correspondingsidesareproportional.Therefore, weget 8 1050 850 10 40 x x x
Inaddition,correspondinganglesmusthavethe sameanglemeasure.Therefore,wehave 50 A ,125 B ,and5 C .
47. Thetotaldistancetraveledis4timesthe circumferenceofthewheel. TotalDistance44()416 64201.1inches16.8feet
48. Thedistancetraveledinonerevolutionisthe circumferenceofthedisk4 Thenumberofrevolutions= dist.traveled2051.6revolutions circumference4
49. Areaoftheborder=areaofEFGH–areaof ABCD222 1061003664ft
50. FG=4feet;BG=4feetandBC=10feet,so CG=6feet.TheareaofthetriangleCGFis: 12 2(4)(6)12ft
51. Areaofthewindow=areaoftherectangle+ areaofthesemicircle. 122 (6)(4)224230.28ft 2 A
Perimeterofthewindow=2heights+width+ one-halfthecircumference. 1 2(6)4(4)1242 2 16222.28feet P
52. Areaofthedeck=areaofthepoolanddeck–areaofthepool. 22 22 (13)(10)169100 69ft216.77ft
Theamountoffenceisthecircumferenceofthe circlewithradius13feet. 2(13)26ft81.68ft
53. WecanformsimilartrianglesusingtheGreat Pyramid’sheight/shadowandThales’ height/shadow: h 126114 240 {{ 2 3
Thisallowsustowrite 2 2403 2240
TheheightoftheGreatPyramidis160paces.
54. Let x =theapproximatedistancefromSanJuan toHamiltonand y =theapproximatedistance fromHamiltontoFortLauderdale.Usingsimilar triangles,weget 1046
5853.5 104653.5 58 964.8 x x x
1046 5857 104657 58 1028.0 y y y
TheapproximatedistancebetweenSanJuanand Hamiltonis965milesandtheapproximate distancebetweenHamiltonandFortLauderdale is1028miles.
55. Convert20feettomiles,andsolvethe PythagoreanTheoremtofindthedistance:
201milefeet20feet0.003788miles 5280feet (39600.003788)396030 5.477miles
222sq.miles
56. Convert6feettomiles,andsolvethe PythagoreanTheoremtofindthedistance:
61mile feet6feet0.001136miles 5280feet (39600.001136)39609 3miles
222sq.miles
57. Convert100feettomiles,andsolvethe PythagoreanTheoremtofindthedistance: 1001milefeet100feet0.018939miles 5280feet
222sq.miles (39600.018939)3960150 12.2miles
Convert150feettomiles,andsolvethe PythagoreanTheoremtofindthedistance: 1501milefeet150feet0.028409miles 5280feet
222sq.miles (39600.028409)3960225 15.0miles
58. Given0,0andmnmn , if2222 ,2and amnbmncmn ,then
and 2 2224224 2 cmnmmnn
222,and abcabc representthesides ofarighttriangle.
Ifyoudoubletheradiusthevolumeis8times theoriginalvolume.
63. Let l= lengthoftherectangle and w =widthoftherectangle. Noticethat 22 ()() [()()][()()] (2)(2)44 lwlw lwlwlwlw lwlwA
Since2()0 lw ,thelargestareawilloccur when l–w =0or l=w;thatis,whenthe rectangleisasquare.But 1000222() 5002 250 lwlw lwl lw
Vrh So,
59. 2 2 3 (10)(4.5) 450ft
3 33 1ft7.48052galso 450ft7.48052gal/ft10,575gal
60. 3 2 2 10000(5.61458)56145.8ft 56145.8(25) 56145.828.6ft 625
61. 2 2 2 2 2 (2) 4 44
Ar Ar r rA
Ifyoudoubletheradius,theareaisfourtimes theoriginalarea.
Thelargestpossibleareais225062500 sqft. Acircularpoolwithcircumference=1000feet yieldstheequation:500 21000rr
Theareaenclosedbythecircularpoolis: 22 22 50050079577.47ft Ar
Thus,acircularpoolwillenclosethemostarea.
64. Considerthediagramshowingthelighthouseat pointL,relativetothecenterofEarth,usingthe radiusofEarthas3960miles.LetPrefertothe furthestpointonthehorizonfromwhichthe lightisvisible.Notealsothat 362362feetmiles. 5280
ApplythePythagoreanTheoremto CPL :
2 22 1 39603960362 5280 d
2 22 1 22 1 362 5280 362 5280 39603960 3960396023.30mi. d d
Therefore,thelightfromthelighthousecanbe seenatpointPonthehorizon,wherepointPis approximately23.30milesawayfromthe lighthouse.Brochureinformationisslightly overstated.
Verifytheshipinformation:
LetSrefertotheship’slocation,andlet x equal theheight,infeet,oftheship.
Weneed1240 dd
Since123.30miles d weneed 24023.30=16.70miles. d
ApplythePythagoreanTheoremto CPS :
396016.73960
222 22 22
396016.73960
396016.73960
Theshipwouldhavetobeatleast186feettallto seethelighthousefrom40milesaway.
Verifytheairplaneinformation:
LetArefertotheairplane’slocation.The distancefromtheplanetopointPis2 d Wewanttoshowthat12120 dd Assumethealtitudeoftheairplaneis 10,000feet=10000miles. 5280
ApplythePythagoreanTheoremto CPA :
2 22 2 3960396010000 5280 d
2 22 2 2 2 2 10000 39603960 5280 10000 39603960 5280 122.49miles. d d
61Therefo re,1223.30122.49145.79120. dd
Thebrochureinformationisslightlyunderstated. Notethataplaneatanaltitudeof6233feet couldseethelighthousefrom120milesaway.
Section R.4
False;monomialscannothavenegativedegrees.
False;thedividend=(quotient)(divisor)+ remainder
9. 23 x Monomial;Variable: x ; Coefficient:2;Degree:3
10. 42 x Monomial;Variable: x ;Coefficient: –4;Degree:2
11. 81 8 x x Notamonomial;whenwrittenin theform k ax ,thevariablehasanegative exponent.
12. 23 x Notamonomial;whenwritteninthe form k ax ,thevariablehasanegativeexponent.
13. 22 xy Monomial;Variables:,xy ; Coefficient:–2;Degree:3
14. 523 xy Monomial;Variables:,xy ; Coefficient:5;Degree:5
15. 81 8 x xy y Notamonomial;whenwritten intheform nm axy ,theexponentonthevariable y isnegative.
16. 2 23 3 22 x xy y Notamonomial;when writtenintheform nm axy ,theexponentonthe variable y isnegative.
17. 22 xy Notamonomial;theexpression containsmorethanoneterm.Thisexpressionis abinomial.
18. 2 34 x Notamonomial;theexpression containsmorethanoneterm.Thisexpressionis abinomial.
19. 2 35 x Polynomial;Degree:2
20. 14 x Polynomial;Degree:1
21. 5Polynomial;Degree:0
22. –πPolynomial;Degree:0
23. 325 x x Notapolynomial;thevariableinthe denominatorresultsinanexponentthatisnota nonnegativeinteger.
24. 32 x Notapolynomial;thevariableinthe denominatorresultsinanexponentthatisnota nonnegativeinteger.
25. 3 22 y Polynomial;Degree:3
26. 102zz Polynomial;Degree:2
27. 2 3 5 1 x x Notapolynomial;thepolynomialin thedenominatorhasadegreegreaterthan0.
28. 3 2 321 1 xx xx Notapolynomial;the polynomialinthedenominatorhasadegree greaterthan0.
29. 22 22 2 (68)(347) (3)(64)(87) 4215
xxxx xxxx xx
30. 322 322 32 (32)(44) (3)(4)(24) 446 xxxx xxxx xxx
31. 322 322 (2510)(243) 2510243 xxxxx xxxxx
322 32 (22)(54)(103) 497 xxxxx xxx
32. 232 232 (34)(35) 3435 xxxxx xxxxx
322 32 (3)(3)(45) 449 xxxxx xxx
33. 53432 542 653 653 xxxxxx xxxx
34. 5232 532 108326 103106 xxxx xxx
35. 22 22 2 (64)3(25) 646315 7311
xxxx xxxx xx
36. 22 22 2 2(1)(52) 22252 73 xxxx xxxx xx
37. 3232 3232 32 6(3)4(23) 6618812 21818 xxxx xxxx xx
38. 323 323 8(431)6(482) 32248244812 xxxx xxxx
32 824484 xxx
39.
40.
222 222 22351 22351 xxxxx xxxxx
2 246 xx
222 222 1452 1452 xxxx xxxx
2 26 xx
41. 22 22 2 75343 73521124 11359
yyy yyy yy
42. 323 323 8141 884444 yyyy yyyy
32 44412 yyy
43. 22432 (25)25 xxxxxx
44. 23532 4(2)448 xxxxxx
45. 2352 2(45)810 xxxx
46. 343 5(34)1520 xxxx
47. 2 22 (1)(24) (24)1(24) xxx xxxxx 322 32 2424 324 xxxxx xxx
48. 2 22 (23)(1) 2(1)3(1) xxx xxxxx 322 32 222333 23 xxxxx xxx
49. 2 2 (2)(4)428 68 xxxxx xx
50. 2 2 (3)(5)5315 815 xxxxx xx
51. 2 2 (27)(5)271035 21735 xxxxx xx
52. 2 2 (31)(21)6321 651 xxxxx xx
53. 2 2 (4)(2)248 28 xxxxx xx
54. 2 2 (4)(2)248 28 xxxxx xx
55. 2 2 (6)(3)6318 918 xxxxx xx
56. 2 2 (5)(1)55 65 xxxxx xx
57. 2 2 (23)(2)2436 26 xxxxx xx
58. 2 2 (24)(31)62124 6104 xxxxx xx
59. 2 2 (34)(2)3468 3108 xxxxx xx
60. 2 2 (31)(1)331 341 xxxxx xx
61. 2 2 (5)(27)210735 21735 xxxxx xx
62. 2 2 (23)(3)6293 239 xxxxx xx
63. 22 22 (2)()22 2 xyxyxxyxyy xxyy
64. 22 22 (23)()2233 23 xyxyxxyxyy
65. 22 22 (23)(32)6496 6136 xyxyxxyxyy xxyy
66. 22 22 (3)(2)263 273 xyxyxxyxyy xxyy
67. 222 (7)(7)749 xxxx
68. 222 (1)(1)11 xxxx
69. 222 (23)(23)(2)349 xxxx
70. 222 (32)(32)(3)294 xxxx
71. 2222 (4)244816 xxxxx
72. 2222 (5)2551025 xxxxx
73. 2222 (4)244816 xxxxx
74. 2222 (5)2551025 xxxxx
75. 222 (34)(34)(3)4916 xxxx
76. 222 (53)(53)(5)3259 xxxx
77. 222 2 (23)(2)2(2)(3)3 4129 xxx xx
78. 222 2 (34)(3)2(3)(4)4 92416 xxx xx
79. 2 222 ()()() xyxyxyxy
80. 2 222(3)(3)()39 xyxyxyxy
81. 2 222(3)(3)(3)9 xyxyxyxy
82. 2 222(34)(34)(3)4916 xyxyxyxy
83. 222()2 xyxxyy
84. 222()2 xyxxyy
85. 222 22 (2)222 44 xyxxyy xxyy
86. 222 22 (23)22233 4129 xyxxyy xxyy
87. 33223 32 (2)32322 6128 xxxx xxx
88. 33223 32 (1)31311 331 xxxx xxx
89. 33223 32 (21)(2)3(2)(1)3(2)11 81261 xxxx xxx
90. 33223 32 (32)(3)3(3)(2)3(3)22 2754368 xxxx xxx
91. 2 32 32 2 2 41123 2431 48 11 1122 231 2346 45 xx xxxx xx xx xx x x
2 322 32 Check: (2)(41123)(45) 411238224645 431 xxx xxxxx xxx Thequotientis241123 xx ;theremainder is–45.
92. 2 32 32 2 2 3715 232 36 7 714 152 1530 32 xx xxxx xx xx xx x x 2 322 32
Check:
(2)(3715)(32) 37156143032 32 xxx xxxxx xxx
Thequotientis23715 xx ;theremainderis –32.
93. 232 3 2 2 43 431 4 31 3 1 x xxxx x xx x x
Check: ()(43)(1)431 xxxxxx Thequotientis43 x ;theremainderis1 x
232
94. 232 3 2 2 31 32 3 2 2 x xxxx x xx x x
232 Check: ()(31)(2)32 xxxxxx
Thequotientis31 x ;theremainderis2 x
95. 2 2432 42 2 2 513 25031 510 131 1326 27 x xxxxx xx xx x x
Check:
22 422 42 251327 510132627 531 xxx xxxx xxx
Thequotientis2513 x ;theremainderis 27 x
96. 2 2432 42 2 2 511 2502 510 112 1122 20 x xxxxx xx xx x x
Check:
22 422 42 251120 510112220 52 xxx xxxx xxx
Thequotientis2511 x ;theremainderis 20 x
97. 2 35432 52 2 2 2140031 42 1 x xxxxxx xx xx
Check: 322 52252 2121 421431 xxxx xxxxxxx
Thequotientis2 2 x ;theremainderis 21xx
22 432322
42 Check: (1)(1)(22) 122 1 xxxxx xxxxxxxx x xx
Thequotientis22 xaxa ;theremainderis0.
Thequotientis432234 xaxaxaxa ;the remainderis0.
108. Theproducts()() xyxy and()() zwzw willeachresultinabinomialthatisthe differenceofsquares.Theproductofthose resultingbinomialswillhave4terms.
109. Whenwemultiplypolynomials 1 px and 2 px ,eachtermof 1 px willbemultiplied byeachtermof 2 px .Sowhenthehighestpoweredtermof 1 px multipliesbythehighest poweredtermof 2 px ,theexponentsonthe variablesinthosetermswilladdaccordingtothe basicrulesofexponents.Therefore,thehighest poweredtermoftheproductpolynomialwill havedegreeequaltothesumofthedegreesof 1 px and 2 px .
110. Whenweaddtwopolynomials 1 px and 2 px ,wherethedegreeof 1 px thedegree of 2 px ,eachtermof 1 px willbeaddedto eachtermof 2 px .Sinceonlythetermswith equaldegreeswillcombineviaaddition,the degreeofthesumpolynomialwillbethedegree ofthehighestpoweredtermoverall,thatis,the degreeofthepolynomialthathadthehigher degree.
111. Whenweaddtwopolynomials 1 px and 2 px ,wherethedegreeof 1 px =thedegree of 2 px ,thenewpolynomialwillhavedegree thedegreeof 1 px and 2 px .
112. Answerswillvary.
113. Answerswillvary.
Section R.5
1. 322 xxx
2. prime
3. c 4. b
5. d
6. c
7. True;24 x isprimeoverthesetofreal numbers.
8. False; 322 3264322 xxxxx
9. 363(2) xx
10. 7147(2) xx
11. 22(1)axaax
12. (1)axaax
13. 322(1)xxxxxx
14. 322(1)xxxxxx
15. 2 222(1) xxxx 16. 2 333(1) xxxx
17. 22 36123(24) xyxyxyxyxy
18. 2232 60487212(546) xyxyxyxyxyx
19. 22211(1)(1)xxxx
20. 22242(2)(2)xxxx
21. 222 41(2)1(21)(21) xxxx
22. 222 91(3)1(31)(31) xxxx
23. 222164(4)(4)xxxx
24. 222255(5)(5)xxxx
25. 2 254(52)(52) xxx
26. 22 3699419(21)(21) xxxx
27. 22 21(1)xxx
28. 22 44(2)xxx
29. 22 44(2)xxx
30. 22 21(1)xxx
31. 22 1025(5)xxx
32. 22 1025(5)xxx
33. 22 441(21) xxx
34. 22 961(31) xxx
35. 22 1681(41) xxx
36. 22 25101(51) xxx
37. 3332273(3)(39)xxxxx
38. 33321255(5)(525)xxxxx
39. 3332273(3)(39)xxxxx
40. 333 2 2 2783(2) (32)(964) 23469 xx xxx xxx
41. 333 2 827(2)3 (23)(469) xx xxx
42. 333 2 2 64274(3) (43)(16129) 3491216 xx xxx xxx
43. 256(2)(3) xxxx
44. 268(2)(4) xxxx
45. 276(6)(1) xxxx
46. 298(8)(1) xxxx
47. 2710(2)(5) xxxx
48. 21110(10)(1) xxxx
49. 21016(2)(8) xxxx
50. 21716(16)(1) xxxx
51. 278(1)(8) xxxx
52. 228(2)(4) xxxx
53. 278(8)(1) xxxx
54. 228(4)(2) xxxx
55. 2 24362(2)3(2) (2)(23) xxxxxx xx
56. 2 33223(1)2(1) (1)(32) xxxxxx xx
57. 2 51535(3)1(3) (3)(51) xxxxxx xx
58. 2 3623(2)1(2) (2)(31) xxxxxx xx
59. 2 6218283(27)4(27) (27)(34) xxxxxx xx
60.
2 9632332132 3231 xxxxxx xx
61. 2 341(31)(1) xxxx
62. 2 231(21)(1) xxxx
63. 2 297(27)(1) zzzz
64. 2 651(31)(21) zzzz
65. 2 568(54)(2) xxxx
66. 2 3108(34)(2) xxxx
67. 2 568(54)(2) xxxx
68. 2 3108(34)(2) xxxx
69. 2 5228(52)(4) xxxx
70. 2 3148(32)(4) xxxx
71. 2 5188(52)(4) xxxx
72. 2 3108(32)(4) xxxx
73. Since b is10thenweneedhalfof10squaredto bethelastterminourtrinomial.Thus
12 2 22 (10)5;(5)25 1025(5)xxx
74. Since b is14thenweneedhalfof14squaredto bethelastterminourtrinomial.Thus
12 2 22 (14)7;(7)49 1449(7)ppp
75. Since b is-6thenweneedhalfof-6squaredto bethelastterminourtrinomial.Thus
12 2 22 (6)3;(3)9 69(3)yyy
76. Since b is-4thenweneedhalfof-4squaredto bethelastterminourtrinomial.Thus 12 2 22 (4)2;(2)4 44(2)xxx
77. Since b is12thenweneedhalfof12squared tobethelastterminourtrinomial.Thus 2 11111 224416 22 111 2164 ();() ()xxx
78. Since b is13thenweneedhalfof13squaredto bethelastterminourtrinomial.Thus 2 11111 236636 22 111 3366 ();() ()xxx
79. 236(6)(6) xxx
80. 29(3)(3) xxx
81. 22 282(14)21212 xxxx
82. 22 3273(19)31313 xxxx
83. 21110(1)(10) xxxx
84. 254(4)(1) xxxx
85. 2102173 xxxx
86. 268(2)(4) xxxx
87. 22 4832428 xxxx
88. 22 31215345 xxxx
89. 2416xx isprimeovertherealsbecause therearenofactorsof16whosesumis4.
90. 22 1236(6)xxx
91. 22 152(215)(5)(3) xxxxxx
92. 22 146(614) xxxx isprimeoverthe integersbecausetherearenofactorsof–14 whosesumis–6.
93. 22 312363(412) 3(6)(2) xxxx xx
94. 322820(820) (10)(2) xxxxxx xxx
95. 43222 2 1130(1130) (5)(6) yyyyyy yyy
96. 322 318483(616) 3(2)(8) yyyyyy yyy
97. 22 4129(23) xxx
98. 22 9124(32) xxx
99.
22 6822341 2311 xxxx xx
100.
22 8622431 2411 xxxx xx
101. 2 42222 2 819(9)(9) (3)(3)(9) xxxx xxx
102. 2 42222 2 11(1)(1) (1)(1)(1) xxxx xxx
103. 6332 22 222 21(1) (1)(1) (1)(1) xxx xxx xxx
104. 6332 22 222 21(1) (1)(1) (1)(1) xxx xxx xxx
105. 75525(1)(1)(1)xxxxxxx
106. 855352(1)(1)(1)xxxxxxxx
107. 22 1624943 xxx
108. 22 9241634 xxx
109. 22 51616(16165) (45)(41) xxxx xx
110. 22 51116(16115) (165)(1) xxxx xx
111. 2 41615(25)(23) yyyy
112. 2 994(34)(31) yyyy
113. 2442 22 2 189(981) (91)(1) (31)(31)(1) xxxx xx xxx
114. 2442 22 2 41482(472) 2(41)(2) 2(21)(21)(2) xxxx xx xxx
115. (3)6(3)(3)(6) xxxxx
116. 5(37)(37)(37)(5) xxxxx
117.
118.
2 (2)5(2)(2)(2)5 (2)(3) xxxx xx
2 (1)2(1)(1)(1)2 (1)(3) xxxx xx
119.
3 33 2 2 2 3227 323 323323329 359124969 35937 x x xxx xxxx xxx
120.
121.
3 33 2 2 2 511 511 511511511 525101511 525153 x x xxx xxxx xxx
2 2 3102545 3545 5354 53154 5311
2 2 76953 7353 3735 37215 3716 xxx xx xx xx xx
123.
124.
322 2 22(2)12 (2)(1) (2)(1)(1) xxxxxx xx xxx
322 2 33(3)13 (3)(1) (3)(1)(1) xxxxxx xx xxx
125. 433 3 2 1(1)11 (1)(1) (1)(1)(1) xxxxxx xx xxxx
126. 433 3 2 22 1(1)11 (1)(1) (1)(1)(1) (1)(1) xxxxxx xx xxxx xxx
127. 2 234232343 23434233 2343469 234913 xxx xxx xxx xx
128. 2 521562212 21521564 211052024 213019 xxx xxx
129. 2 2252225 225 235 xxxxxx xxx xx
130.
232 2 2 38383838 2498 329 xxxxxx xxx xx
132.
322 2 2 2 2 232332 322233 322439 3255 5321 xxxx xxxx xxxx xxx xxx
324 3 3 3 3 3 451521 251215 251225 25133 23511 6511 xxxx xxxx xxxx xxx xxx xxx
2 432434 43438 43438 43123 34341 xxx xxx xxx xx
134.
2 23 2 2 2 3342343 334342 334342 33454 xxxx xxxx xxxx
135.
136.
223 2 2 2 345451452515 24551651545 245513062025 245515031 xxxx xxxx xxxx xxx
137. Thepossiblefactorizationsare 2 1454xxxx or 2 2244xxxx ,noneofwhich equals24 x
138. Thepossibilefactorizationsare 22 121xxx ,neitherofwhichequals
21xx
139. Answerswillvary.
140. Answerswillvary.
Section R.6
1. quotient;divisor;remainder 2. 32051
3. d 4. a 5. True
6. True
7. 217510 21010 1550
Quotient:255 xx
Remainder:0
8. 11231 114 1145
Quotient:24 xx
Remainder:5
9. 33213 93396 3113299
Quotient:231132 xx
Remainder:99
10. 24211 82042 4102143
Quotient:241021 xx
Remainder:43
11. 3104010 391545138 1351546138
Quotient:432351546
Remainder:138 xxxx
12. 210102 241020 1251022
Quotient:322510xxx Remainder:22
13. 14030105 441122 4411227
Quotient:54324422 Remainder:7 xxxxx
14. 11050010 11666 1166616
Quotient:432666xxxx Remainder:–16
15. 1.10.100.20
0.110.1210.3531
0.10.110.3210.3531
Quotient:20.10.110.321 xx Remainder:–0.3531
16. 2.10.100.2
0.210.441
0.10.210.241
Quotient:0.10.21 x Remainder:0.241
17. 21000032 2481632 1248160
Quotient:43224816xxxx Remainder:0
18. 1100001 11111 111110
Quotient:4321 xxxx Remainder:0
19. 24384 8104 4528
Remainder=8≠0.Therefore,2 x isnota factorof324384 xxx .
20. 34508 1251153 41751161
Remainder=161≠0.Therefore,3 x isnota factorof32458 xx
21. 3260721 60021 20070
Remainder=0.Therefore,3 x isafactorof 43 26721 xxx
22. 2401504 81624 48120
Remainder=0.Therefore,2 x isafactorof 42 4154 xx
23. 2500430024 10204061224 3102036120
Remainder=0.Therefore,3 x isafactorof 63 54324 xx
24. 320180109 6180039 2600130
Remainder=0.Therefore,2 x isafactorof 642 2189 xxx
25. 410161019 4160416 140143
Remainder=1≠0.Therefore,4 x isnota factorof5321619xxx .
26. 4101601016 41600416 1400140
Remainder=0.Therefore,4 x isafactor 6421616xxx
27. 131062 3 1002 30060
Remainder=0;therefore1 3 x isafactorof 43 362 xxx
28. 131031 3 1001 30032
Remainder=20 ;therefore1 3 x isnota factorof43331 xxx
29. 21235 2822 141117 32 2 23517 411 22 xxx xx xx
1411179 abcd
30. 23 23 2 1322 3 130 hhhhh hhhh hh 322 3 xxhxh isthequotientand0isthe remainder.
31. 234 233 23 1334 254 12540 yyyyy yyyy yyy
Yes, xy isafactorof 432234 334 xxyxyxyy .
32. Answerswillvary.
Section R.7
1. lowestterms
2. LeastCommonMultiple
3. d
4. a
5. True; 113 33 115 55 355(3) 353(5) x xx x xx xxx xxx
6. False; 322 433 3 2623 64232 2332 xxxx xxxx LCMxxx
7. 2 3(3) 393 9(3)(3)3 xx xxxx
8. 24(2)48 122412(2)3 xxxxx xx
9. 2(2) 2 363(2)3 xxxxx xx
10. 2 22 3(58) 152458 33 xx xxx xxx
11. 22 2 24244 1266(21)21 xxx xxxxx
12. 2 2 (2)2 442 4(2)(2)2 xx xxx xxxx
15. 2 2 (6)(2) 4126 44(2)(2)2
16. 2 2 (1) 2(2)(1)22 xxxxxx xxxxxx
254 20 42 54 14 5
18. 2(21)(3) 253(3)3 121(21) xxxx xx xx
19. 222 363(2) 545(2)(2) 3 5(2) x xxx xxxxx xx
20. 2 333 261022(35)4(35) xxx xxxx
23 2 22 2 2 464 162 44416 (4)(4)2 224416 244 2416 4 xx xx xxxx xxx xxxxx xxx xxx
26.
xxxx xxxx xxxx xxxx x x
22 22 625 45215 2355 5153 235 513 xxx xxxx xxxx xxxx xxx xxx
27. 2 2 6 4624 3939 4 24 62(2) (2)(2)3(3) 4 (2)(3) x xxx xx x x xx xxx x xx
32. 3 22 3 3 339 3 99 9 x xxx x xx x
33. 2 222 222 2 2 2 712 71271212 1271212 12 (3)(4)(4)(3) (3)(4)(4)(3) (3) (3) xx xxxxxx xxxxxx xx xxxx xxxx x x
34. 2 222 222 2 76 67656 56656 56 (6)(1)(2)(3) (3)(2)(6)(1) (1)(2) (2)(1) xx xxxxxx xxxxxx xx xxxx xxxx xx xx
35. 2 222 222 2 576 23557621320 1514323515143 21320 (53)(2)(25)(4) (1)(25)(53)(31) (2)(4) (1)(31)
xx xxxxxx xxxxxx xx xxxx xxxx xx xx
36.
(32)(41)(31)(31) (41)(23) (41)(31)
43. 3524(35)(24) 212121 3524 21 9 21 xxxx
44. 541(54)(1) 343434 541 34 45 34
666 11111
47. 737(1)3(3) 31(3)(1)(1)(3) 7739 (1)(3) 416 (1)(3) 4(4) (1)(3)
48. 252(5)5(5) 55(5)(5)(5)(5) 210525 (5)(5) 335 (5)(5) 335 (5)(5) xx xxxxxx xx xx x xx x xx
49. 22 2 23(1)(23)(1) 11(1)(1)(1)(1) 23 (1)(1) 323 (1)(1) xxxxxx xxxxxx xxxx xx xx xx
50.
22 2 323(3)2(4) 43(4)(3)(4)(3) 3928 (4)(3) 5 (4)(3) 51 (4)(3) xxxxxx xxxxxx xxxx xx xx xx xx xx
51. 22 22 34(3)(2)(4)(2) 22(2)(2)(2)(2) 56(68) (2)(2) 5668 (2)(2) 112(112) or (2)(2)(2)(2) xxxxxx xxxxxx xxxx xx xxxx xx xx xxxx
52.
55. 2422 xxx
2212 xxxx Therefore,
56. 21234 xxxx
281644 xxxx Therefore, LCM342 xx .
57. 32111 xxxxxxx
21 xxxx
Therefore, LCM11 xxx .
58. 22 32739333 xxxx 2 215253 xxxx
Therefore, LCM32533 xxx . 59.
322 44441 2121 xxxxxx xxx 322 3 221 xxxx x Therefore, 32LCM21 xx .
60. 3 x
2 32 33 9933 xxxx xxxxxxx Therefore, LCM33 xxx .
61. 32111 xxxxxxx
3222 32 2211 111 xxxxxxxx xxxx
Therefore, 22 LCM111 xxxxx
62. 22 442xxx 322 3 22 2 xxxx x Therefore, 23LCM2 xx .
63. 22 22 76224 (6)(1)(6)(4) (4)(1) (6)(1)(4)(6)(4)(1) 45 (6)(4)(1)(6)(4)(1) xx xxxx xx xxxx xxxx xxxxxx xxxxx xxxxxx
64. 2 22 1 3524 1 (3)(3)(8) (8)1 (3)(8)(3)(8) 8171 (3)(8)(3)(8) xx xxx xx xxx xxx xxxx xxxxx xxxx
65. 22 42 46 42 (2)(2)(3)(2) x xxx x xxxx
2 2 2 4(3)2(2) (2)(2)(3)(3)(2)(2) 41224 (2)(2)(3) 4104 (2)(2)(3) 2(252) (2)(2)(3) xxx xxxxxx xxx xxx xx xxx xx xxx
66. 22 2 2 2 2 2 3434 1(1)21(1) 3(1)4 (1)(1)(1) 334 (1) 344 (1) xxxx xxxxx xxx xxx xxx x xx x
22 22 22 22 32 1111 3121 11 3322 11 51 11 xxxx xx xx xx xx x
68.
22 26 2121xxxx
69. 22 22 2 423 228 423 (2)(1)(4)(2) (4)(4)(23)(1) (2)(1)(4)(4)(2)(1) 816(253) (2)(1)(4) 313 (2)(1)(4) xx xxxx xx xxxx xxxx xxxxxx xxxx xxx xx xxx
70. 22 232 87(1) xx xxx 2 2 22 2 2 2 232 (1)(7)(1) (23)(1)(2)(7) (1)(7)(1)(1)(7) 23(514) (1)(7) 611 (1)(7) xx xxx xxxx xxxxx xxxx xx xx xx
71. 232 123 xxxxx
2 2 22 2 32 2 32 2 123 11 112131 11 12233 11 253 11 243 11 xxxxx xxxxxx xxx xxxxx xxx xxxx xxx xxx xxx
73. 111111() ()() 1 () () 1 () xxh hxhxhxhxxxh xxh hxxh h hxxh xxh
74. 22 111 () hxhx
22 2222 222 22 2 22 22 22 22 111() ()() 1(2) () 2 () (2) () 2 () 2 () xxh hxhxxxh xxxhh hxxh xhh hxxh hxh hxxh xh xxh xh xxh
78. 2 11 1111 1 21211 1 11 (1) xxx xxxx xxxx xxxx x xx x x
79. 43 21 1 xx xx x
22 2 (4)(1)(3)(2) (2)(1)(1)(2) 1 54(56) (2)(1) 1 1021 (2)(1)1 2(51) (2)(1) xxxx xxxx x xxxx xx x x xxx x xx
80. 2 12 3 xx xx x
22 (2)(2)(1) (1)(2)(2)(1) 3 44() (2)(1) 3 541 (2)(1)3 54 (2)(1)(3) 54 (2)(1)(3) xxxx xxxx x xxxx xx x x xxx x xxx x xxx
81. 21 21 23 1 xx xx xx xx
2 22 22 2 2 2 (2)(1)(1)(2) (2)(1)(1)(2) (23)(1) (1)()(1) 22 (2)(1) (23) (1) 24 (2)(1) 3 (1) 2(2)( (2)(1) xxxx xxxx xxx xxxx xxxx xx xxx xx x xx xx xx xxx xx
2 2 2 2 2 1) (3) 2(2) (2)(3) 2(2) (2)(3) xx xx xxx xx xxx
82. 22 25 3 (1) 33 xx xx xx xx
22 22 3232 2 2 2 (25)(3)() (3)(3) (3)(3)(1) (3)(3)(3)(3) 215 (3) 3(53) (3)(3) 15 (3) 453 (3)(3) 15( (3) xxxx xxxx xxxx xxxx xxx xx xxxxx xx xx xx xx xx xxx xx
2 2 2 3)(3) 453 (15)(3) (453) x xx xxx xxx
83. 1111 111 1 1 1 1 1 1 x xx x x xx x x
84. 111111 1111 111 1111 1 1 xx xxx xx xx xx x x
2 7121 236 (3)7(2)121 3714121 (7)314121 712 5
11 111 1111 121 11 2,1,1 x xx xx xxx xx abc
111111 121121 111 21132 2121 3,2,1 x xx x x xxx xx abc
1121111 132132 1121 11
322153 3232 5,3,2 x xx x x xxx xx abc
Ifwecontinuethisprocess,thevaluesof a, b and c producethefollowingsequences: :1,2,3,5,8,13,21,.... :1,1,2,3,5,8,13,21,..... :0,1,1,2,3,5,8,13,21,..... a b c Ineachcasewehavea Fibonaccisequence, wherethenextvalueinthelistisobtainedfrom thesumoftheprevious2valuesinthelist.
21. 33433882 xxxxx
22. 3335322 19264343 xxxxx 23. 444 24381333
24. 44544 4816323 xxxxx
44 41283232 4 xyxyxy 26.
5 5105252 5 xyxyxy
27. 97 44842 3 xy xyxy xy
28. 23 33 42333 3111 81273 27 xy xyxx x
29. 648 xx
30. 542933 xxxxx
31.
444491223 4 234 16223 32 xyxxy xyx 32.
33314103243 3 432 3 405(2) 25 xyxxyy xyxy
33. 222 1557525353 xxxxxxxx
34. 34252010010 xxxx 35.
22233 32333 5959 59581533153
44433 34233 310310 310331003003
37. 3622612643123
51.
445444 44 44 3221622 222 22or22 xxxxx xxx xxxx
32 8350423252 22152 2152 xxxxx xxx xx
52.
39425920 920 xyyxyy xy
53.
54.
4433 3 3333 3 333 3 33 163252 823252 223252 2352 52or52 xyxxyxy xxyxxyyxy xxyxxyyxy xxyxy xyxyxyxy
2233 3 8258852 852 5 xyxyxyxyxyxy xy xy
55. 1122 2222
56. 22323 3333
57. 33515 5555
58. 333266 822222224
59.
60.
3352 525252 352 252 352536 or 2323
2272 727272 272 74 2721422 or 33
61. 2525235 235235235 4256515 445 19858519 4141
62. 3131233 233233233 623333953 1293
63. 5521 212121 525525 21
64. 3354 545454 35123512 51611 3512 11
65. 33 333 55454 2242
66. 33 333 22323 9933
2 2 2 2 22 xhxxhxxhx xhxxhxxhx xhxxhx xhx xhxxhx xhx xhxxh h
22 22 22 2 2 22 2 xhxhxhxh xhxhxhxh xhxhxhxh xhxh xhxhxh xhxh xxh h xxh h
111111111 22111 11110 21112111 5 111
615615615 1515615 6159 901531015 93 3105105
72.
80. 3 3/43 4 161628
3/2 3/233 41111 428 4
3/2 3/233 161111 16464 16
3/2333 33 9933 882222 2727272 8221621622 272
2 2/32 3 272739 8824
1/31/31/3 36362 xyxyxy
3/43/43/4 484836 xyxyxy
1/32/31/32/3 1/32/3 2222
2/32/32/32/3
2/31/32/34/3 2/32/3
3/43/43/421/33/421/3
1/41/4 21/42 433/21/4
1/41/2
33/21/41/41/2 5/43/4 5/4 3/4 1616 16 2 8 8 xyxy xyxy xy xy xy xy x y
3/23/23/211/33/211/3
3/23/23/2
33/21/2
3/23/2
33/23/21/23/2 31 3 44 4 2 8 8 xyxy xyxy xy xy xy xy xy
2/32/32/31/34/32/3
2/312/3 xyxyxyxy xyxy xyxy xy xy xyxy
1/41/21/21/2 221/41/422
3/43/4 223/4
1/41/4
3/23/4
1/413/21/413/4
1/2 1/41/2 1/4 xyxyxyxy xyxy xyxy xy xy y xy x
1/21/2 1/2 1/21/2 1/2 1/2 1/2 211 21 (1)(1) 21 (1) 22 (1) 32 (1) xxxx x xx xx x xx x x x
102. 1/21/2 1/2 1/21/2 1/21/2 112 22 1231 22 xxxx x xx xxx xx
1/32/3 1 11,1 3 xxxx
106.
33 2233 8121 3,2,8 22481 xx xx xx
24281 8812 24281 88+12 24281 6482 24281 656 24281 xxxx xx xx xx xx xx xx xx x xx
112.
21/2 2 21/2 2 1 1 ,1or1 x x x xx x
1/21/2222 21/2 2 1/21/2222 21/22 11 1 111 1 xxx x x xxx xx
22 21/22 22 21/22 221/2 11 1 11 1 1 1 xx xx xx xx xx
1/21/2222 2 44 4 xxx x
21/22 21/2 2 1/21/2 222 21/2 2 1/21/2 222 21/22 22 1/223/222 4 4 4 44 4 4 441 44 414 4 44 x x x x xxx x x xxx xx xx x xx
119.
120.
1/223/21/2 1/22 1/22 1/2 688 23()44 234 2(34)(1) xxxxx xxxx xxx xxx
1/23/2 1/2 1/2 6238 23(23)4 2109 xxx xxx xx
4/31/322 1/3 222 1/3 222 1/3 22 34442 4348 43128 41112 xxxx xxx xxx xx
4/31/3 2 1/3 1/3 234434 234342 23454 xxxx xxxx xxx
121.
1/33/24/31/2 4352333523 xxxx
1/31/2 1/31/2 1/31/2 3523423335 3523812915 35231727 xxxx xxxx xxx
3 where 2 x
136. 161222 3222 24.44seconds T
137. 314313 334312 134431
Thequotientis2(34)43 xx Theremainderis1
138. 1219137 126727 1387720
Yes,12 isafactorof329137xxx
139. Answersmayvary.Onepossibilityfollows:If 5 a ,then 225255 aa Sinceweusetheprincipalsquareroot,whichis alwaysnon-negative, 2if0 if0 aa a aa
whichisthedefinitionof a ,so 2 aa
SectionR.8: nth Roots; Rational Exponents