1.2 Fundamental Operations of Algebra
1. ()() 162216416420 −×−=−−=+=
2. ()()() 18 5233568614 6 +−−=+−−=+=
3. () 1251124 224 822(1)62 +=+=−+−=−
4. 7642 isundefined 000 × == × ,notindeterminate.
5. () 58583 +−=−=−
6. () 474711 −+−=−−=−
7. 396−+= oralternatively ()() 399366 −+=+−=+=
8. 18213 −=− oralternatively 1821(2118)(3)3 −=−−=−=−
9. () 191619163 −−−=−+=−
10. () 8108102 −−−=−+=
11. () 74(74)28 −=−×=−
12. () 9327−=−
13. () 75(75)35 −−=+×=
14. 9 3 3 =−
15. 6(2010)6(10)6020 333 ===
16. 282828 4 7(56)7(1)7 ===−
17. ()()() 2458540 −−=−−=
18. ()()() 34616272−−−==−
19. ()() 22710251010101 −÷=−÷=−÷=−
20. 64646464 8 248242(4)8 ====
21. 162(4)8(4)32 ÷−=−=−
22. 205(4)4(4)16 −÷−=−−=
23. 9210989817 −−−=−−−=−−=−
24. ()()() 7757020 −÷−=÷−=
25. 17710isundefined 770 =
26. (77)(2)0(2)0isindeterminate (77)(1)0(1)0 ==
27. () 83481220 −−=+=
28. 208420218 −+÷=−+=−
29. () 8 2612412416 2 −−+=+−=+=
30. |2|21 22 ==−
31. ( ) ( ) 1083(1050)10(8)(3)(40) 80(3)(40) 240(40) 6 −−÷−=−−÷−
32. 75752 1 1(2)22 ===
33. () 2424 49(49)1236243(5)2 −−=+×=−+= +−−
34. 184|6|184626628 31311 −=−=−−=−−=−
35. () () () 1414 7368732 22321 14 73(2) 2 776 776 6 −−−−=−−−− =−−− =−−−− =−+− =−
36. () 6 73|9|(73)29 3 2129 14 −−+−−=+×+− =+− =
37. 3|92(3)|3|96| 1109
38. () 201240(15)240600360isundefined 989898980 −+ ===
39. ()() 6776 = demonstratesthecommutativelawofmultiplication.
40. 6886 +=+ demonstratesthecommutativelawofaddition.
41. ()()() 6316361 +=+ demonstratesthedistributivelaw.
42. () 45(45)×=×ππ demonstratestheassociativelawofmultiplication.
43. ()() 359359 ++=++ demonstratestheassociativelawofaddition.
44. ()()() 8328382 −=− demonstratesthedistributivelaw.
45. ()5395(39) ××=×× demonstratestheassociativelawofmultiplication.
46. ()3677(36) ××=×× demonstratesthecommutativelawofmultiplication.
47. () abab −+−=−− ,whichisexpression(d).
48. () babaab −−=+=+ ,whichisexpression(a).
49. () babaab −−−=−+=− ,whichisexpression(b).
50. () ababba −−−=−+=− ,whichisexpression(c).
51. Since|5(2)||52||7|7 −−=+== and|5(2)||52||3|3 −−−=−+=−= , |5(2)||5(2)| −−>−−−
52. Since|3|7|||37||10|10 −−−=−−=−= and||3|7||37||4|4 −−=−=−= , |3|7||||3|7| −−−>−− .
53. (a) Thesignofaproductofanevennumberofnegativenumbersispositive. () Example:3618 −−= (b) Thesignofaproductofanoddnumberofnegativenumbersisnegative. Example: ()() 54240−−−=−
54. Subtractionisnotcommutativebecause xyyx −≠− .Example:752doesnotequal572 −=−=−
55. Yes,fromthedefinitioninSection1.1,theabsolutevalueofapositivenumberisthenumberitself,andtheabsolute valueofanegativenumberisthecorrespondingpositivenumber.Soforvaluesof x where0 x > (positive)or0 x = (neutral)then xx = .
Example:44 =
Theclaimthatabsolutevaluesofnegativenumbers xx =− isalsotrue.
Example: () ifis6,then666. x −−=−−=
56. Theincorrectanswerwasachievedbysubtractingbeforemultiplyingordividingwhichviolatestheorderofoperations.
2462318239327 −÷×≠÷×=×=
Thecorrectvalueis: 24623243324915 −÷×=−×=−=
57. (a) 1 xy −= istrueforvaluesof x and y thatarenegativereciprocalsofeachotheror 1 y x =− ,providingthatthe number x inthedenominatorisnotzero.Soif12 x = ,then
(b) 1 xy xy = istrueforallvaluesof x and y,providingthat xy ≠ topreventdivisionbyzero.
58. (a) xyxy +=+ istrueforvalueswhereboth x and y havethesamesignoreitherarezero: xyxy +=+ ,when0and0 xy≥≥ orwhen0and0 xy≤≤
Example: 63639and 63639 Also, 6(3)99 63639 +=+= +=+= −+−=−= −+−=+= xyxy +=+ isnottruehowever,whenxandyhaveoppositesigns xyxy +≠+ ,whenx0and0;or0and0 yxy ><<> .
Example: 2161515, 2162162715 −+=−= −+=+=≠
4(5)11, 454591 +−=−= +−=+=≠
(b) Inorderfor xyxy −=+ itisnecessarythattheyhaveoppositesignsoreithertobezero. Symbolically, xyxy −=+ when0and0 xy≥≤ ;orwhen0and0 xy≤≥ .
Example: 6(3)639and 63639 −−=+= +−=+=
Example: 1171818 11711718 −−=−= −+−=+=
xyxy −=+ isnottrue,however,when x and y havethesamesigns. xyxy −≠+ ,whenx0and0;or0and0 yxy >><< .
Example: 2161515, 2162715 −== +=≠
59. Thetotalchangeinthepriceofthestockis0.680.420.06(0.11)0.020.29 −+++−+=− .
60. Thedifferenceinaltitudeis86(1396)1396861310m −−−=−=
61. Thechangeinthemeterenergyreading E wouldbe:
2.1kWh4.5kWh 2.4kWh changeusedgenerated change change change EEE E E E =−
()2.1kWh1.5kW3.0h
62. Assumingthatthisbattingaverageisforthecurrentseasononlywhichisjuststarting,thenumberofhitsiszeroand thetotalnumberofat-batsisalsozerogivingusa numberofhits0 battingaverageatbats0 == whichisindeterminate,not 0.000.
63. Theaveragetemperaturefortheweekis: 7(3)231(4)(6)C 7 7323146 C 7 14 C2.0C 7 avg avg
64. Theverticaldistancefromtheflaregunis ()()()() () 7051625 350400 350400 50m d d d d =+− =+− =− =− Theflareis50mbelowtheflaregun.
65. Thesumofthevoltagesis ()() 6V2V8V5V3V 6V2V8V5V3V 10V
66. (a) Thechangeinthecurrentforthefirstintervalisthesecondreading–thefirstreading 222 12lb/in7lb/in9lb/inChange =−−=−
(b) Thechangeinthecurrentforthemiddleintervalsisthethirdreading–thesecondreading () 22222 29lb/in2lb/in9lb/in2lb/in7lb/inChange =−−−=−+=− .
(c) Thechangeinthecurrentforthelastintervalisthelastreading–thethirdreading () 22222 36lb/in9lb/in6lb/in9lb/in3lb/inChange =−−−=−+=
67. Theoildrilledbythefirstwellis100m200m300m += whichequalsthedepthdrilledbythesecondwell 200m100m300m += 100m200m200m100m +=+ demonstratesthecommutativelawofaddition.
68. Thefirsttankleaks () L 127h84L h = .Thesecondtankleaks () L 712h h 84L. = 127712 ×=× demonstratesthecommutativelawofmultiplication.
69. Thetotaltimespentbrowsingthesewebsitesisthetotaltimespentbrowsingthefirstsiteoneachday+thetotaltime spentbrowsingthesecondsiteoneachday minutesminutes 7days257days15dayday
280min t t t t t t
175min105min
280min OR minutes 7days(2515)day minutes 7days40day
whichillustratesthedistributivelaw.
70. Distance=rate×time () () kmkm
600503h hh kmkm
6003h+503h hh
1800km150km1950km OR kmkm 600503h hh km 6503h h 1950km d d d d d d
Thisillustratesthedistributivelaw.
1.3 Calculators and Approximate Numbers
1. 0.390hasthreesignificantdigitssincethezeroisafterthedecimal.Thezeroisnotnecessaryasaplaceholderand shouldnotbewrittenunlessitissignificant.
2. 35.303roundedofftofoursignificantdigitsis35.30.
3. Infindingtheproductoftheapproximatenumbers,2.530.576.25 ×= ,butsince2.5has2significantdigits,theanswer is76.
4. 38.321.9(3.58)116.702 −−= usingexactnumbers;ifweestimatetheresult,4020(4)120 −−=
5. 8cylindersisexactbecausetheycanbecounted.55km/hisapproximatesinceitismeasured.
6. 0.002mmthickisameasurementandisthereforeanapproximation.$7.50isanexactprice.
7. 24hrand1440min(60min/h×24h=1140min)arebothexactnumbers.
8. 50keysisexactbecauseyoucancountthem;50hofuseisapproximatesinceitisameasurementoftime.
9. Both1cmand9garemeasuredquantitiesandsotheyareapproximate.
10. Thenumbers90and75areexactcountsofwindowswhile15yearsisameasurementoftime,henceitisapproximate.
11. 107has3significantdigits;3004has4significantdigits;1040has3significantdigits(thefinalzeroisaplaceholder.)
12. 3600has2significantdigits;730has2significantdigits;2055has4significantdigits.
13. 6.80has3significantdigitssincethezeroindicatesprecision;6.08has3significantdigits;0.068has2significant digits(thezerosareplaceholders.)
14. 0.8730has4significantdigits;0.0075has2significantdigits;0.0305has3significantdigits.
15. 3000has1significantdigit;3000.1has5significantdigits;3000.10has6significantdigits.
16. 1.00has3significantdigitssincethezerosindicateprecision;0.01has1significantdigitsinceleadingzerosarenot significant;0.0100has3significantdigits,countingthetrailingzeros.
17. 5000has1significantdigit;5000.0has5significantdigits;5000has4significantdigitssincethebaroverthefinal zeroindicatesthatitissignificant.
18. 200has1significantdigit;200has3significantdigits;200.00has5significantdigits.
19. (a) 0.010hasmoredecimalplaces(3)andismoreprecise. (b) 30.8hasmoresignificantdigits(3)andismoreaccurate.
20. (a) Both0.041and7.673havethesameprecisionastheyhavethesamenumberofdecimalplaces(3). (b) 7.673ismoreaccuratebecauseithasmoresignificantdigits(4)than0.041,whichhas2significantdigits.
21. (a) Both0.1and78.0havethesameprecisionastheyhavethesamenumberofdecimalplaces. (b) 78.0ismoreaccuratebecauseithasmoresignificantdigits(3)than0.1,whichhas1significantdigit.
22. (a) 0.004ismoreprecisebecauseithasmoredecimalplaces(3). (b) 7040ismoreaccuratebecauseithasmoresignificantdigits(3)than0.004,whichhasonly1significantdigit.
23. (a) 0.004ismoreprecisebecauseithasmoredecimalplaces(3). (b) Bothhavethesameaccuracyastheybothhave1significantdigit.
24. Theprecisionandaccuracyof8.914and8.914arethesame.
(a) Both50.060and8.914havethesameprecisionastheyhavethesamenumberofdecimalplaces(3). (b) 50.060ismoreaccuratebecauseithasmoresignificantdigits(5)than8.914,whichhas4significantdigits.
25. (a) 4.936roundedto3significantdigitsis4.94. (b) 4.936roundedto2significantdigitsis4.9.
26. (a) 80.53roundedto3significantdigitsis80.5. (b) 80.53roundedto2significantdigitsis81.
27. (a) -50.893roundedto3significantdigitsis-50.9. (b) -50.893roundedto2significantdigitsis-51.
28. (a) 7.004roundedto3significantdigitsis7.00. (b) 7.004roundedto2significantdigitsis7.0.
29. (a) 5968roundedto3significantdigitsis5970. (b) 5968roundedto2significantdigitsis6000.
30. (a) 30.96roundedto3significantdigitsis31.0. (b) 30.96roundedto2significantdigitsis31.
31. (a) 0.9449roundedto3significantdigitsis0.945. (b) 0.9449roundedto2significantdigitsis0.94.
32. (a) 0.9999roundedto3significantdigitsis1.00. (b) 0.9999roundedto2significantdigitsis1.0.
33. (a) Estimate:131212 +−= (b) Calculator:12.781.04951.63312.1965, +−= whichis12.20to0.01precision
34. (a) Estimate:41768 ×= (b) Calculator:3.64(17.06)62.0984, = whichis62.1to3significantdigits
35. (a) Estimate0.7496 ×−=− (b) Calculator:0.65723.948.6516.061632, ×−=− whichis6.06to3significantdigits
36. (a) Estimate40264406.534 −÷=−= (b) Calculator:41.526.43.734.3648649, −÷= whichis34to2significantdigits
37. (a) Estimate9(1)(4)9413 +=+= (b) Calculator:8.75(1.2)(3.84)13.358, += whichis13to2significantdigits
38. (a) Estimate2030301020 2 −=−=
(b) Calculator:28, 20.955 18.475 2.2 = whichis18to2significantdigits
39. (a) Estimate9(15)1356, 91524 == + to1significantdigit
(b) Calculator:, 8.75(15.32)5.569173 8.7515.32 = + whichis5.57to3significantdigits
40. (a) Estimate9(4)365, 257 == + to1significantdigit
(b) Calculator:, 8.97(4.003)5.296 2.04.78 = + whichis5.3to2significantdigits
41. (a) Estimate2(300)4.53.0, 400 −= to2significantdigits
(b) Calculator:2.056(309.6)4.522.9093279, 395.2 −= whichis2.91to3significantdigits
42. (a) Estimate15812, 22 += + to2significantdigits
(b) Calculator:14.98.19512.1160526, 1.72.1 += + whichis12to2significantdigits
43. 0.978814.915.8788 += sincetheleastprecisenumberinthequestionhas4decimalplaces.
44. 17.31122.985.669 −=− sincetheleastprecisenumberinthequestionhas3decimalplaces.
45. 3.142(65)204.23 −=− ,whichis-204.2becausetheleastaccuratenumberhas4significantdigits.
46. 8.6217280.004988 ÷= ,whichis0.00499becausetheleastaccuratenumberhas3significantdigits.
47. Withafrequencylistedas2.75MHz,theleastpossiblefrequencyis2.745MHz,andthegreatestpossiblefrequencyis 2.755MHz.Anymeasurementsbetweenthoselimitswouldroundto2.75MHz.
48. Foranenginedisplacementstatedat2400cm3,theleastpossibledisplacementis2350cm3,andthegreatestpossible displacementis2450cm3.Anymeasurementsbetweenthoselimitswouldroundto2400cm3 .
49. Thespeedofsoundis3.25mi15s0.21666...mi/s=1144.0...ft/s ÷= .However,theleastaccuratemeasurementwas timesinceithasonly2significantdigits.Thecorrectansweris1100ft/s.
50. 4.4s2.72s1.68s −= , buttheanswermustbegivenaccordingtoprecisionoftheleastprecisemeasurementinthe question,sothecorrectansweris1.7s.
51. (a) 2.23.84.52.2(3.84.5)19.3 +×=+×= (b) (2.23.8)4.56.04.527 +×=×=
52. (a) 6.032.251.77(6.032.25)1.774.45 ÷+=÷+= (b) 6.03(2.251.77)6.034.021.5 ÷+=÷=
53. (a) 202 += (b) 202 −= (c) 022 −=− (d) 200 ×= (e) 20÷= error;fromSection1.2,anequationthathas0inthedenominatorisundefinedwhenthenumeratorisnot also0.
54. (a) 20.000120000 ÷= ;20÷= error (b) 0.00010.00011 ÷= ;00÷= error
(c) Anynumberdividedbyzeroisundefined.Zerodividedbyzeroisindeterminate.
55. 3.14159265... π = (a) 3.1416 π < (b) 2273.1428 (227) π ÷= <÷
56. (a) 8330.2424...0.24 ÷== (b) 3.14159265... π =
57. (a) 130.333... ÷= Itisarationalnumbersinceitisarepeatingdecimal. (b) 5110.454545... ÷= Itisarationalnumbersinceitisarepeatingdecimal. (c) 250.400... ÷= Itisarationalnumbersinceitisarepeatingdecimal(0istherepeatingpart).
58. 1249900.12525.... ÷= thecalculatormayshowtheansweras0.1252525253becauseithasroundedupforthenext5 thatdoesn’tfitonthescreen.
59. 32.4MJ26.704MJ36.23MJ95.334MJ ++= .Theanswermustbetothesameprecisionastheleastprecise measurement.Theansweris95.3MJ.
60. Wewouldcompute8(68.6)5(15.3)625.3 += androundtothreesignificantdigitsforatotalweightof625lb.The values8and5areexact.
61. Wewouldcompute12(129)16(298.8)6328.8 += androundtothreesignificantdigitsforatotalweightof6330g.The values12and16areexact.
62. (15.25.64101.23)3.55A 122.073.55A 433.3485V 433Vto3significantdigits V
63. 100(40.6352.96)59.1386%59.14%to4signficiantdigits 105.3052.96 + == +
64. 50.45(9.80)91.779N=92Nto2significantdigits 1100.923 T == +÷
65. (a) Estimate851030, ×−= to1significantdigit. (b) Calculator:7.844.93211.31727.34988 ×−= whichis27.3to3significantdigits.
66. (a) Estimate20501015 −÷= to2significantdigit. (b) Calculator:21.653.149.6416.0875519 −÷= whichis16.1to3significantdigits.
1.4 Exponents and Unit Conversions
1. 2 32323266 ()(1)(1)()(1) xxxxx
−=−=−==
2. 0 22(1)2 x ==
3. 34347 xxxx + ==
4. 27279 yyyy + ==
5. 42426222 bbbb + ==
6. 5516333 kkkk + ==
7. 5 532 3 m mm m ==
8. 6 2615 22 x xx x =−=−
9. 5594 94 1 77 77 nnn nn =−=−=−
10. 143 43 33 33 s ss ss ===
11. () 4 22(4)8 PPP ==
12. ()3 88(3)24 xxx ==
13. ()30 2302(30)3060 aTaTaT ==
14. ()3 232(3)6 3(3)27 rrr ==
==
15. 33 33 2(2)8 bbb
16. 2020 20 FF tt
=
17. 4 22(4)8 4 21(2)6 xxx
==
18. 33 33(3)9 3(3)27 nnn
==
19. () 0 81 a =
20. 01 v −=−
21. 0 33(1)3 x −=−=−
22. 0 (2)1(1)1 −−=−=−
23. 1 1 11 6 66 ==
24. 5 5 1 w w −=−
25. 2 2 1 R R =
26. 48 48 1 t t
27. 7 27272(7)1414 ()(1)()(1)(1) ttttt
28. 5 35353(5)1515 ()(1)()(1)(1) yyyyy
29. 3 3(5)2 5 L LL L −=−=−
30. 407040(70)30 30 2 222 iiii i
31. 444 4444 2221 (2)(2)()168 vvv vvv
32. 23235 232(3)6 1 () xxxx xxxx + ===
33. 242(4)8 424(2)8 ()1 () nnn nnn ===
34. () 111 11 (3)(3)1 33339 ttt ttt ===
35. 02110(1)2(1)1(1)021 ()2 a xaxaxa x πππ===
36. 4 24222(2)4(2)248 (3)(3)8 3 9 m mnmnmn n ===
37. 6 13221(2)3(2) 2 (8)(8)64 s gsgs g −=−=
38. 63 223232(3)316327 2 ()(1)()()aax axaxaxaxaxax x + −=−=−=−=−
39. 3 131(3)3 11(3)3 4(4) 64 xxx aaa
==
40. 2 222(2)410 55(2)104 2(2) 44 bbby yyyb ===
41. 252(1)3 16 1555 3 nTnn nTTT ==
42. 2323232(2)2(32)32346432343234 232323232(64)96 ()nRTnRTnRTnRnR RTTTTT ====
43. () 2 74(5)282553 −−−=−−=−
44. 5 62(2)(8)632(16)6321610 −−−−=−−−=−+=−
45. 23 (26.5)(9.85)(702.25)(955.671625)253.421625 −−−−=−−−= whichgetsroundedto253because702.25and–955.671625arebothaccuratetoonly3significantdigitsduetothe originalnumbershavingonly3significantdigits.
46. 2626 0.711(0.809)(1)(0.711)(0.809)(1)(0.505521)(0.2803439122)0.7858649122 −−−−=−−−=−−=− whichgetsroundedto3significantdigits:–0.786.
47. 4 3.07(1.86)5.71025.71020.420956185 (1.86)1.59611.968832161.59613.56483216 === + −+ whichgetsroundedto3significantdigits:–0.421.
48. 24 15.66(4.017)245.2356260.37982269215.1442226923.941837074 1.044(3.68)3.841923.84192 === whichgetsroundedto3significantdigits:3.94.
49. 2 3 254 2.38(60.7)1.17 254 2.38(3684.49)1.601613=− 8769.0862158.5901213339=− 8610.4960786661 = whichgetsroundedto3significantdigits:8610.
50. 2 0.889 4.2(4.6)1.891.09 + 0.889 19.32 1.891.1881 =+ 0.889 19.32 0.7019
19.320.889880728 20.209880728 =+ =+ = whichgetsroundedto2significantdigits:20
51. 11 11(1) 111 xxx
,whichisthereciprocalof x
52.
53. If35 a = ,then () () 123(4) 1234 124 12 5 625 aa aa a a = = = =
,since01 a = requiresthat0 a ≠
54. Foranynegativevalueof a , a willbenegative,and2 a willbepositive,makingallvaluesof2 1 a greaterthan 1 a Therefore,itisneverthecasefornegativevaluesof a , 21 . aa <
55. 55050(5)0 ()()()1 aaaa xxxxxx ⋅===== ,providedthat0 x ≠
56. 2()22222(2)4()((1))(1)() ababababaaa yyyyyy −+ −++ −⋅=−=−== .
57.
58. 12 ()()123 GmMGM GmMmrr mrr + ==
59.
60. 2
61. () 24 250010.04224 $25001.0105 4
$2500(1.28490602753)
$3212.26700688
$3212.27
62. 23 6.85(100020(6.85)(6.85))6.85(100020(46.9225)321.419125) 18501850
6.85(1321.419125938.45) 1850 6.85(382.969125) 1850 2623.33850625 1850 1.418020814 1.42cm
63. 110 1001000 1 If()1then(1)22.000,(10)1.12.594, (100)1.012.705,and(1000)1.0012.717.
64. We10101010102010102040 have1TB2GB2(2MB)2(2(2bytes))2bytes2bytes ++ =====
=
65. () ft 28.29.81s276.642ftwhichisroundedto277ft. s
66. () mi 40.53.7gal149.85miwhichisroundedto150mi. gal
=
67. 2 222 m1ft60sftft 7.2585,629.92whichisroundedto85,600. 0.3048m1min sminmin
=
68. 3 33 238kg1000g1mg0.238. 1kg100cm mcm
=
69. 1L 15.7qt15.7qt14.8533586Lwhichisroundedto14.9L. 1.057qt
=×=
=×=
70. 7.501hpW7.50W0.01005362hpwhichisroundedto0.0101hp. 746.0W
71. 2 2222 1in 245cm245cm37.975076inwhichisroundedto38.0in. 2.54cm
=×=
72. 2 2222 1km 85.7mi85.7mi221.941401kmwhichisroundedto222km. 0.6214mi
73. mm60s1ftftft 65.265.212834.6457whichisroundedto12800. ss1min0.3048mminmin
74. mimi1km1galkmkm 25.025.010.6292562whichisroundedto10.6. galgal0.6214mi3.785LLL
75. 15.62.54cmin15.6in39.624cmwhichisroundedto39.6cm. 1in
76. 1km 12,500mi12,500mi20,115.8674kmwhichisroundedto20,100km. 0.6214mi
77. galgal1day3.785LLL 575,000575,00090,682.2917whichisroundedto90,700. dayday24hr1galhrhr
78. galgal1min3.785LLL 85855.3620833whichisroundedto5.4. minmin60s1galss
79. ftft60s60min0.3084m1kmkmkm 113011301254.5712whichisroundedto1250. ss1min1hr1ft1000mhrhr
=×××=
80. kmkm1hr1min1000mm 720072002000. hrhr60min60s1kms
81. 22 222 lblb4.448N1in100cmN 14.714.7101,347.883whichisroundedto101,000Pa. inin1lb2.54cm1mm
=×××=
82. 3 3333 3 lblb1kg1ftkgkg 62.462.4999.381whichisroundedto999. ftft2.205lb0.3048mmm (Thekg actualvalueis1000.) m =××=
1.5 Scientific Notation
1. 3 8.06108060 ×=
2. -111-1 -1-11 -11 -12 750000000000(7.510) 7.510 0.1333...10 1.3310 =× =× =× =× roundedto3significantdigits.
3. 4 4.51045,000 ×= 4. 7 6.81068,000,000 ×=
5. 3 2.01100.00201 ×=
6. 5 9.61100.0000961 ×=
7. 0 3.23103.2313.23 ×=×=
8. 0 810818 ×=×= 9. 1.861018.6 ×= 10. 1 1100.1 ×= 11. 34000410 =×
12. 564 0005.610 =× 13. 0.00878.7103 =× 14. 0.000747.4104 =× 15. 8609,000,0006.0910 =×
16. 110110 =×
17. 0.05285.28102 =×
18. 0.00009089.08105 =×
19. 4913 28,000(2,000,000,000)2.810(210)5.610 =××=×
20. 432 50,000(0.006)510(610)300310 =××==×
21. 4 8 4 88,0008.8102.210 0.0004410 × ==× ×
22. 5 12 6 0.00003310 510 6,000,000610 × ==× ×
23. 635,600,00035.610 =×
24. 0.00000565.6106 =×
25. 0.097397.3103 =×
26. 9925,000,000,00092510 =×
27. 0.000000475475109 =×
28. 3370,00037010 =×
29. 3534343434 2103100.2103103.210 ×+×=×+×=×
30. 121010101012 5.3103.710530103.710526.3105.26310 ×−×=×−×=×=×
31. 293329(3)87 (1.210)1.2101.72810 ×=×=×
32. 165516(5)8078 (210)2100.03125103.12510 ×=×=×=×
33. 101320(649,000)(85.3)7.307480410 =× whichgetsroundedto10 7.3110 ×
34. 0.0000569(3,190,000)181.511 = whichgetsroundedto2 1.8210 ×
35. 0.0732(6710)491.1727 1.586780310 0.00134(0.0231)0.000030954 ==× whichgetsroundedto7 1.5910 × .
36. 0.004520.0045211 1.91563574110 2430(97,100)235,953,000 ==× whichgetsroundedto11 1.9210 ×
37. 853 (3.64210)(2.73610)9.96451210 ××=× whichgetsroundedto3 9.96510 × .
38. 12 18 2019 (7.30910)0.534214813.5665672339410 5.9843(2.503610)1.4978702910 × ==× ××
whichgetsroundedto18 3.56710 ×
39. 72115 (3.6910)(4.6110)1.701091016 3.3751785714210 0.05040.0504 ××× ==× whichgetsroundedto16 3.3810 × .
40. 712272432 33 5 (9.90710)(1.0810)(9.90710)(1.166410)1.01555248101.5614388204510 (3.60310)(2054)0.074005620.07400562 ××××× ===× × whichgetsroundedto33 1.5610 ×
41. 500,000,0008tweets510tweets =×
42. 17,200,000,00010bytes1.7210bytes =×
43. 0.0000036W310W =×
44. 0.00753mm7.510mm =×
45. 1,200,000,0009Hz=1.2×10Hz
46. 12 1.84101,840,000,000,000 ×=
47. 21 12,000,000,00002 m1.210m =×
48. 16 3.08610m=30,860,000,000,000,000m ×
49. 12 1.610W=0.0000000000016W ×
50. 43 2.4100.00000000000000000000000000000000000000000024 ×=
51. (a) 3 23002.310=2.3kW =× (b) 3 0.2323010230mW =×=
(c) 6 2,300,0002.3102.3MW =×= (d) 6 0.0002323010230W μ =×=
52. (a) 6 80900008.09108.09M =×=Ω (b) 809300080910809k =×=Ω (c) 3 0.080980.91080.9m =×=Ω
53. (a) 100100googol11010 =×= (b) 100googol10googolplex1010 ==
54. 100 1001007921 79 10 googol10,sotofindtheratio1010 10 == = Agoogolis2110timeslargerthanthenumberofelectronsintheuniverse.
55. 9 sun's77 diameter1.410m earth'sdiameter1.2727210mwhichisroundedto1.310m. 110110 × ===××
56. 3099 21,073,741,8241.07374182410110 ==×≈ ×
57. 15 7.510s68 5.610additions4.210s addition × ××=×
58. 0.000000039%0.00000000039 = 0.000000000390.0851111 mg3.31510mg=3.310mg×=××
59. 8770.078ms2.998102.338440010mwhichroundsto2.310m s ××=× ×
60. (a) 244 h60min60s 1day××86400s8.6410s dayhmin ×== × (b) 365.259100day24h60min60s year××××3155760000s3.1557600×10s yeardayhmin ==
61. -271 81 1.66×108 kg1.6×10amu×1.2510oxygenatoms3.3210kg amuoxygenatoms ×× =×
62. 4 -8424 -8494 2 2 5.710WK×(3.03×10K) 5.710WK×8.428892481×10K =4.80446871417×10W =4.8×10W WkT W W W W = =× =×
63. () -82-82 22--592 2.196×10 Ω m2.196×10 Ω m ==3.43296626857 Ω =3.433 Ω 7.998×10m6.3968004×10m k R d ==
64. 8 55 2 1.49610kmAU2.99799599198×10kms2.99810kms AU4.9910s × ×==× × ThisisthesamespeedmentionedinQuestion56asthespeedofradiowaves.
1.6 Roots and Radicals
1. 33 3 64(4)4 −=−=−
2. (15)(5)
Neither15nor5isaperfectsquare,sothisexpressionisnotasuseful.However,ifwefurtherfactorthe15 to2(3)(5)(5)3(5)53 == ,theresultcanstillbeobtained.
3. 2 1691441212 ×===
4. 64isstillimaginarybecauseanevenroot(inthiscase n =2)ofanegativenumberisimaginary,regardlessofthe numericalfactorplacedinfrontoftheroot.
5. 2 4977 ==
6. 225(25)(9)2595315 ==×=×=
7. 2 1211111−=−=−
8. 2 3666 −=−=−
9. 2 6488 −=−=−
10. 111 0.250.5 42 4 ====
11. 993 0.090.3 10010 100 ====
12. 900(9)(100)910031030 −=−=−×=−×=−
13. 33312555 == 14. 4441622 == 15. 4448133 == 16. () 55 5 32(2)22 −−=−−=−−=
17. () 2 5555 =×=
18. ()3 3333 3131313131 =××=
19. () () () 33 3 3347147(1)(47)47 −−=−−=−−=
20. ()5 52323 −=−
21. () () () 44 4 44 53153(1)(53)53 −=−==
22. 75(25)(3)25353 ==×=
23. 18(9)(2)9232 ==×=
24. 32(16)(2)16242 −=−=−×=−
25. 1200(100)(4)(3)100431023203 ==××=××=
26. 50(25)(2)2525252 ==×=×=
27. 2842(4)(21)24212221421 ==××=××=
28. (36)(3) 10836363 33 2222 ×× ====
29. 8080 2045452525 374 ===×=×=×=
30. 22 8110811091090 ×=×=×=
31. 323 33 864(4)4 −=−=−=−
32. 4424 4 98133 ===
33. 2 2 781(49)(9)(49)(9) (3)49(9)(7) == (9) 7 (7) =
34. 5 55 5 2 2243323(32)(3) 3144312 =−=− (3) 8 (12)3 =−
35. 2 36641001010 +===
36. 2 251441691313 +===
37. 22 3998190(9)(10)910310 +=+===×=
38. 22 84641648(16)(3)16343 −=−===×=
39. 85.49.24121204171 = ,whichisroundedto9.24
40. 376261.3351449007 = ,whichisroundedto61.34
41. 0.81520.9028842672 = ,whichisroundedto0.9029
42. 0.06270.25039968051 = ,whichisroundedto0.250
43. (a) 12962304360060 +== ,whichisexpressedas60.00 (b) 12962304364884 +=+= ,whichisexpressedas84.00
44. (a) 10.62762.160912.78853.57610122899 +== ,whichisroundedto3.57610 (b) 10.62762.16093.261.474.73 +=+= ,whichisexpressedas4.7300
45. (a) 22 0.04290.01830.001840410.00033489 0.00150552 0.03880103091 0.0388 −=− = = = (b) 22 0.04290.01830.04290.0183 0.0246 −=− =
46. (a) 22 3.6250.61413.1406250.376996 13.517621 3.67663174658 3.677 +=+ = = = (b) 22 3.6250.6143.6250.614 4.239 +=+ =
47. 24(24)(150)360060mi/h s ===
48. 2222 22 2 (5.362)(2.875) 28.7510448.265625 20.485419 4.52608208056 4.526 ZX−=Ω−Ω =Ω−Ω =Ω
49. 9 33 2.1810Pa 1.0310kg/m B d × = ×
2116504.85436 kg/m kgm/s/m
2116504.85436 kg/m
2116504.85436m/s 1454.82124481m/s 1450m/s
50. 40(40)(75) 3000 54.7722557505
51. 2222 22 2 (52.3in)(29.3in)
2735.29in858.49in
3593.78in 59.948144258in 59.9in
52.
53. (9.8)(3500) 34300 185.20259 190m/s
54. 4242 87 8 1.27101.2710(9500)(9500) =1.2065109.02510 =2.10910 =14522.3965 whichisroundedto15000km
55. 2 aa = isnotnecessarilytruefornegativevaluesof a because a2willbeapositivenumber,regardlesswhether a is negativeorpositive.Theprincipalrootcalculatedisassumedtobepositive,buttherearealwaystwosolutionstoa squareroot,2aa =± since22 ()aa+= and22 ()aa−= (seetheintroductiontothischaptersection),soitissometimes trueandsometimesfalsefornegativevaluesof a,dependingonwhichrootsolutionisdesired.If onlyprincipalroots areconsidered,thenitwill not betruefornegativevaluesof a.Forexample,2(4)1644 −==≠− .
56. (a) xx > when1 x > .Anynumbergreaterthan1willhaveasquarerootthatissmallerthanitself.For example,221.41 >= (b) xx = when1 x = or0 x = becausetheonlynumbersthataretheirownsquaresare0and1(i.e.,200 = and 2 11 = ).
(c) xx < when01 x << .Anynumberbetween0and1willhaveasquarerootlargerthanitself.For example,0.250.250.5 <=
57. (a) 3214012.8865874254 = ,whichisroundedto12.9 (b) 30.2140.59814240297 −=− ,whichisroundedto–0.598
58. (a) 70.3820.87155493458 = ,whichisroundedto0.872 (b) 73822.33811675837 −=− ,whichisroundedto–2.34
59. 6 11 22(3.1416)0.250(40.5210) f
6.2832(0.003172144385) 1 0.0199312175998 50.172549 whichisroundedto50.2Hz = × = = =
6.283210.062510
60. 2 standarddeviation=variance 80.5kg 8.972179222kg whichisroundedto8.97kg = =
1.7 Addition and Subtraction of Algebraic Expressions
1. 32533 xyyxy +−=−
2. 3(2)3224 cbccbcbc −−=−+=−+
3. 3[(5)2]3[52] 3[5] 35 45 axaxsaxaxaxsax axaxs axaxs axs −−−=−−− =−−− =++ =+
4. 2222 22 22 22 2 3{[2(2)]}3{[22]} 3{22} 3{222} 3222 522 abaabababaabab abaabab abaabb abaabb abab −−−+=−−−− =−−++ =−−+ =−+− =−−
5. 5748 xxxx +−=
6. 634tttt −−=−
7. 244 yyxyx −+=+
8. 4610 CLCCL −+−=−+
9. 343055 tststss −−−=−=−
10. 812404 abababa −−++=+=
11. 2223532 FTFTFT −−+−=−−
12. 2323 xyxyzxyz −−−+=−−+
13. 2222222 2 ababababab −−=−−
14. 2222222 323 xyxyxyxyxy −−+=−
15. 2(62)266 pppppp +−−=−−=−
16. 5(34)53448 npnpnp +−+=+−+=−++
17. (792)79297 vxvvxvvx −−+=−+−=−+−
18. 111312()222222 abaabaab −−−=−−+=−−
19. 23(45)14555 aaa −−−=−−+=−
20. (2)3234 AhAAAhAAAh +−−=+−−=−+
21. (3)(56)35652 aaaaa −+−=−+−=−+
22. (4)(24)42463 xyxyxyxyxy −−−−=−++=+
23. (2)(3)2325 tuuttuuttu −−+−=−++−=−+
Section 1.7 AdditionandSubtractionofAlgebraicExpressions33
24. 2(63)(54)126548 xyyxxyyxxy −−−−=−+−+=−+
25. 3(2)(5)63578 rssrrssrrs +−−−=+++=+
26. 3()2(2)3324 ababababab −−−=−−+=+
27. 7(63)2(4)4221281950 jjjjj −−−+=−+−−=−
28. 22222 (5)2(32)564745 taasttaastastt −+−−=−−−+=−+−
29. [(46)(3)][463] [77] 77 nnnn n n −−−−=−−−+ =−−+ =−
30. [()()][] [22] 22 ABBAABBA AB AB −−−−=−−−+ =−− =−+
31. 22 2 2 2[4(5)]2[45] 2[9] 218 tt t t −−=−+ =−+ =−+
32. 228 3[3(4)]3[3] 333 21 3[] 33 21 aa a a −−−−−=−−++ =−− =−+
33. 2[2()]2[2] 2[3] 6 xaaxxaax a a −−−−−=−−−−+ =−− =
34. 2[3(2)4]2[364] 2[310] 620 xyyxyy xy xy −−−+=−−++ =−−+ =−
35. [3(4)][34] [1] 1 21 aZaZaZaZ aZaZ aZaZ aZ −−+=−−− =−−− =++ =+
36. 9[6(4)4]9[644] 9[510] 9510 410 vvvvvv vv vv v −−−−+=−+++ =−+ =−− =−
Copyright©2018PearsonEducation,Inc.
37. 5{8[4(21)]}5{8[421]} 5{8421} 5{52} 552 35 zzzz zz zz zz z −−−+=−−−− =−−++ =−+ =−− =−
38. 7{[2()]}7{[2]} 7{[3]} 7{3} 7{2} 72 9 yyyxyyyyxy yyyx yyyx yyx yyx xy −−−−=−−−+ =−−− =−−+ =−−+ =+− =−+
39. 5(2)[3()]52[3] 52[4] 74 85 pqpqpqpqpqpq pqpqp pqqp pq −−−−−=−+−−+ =−+−− =−−+ =−
40. (4)[(57)(62)]4[5762] 4[9] 49 25 LCLCLCLCLCLC LCLC LCLC LC −−−−−+=−+−−−− =−+−−− =−+++ =+
41. 2222 22 2 2 2{(4)[3(4)]}2{4[34]} 2{434} 2{211} 422 xxxx xx x x −−−−+−=−−+−+− =−−+−−+ =−− =−+
42. {[(2)]()}{[2]} {2} {32} 32 xabaxxabax xabax abx abx −−−−−−−=−−−+−−+ =−−+−+ =−−++ =−−
43. 2222 22 22 2 5(6(23))5(623) 5(23) 523 73 VVVV VV VV V −−+=−−− =−−+ =+− =−
44. 22((21)5)22(215) 22(26) 2412 212 FFFF FF FF F −+−−=−+−− =−+− =−+− =−
Copyright©2018PearsonEducation,Inc.
Section 1.7 AdditionandSubtractionofAlgebraicExpressions35
45. (3(72(56)))(3(7256)) (3(313)) (3313) (613) 613 tttttt tt tt t t −−+−−=−−+−+ =−−−+ =−+− =−− =−+
46. 2222 22 22 22 22 2 2(5(72(2)3))2(5(7243)) 2(5(72)) 2(572) 2(212) 424 324 axaxxaxaxx axax axax aa aa a −−−−−−=−−−−+− =−−−−+ =−−−+− =−− =−+ =−+
47. ( ) 4[42.5(2)1.52]4[42.5531.5] 4[6] 244 RZRRZRZRRZ RZ RZ −−−−−=−−+−+ =−− =−+
48. 3{2.11.3[2(5)]}3{2.11.3[210]} 3{2.11.3[29]} 3{2.12.611.7} 3{4.711.7} 14.135.1 efefefef eef eef ef ef −−−−−=−−−−+
49. 3()32 DDdDDdDd −−=−+=+
50. 12212212 (23)2342 iiiiiiii −−+=−++=+−
51. [] 42424442 22 33333333 6 3 3
BBBBBBBB
++−−+−−=++−−+−+
αααααααα
52. Distance30km/h(1)h40km/h(2)h 30(1)km40(2)km (30304080)km (7050)km tt tt tt t =×−+×+ =−++ =−++ =+
53. Memory(4terabytes)+(25)(8terabytes) (48200)terabytes (12200)terabytes xx xx x =+ =++ =+
Copyright©2018PearsonEducation,Inc.
$2[4070] $(80140) nn nn n n =+−− =+−+ =+ =+
54. Difference2[(21)($30)(2)($20)]
$2[60302040]
55. (a) 2222 2 (22)(3)223
xyayxbxyayxb xyab −++−−=−++−− =++− (b) 2222
(22)(3)223
xyayxbxyayxb
56. 2332323323
(3)(22)(443)322443
abccbacbabccbacb abc +−+−−−−+=+−+−−−+− =+−−
57. Thefinalshouldbeaddedandthefinal3shouldbesubtracted.Thecorrectfinalansweris222. yxy−−+
58. Thefinaloccurrenceof2shouldbeaddedratherthansubtracted,resultinginthefinalanswerof762. cabc
59. () () 1() 1() 1 abab ba ba ba ba ba −=−−+ =−− =−×− =−×− =×− =−
60. ()abcabc −−=−−
However,()abcabc −−=−+
Sincetheyarenotequivalent,subtractionisnotassociative. Forexample,(105)2523 −−=−= isnotthesameas10(52)1037 −−=−=
1.8 Multiplication of Algebraic Expressions
1. 3432333122 6122 813 2()(4)2(1)(4) 2(4) 8 sststsstst stst st −=− =− =−
2. 22 23 23 2(34)(2)(3)(2)(4) (6)(8) 68 axaxyzaxaxaxyz axaxyz axaxyz −−=−−− =−−− =−+
3. 2 2 (2)(3)()(3)(2)()(2)(3) 326 56 xxxxxx xxx xx −−=+−+−+−− =−−+ =−+
4. 2 22 22 (2)(2)(2) (2)(2)(2)()(2)()()() 422 44 ababab aaababbb aababb aabb −=−− =+−+−+−− =−−+ =−+
5. 23()()aaxax =
6. 2334(2)()2 xyxyxy =
7. 2223433 () acacxacx −=−
8. 222 222 34 (2)(4)(2)(4)(4) (2)(16) 32 cscscscscs cscs cs −−=−−− =− =−
9. 2222 24 35 (2)(2)(2)(2)(2) (4)(2) 8 axaxaxaxax axax ax −=− =− =−
10. 322322 324 37 (6)(3)(6)(3)(3) (6)(9) 54 pqpqpqpqpq pqpq pq = = =
11. 222 33 (2)()()()(2) 2 iRiriiRiiri iRir +=+ =+
12. 2()(2)()(2)() 22 xpqxpxq pxqx −−=−− =−−
13. 22 3 3(5)(3)()(3)(5) 315 sstssst sst −−=−+−− =−+
14. 22 32 3(2)(3)(2)(3)() 63 bbbbbbb bb −−=−+−− =−+
15. 22 32 5(3)(5)()(5)(3) 515 mmnmnmmnmmn mnmn +=+ =+
16. 22222 32232 (23)()(2)()(3) 23 abcacbcabcacabcbc abcabc −=+− =−
17. 2 3(2)(3)()(3)()(3)(2) 336 MMNMMMNM MMNM −−+=−+−+ =−−+
18. 222222 3322 4(92)(4)(9)(4)(2)(4)() 3684 cgccgccgcccg cgccg −−−+=−−+−−+− =+−
19. 2333 333 434 ()()() ()()()() xytxxytxyxy txyxtxyy txytxy +=+ =+ =+
20. 33 33 234 2(3)(34)6(34) (6)(3)(6)(4) 1824 stststst stsstt stst −−−=− =+− =−
21. 2 2 (3)(5)()()()(5)(3)()(3)(5) 5315 215 xxxxxx xxx xx −+=++−+− =+−− =+−
22. 2 2 (7)(1)()()()(1)(7)()(7)(1) 77 87 aaaaaa aaa aa ++=+++ =+++ =++
23. 2 2 (5)(21)()(2)()(1)(5)(2)(5)(1) 2105 295 xxxxxx xxx xx +−=+−++− =−+− =+−
24. 121211122122 22 112122 22 1122 (4)(23)(4)(2)(4)(3)()(2)()(3) 81223 8103 tttttttttttt tttttt tttt +−=+−++− =−+− =−−
25. 2 2 (8)(8)()()()(8)(8)()(8)(8) 8864 64 yyyyyy yyy y +−=+−++− =−+− =−
26. 2 2 (4)(4)()()()(4)(4)()(4)(4) 4416 16 zzzzzz zzz z −+=++−+− =+−− =−
27. 22 22 (2)(23)(2)(2)(2)(3)()(2)()(3) 4623 672 abbaabaabbba ababab aabb −−+=−++−−+− =−++− =−+
28. 22222 242 42 (34)(31)(3)(3)(3)(1)(4)(3)(4)(1) 93124 12133 wwwwww www ww −+−=−+−−++− =−++− =−+
29. 22 22 (27)(35)(2)(3)(2)(5)(7)(3)(7)(5) 6102135 61135 ststsssttstt sststt sstt +−=+−++− =−+− =+−
30. 22 22 (52)(8)(5)()(5)(8)(2)()(2)(8) 540216 53816 pqpqpppqqpqq ppqpqq ppqq −+=++−+− =+−− =+−
31. 222 32 (1)(25)()(2)()(5)(1)(2)(1)(5) 2525 xxxxxx xxx −+=++−+− =+−−
32. 222 32 (32)(29)(3)(2)(3)(9)(2)(2)(9)(2) 627418 yyyyyy yyy +−=+−++− =−+−
33. 22 22 (24)(24)
()()()(2)()(4)(2)()(2)(2)(2)(4)(4)()(4)(2)(4)(4) 242484816 4416 xyxy xxxyxyxyyyxy xxyxxyyyxy xyxy −−−+ =+−++−+−−+−+−+−−+− =−+−+−−+− =+−−
34. 22 22 (231)(231) (2)(2)(2)(3)(2)(1)(3)(2)(3)(3)(3)(1)(1)(2)(1)(3)(1)(1) 462693231 49121 abab aaabababbbab aabaabbbab abab +++− =++−+++−+++− =+−++−++− =++−
35. 2 2 2 2(1)(9)2[()()()(9)(1)()(9)(1)] 2[99] 2[89] 21618 aaaaaa aaa aa aa +−=+−++− =−+− =−− =−−
36. 2 2 2 5(3)(6)5[()()()(6)(3)()(3)(6)] 5[6318] 5[318] 51590 yyyyyy yyy yy yy −−+=−++−+− =−+−− =−+− =−−+
37. 2 2 2 3(32)(32)3[(3)(3)(3)(2)(2)(3)(2)(2)] 3[6946] 3[656] 181518 TTTTTT TTT TT TT −−+=−++−+− =−−+−+ =−−++ =−−
38. 2 2 32 2(5)(65)2[()(6)()(5)(5)(6)(5)(5)] 2[653025] 2[62525] 125050 nnnnnnnn nnnn nnn nnn −++=−+−++ =−−++ =−++ =−++
39. 2 2 32 2(1)(4)2[()(4)()()(1)(4)(1)()] 2[44] 2[34] 268 LLLLLLLL LLLL LLL LLL +−=+−++− =−+−+ =−++ =−++
40. 222 32 432 (4)(7)[()(7)()()(4)(7)(4)()] [4728] 4728 axxxaxxxxx axxxx axaxaxax +−=+−++− =−−++ =−−++
41. 2 2 2 (37)(37)(37)
(3)(3)(3)(7)(7)(3)(7)(7) 9212149 94249 xxx xxxx xxx xx −=−− =+−+−+−− =−−+ =−+
42. 2 22 22 (3)(3)(3) ()()()(3)(3)()(3)(3) 339 69 xyxyxy xxxyyxyy xxyxyy xxyy −=−− =+−+−+−− =−−+ =−+
43. 2 12121211122122 22 112122 22 1122 (3)(3)(3)()()()(3)(3)()(3)(3) 339 69 xxxxxxxxxxxxxx xxxxxx xxxx +=++=+++ =+++ =++
44. 2 2 2 (71)(71)(71) (7)(7)(7)(1)(1)(7)(1)(1) 49771 49141 mmm mmmm mmm mm −−=−−−− =−−+−−+−−+−− =+++ =++
45. 2 222 222 (2)(2)(2)
()()()(2)(2)()(2)(2) 224 44 xyzxyzxyz xyzxyzxyzxyz xyzxyzxyz xyzxyz −=−− =+−+−+−− =−−+ =−+
46. 2222 2222 4222 422 (6)(6)(6) (6)(6)(6)()()(6)()() 3666 3612 xbxbxb xxxbbxbb xbxbxb xbxb −+=−+−+ =−−+−+−+ =−−+ =−+
47. 2 2 2 2 2(8)2[(8)(8)] 2[()()()(8)(8)()(8)(8)] 2[8864] 2[1664] 232128 xxx xxxx xxx xx xx +=++ =+++ =+++ =++ =++
48. 2 2 2 2 3(34)3[(34)(34)] 3[(3)(3)(3)(4)(4)(3)(4)(4)] 3[9121216] 3[92416] 277248 RRR RRRR RRR RR RR −=−− =+−+−+−− =−−+ =−+ =−+
49. 2 2 22 322 32 (2)(3)(1)[(623)](1) (1)[6] ()()()()(6)()(1)()(1)()(1)(6) 66 256 xxxxxxx xxx xxxxxxx xxxxx xxx +−−=−+−− =−−++ =−+++−−+−+− =−+++−− =−++−
50. 23222 2242 2224 222224224 22463224 64 (3)(3)(3)(3) [(3)(3)33)](3) (3)[96] ()(9)()(6)()()(3)(9)(3)(6)(3)() 9627183 927 cxcxcxcx xxcxcxccx cxxcxc cxccxccxxxcxxc cxcxcxcxcx ccx −+=−+−+−+ =−−+−+ =−+−+ =−+−−+−+++−++ =−+−+−+ =−+− 223 27 cxx +
51. 2 2 2 32 3(2)(21)3[()(2)()(1)(2)(2)(2)(1)] 3[242] 3[242] 3[232] 696 TTTTTTTT TTTT TTTT TTT TTT +−=+−++− =−+− =−+− =+− =+−
52. 22 [(2)(2)] xx−+ 22 22 2 [(2)(2)(2)][(2)(2)(2)] [(2)[()()(2)()(2)()(2)(2)]][(2)[()()(2)()(2)()(2)(2)]] [(2)[224]][(2)[224]] [(2)[4]][(2)[4]] [()()(4)()(2) xxxxxx xxxxxxxxxx xxxxxxxx xxxx xxx =−−+−−+ =−+−++−−+−++− =−−+−−−+− =−−−− =+−+− 222 3232 333233232222 32 ()(2)(4)][()()(4)()(2)()(2)(4)] [248][248] ()()()(2)()(4)()(8)(2)()(2)(2)(2)(4)(2)(8) (4)()(4)(2)(4)(4)(4)( xxxxx xxxxxx xxxxxxxxxxxxxx xxxxxxx +−−+−+−+−− =−−+−−+ =+−+−++−+−−+−−+− +−+−−+−−+− 32 6543543243232 65432 8)(8)()(8)(2)(8)(4)(8)(8) 248248164816328163264 4432166464 xxx xxxxxxxxxxxxxxx xxxxxx ++−+−+ =−−+−++−−++−+−−+ =−−+−−+
53. (a) 222 2222 222 ()(34)749 3491625 () 4925 xy xy xyxy +=+== +=+=+= +≠+ ≠
54. Onecanwrite5(3)(3)(3)(3)(3)(3) xxxxxx +=+++++ andthenperformthemultiplicationsusingtherightmost pairoftermsateachstep.
55.
56.
57. 2 2 2 (10.01)(10.01)(10.01) [(1)(1)(1)(0.01)(0.01)(1)(0.01)(0.01)] [10.010.010.0001] 0.00010.02 PrPrr Prrrr Prrr rPrPP +=++ =+++ =+++ =++
58. 2
1000(10.0025)1000(10.0025)(10.0025)
1000[(1)(1)(1)(0.0025)(0.0025)(1)(0.0025)(0.0025)] 1000[10.00250.00250.0000625] 10002.50.0625 rrr
59.
Theroomwillbe5510feetwideand525210feetlong.Itsareais (10)(210)()(2)()(10)(10)(2)(10)(10)
60. 2 (300.01) 300.01 Rxp xx xx = =− =−
61. 22222
(2)()(2)(2)()
62. 3232332 5432 (23)(3)(2)()(2)()(2)(3)(3)()(3)()(3)(3) 226339 TTTTTTTTTT TTTTT +−−=+−+−++−+− =−−+−−
Numberofswitchesforelements
63. 2 2 2
64.
65.
66. 223 2 222 222 222 2724(6)(12)
2724(6)(6)(12)(12)(12)
27246636(12)1212144
27241236(12)24144
2724288864()()()(24)()(144)(12) xxx xxxxxx xxxxxxxx xxxxxx xxxxxxxx =−−−−−−− =−−−+−−−−+ =−−+−−−+ =−+−−+−++− 2 2322 32 ()(12)(24)(12)(144) 328886424144122881728 39144864 xx xxxxxxx xxx +−−+− =+−−+−+−+ =−+−+
1.9 Division of Algebraic Expressions
1. 222211 25523 663 22 axyax axyyy
==
2. 33223322 2222 3121 3122 2121 2 2 482482 2222 2 4 2 4 xyxyxyxyxyxy xyxyxyxy xx xy yy xx x yy −+ =−+ =−+ =−+
3. 2 2 32 21672 63 42 42 0 x xxx xx x x −−+ −+ −+
4. 232 3 2 2 32 22 21 418403 8-2 423 41 22 84322 21 4141 x xxxx xx xx x x xxx x xx −−++ −++ + −++ =−+
5. 32 831212 44 2 xy xyxy xy =−=−
6. 73 71326 2 18 1818 bc bcbc bc =−=−
7. 35514 5532 1644 4 rttt rtrr ==
8. 5523 2221 5133 17 mnnn mnmm ==
9. 23 (15)(2)3031112 33 1010 xyxzxyz xyzxz xyxy ===
10. 2334 33422 3232 (5)(8)4044 1010 sTsTsT sTT sTsT ===
11. 3232 3222 222 (4)(2)4(4)1 (4)16 axax axa axax ===
12. 2222 2224413 121244 (3)933 ababa ababbb ===
13. 2221111 36363622 33333 axxyaxxyaxxy ay xxx + =+=+=+
14. 22 2626211133 222 mnmnmnmn mnmnmnn mmm =−=−+=−+
15. 2222 36361111211122 22 333 rstrstrstrst rstrstrtt rsrsrs =−=−=−+
16. 2222 5105102111112122 555 anananan ananan ananan =−=−−=−−
17. 32253225 2222 113221221152 3 48164816 4444 24 42 pqpqpqpqpqpq pqpqpqpq pqpqpq qpq +− =+− =+− =−++
18. 223223 12111211 1111 2111211311111 1211 22 211 axxaxaxaxxaxax axaxaxax axxaxax axx +− =+− =+− =+−
19. 22 11 1121 22 2 2 fLfRfLfR fRfRfR fLfR R L R R ππππ πππ =− =− =−
20. 4444 333 444 33 41431143 3 9()69()6 333 96 33 32 32 aBaBaBaB aBaBaB aBaB aBaB aBaB aBB =− =−+ =−+ =−+
21. 223223 22222222 2222 322 2121 2 7142171421 14141414 3 22 113 22 ababaababa abababab ab ab ba a bab −+− =−+− =−+− =−+−
22. 22 2 2 2424 222 2 2 nnnn nnn nnnn xaxxax xxx xax xa ++ −+− + =+ =+ =+
23. 2121 21 6464 222 32 32 nnnn nnn nnnn n yayyay yyy yay yay ++ −−+ =− =− =−
25. 2 2 2 5 4920 4 520 520 0 920 5 4 x xxx xx x x xx x x + +++ + + + ++ =+ +
26. 2 2 2 9 2718 2 918 918 0 718 9 2 x xxx xx x x xx x x + −+− +− =+
39. 2 32 32 2 2 39 30027 3 30 39 9y27 9y27 0 yy yyyy yy yy yy −+ ++++ + −+ + + 3 27239 3 y yy y + =−+ +
40.
41. 22 2 2 2 2 0 xy xyxxyy xxy xyy xyy −−+ −+ −+ 22 2 xxyy xy xy −+ =−
42. 22 2 2 2 2 34 3352 39 42 412 14 rR rRrrRR rrR rRR rRR R + −−+ + 222 35214 34 33 rrRRR rR rRrR −+ =++
43. 232 32 2 2 2 24008 24 248 248 0 t ttttt ttt tt tt ++++− ++ 3 2 8 2 24 t t tt =− ++
45. Weknowthat21 x + multipliedby xc + willgiveus2295 xx ,so 2 295 xx dividedby21 x + willgiveus xc + : 2 2 5 21295 2
46.
55.
56.
1. (a) 312 33123 9 x x x −=− −+=−+ =− (b) 312 33123 15 x x x +=− +−=−− =− (c) 12 3 33(12) 3 36 x x x =−
17. 523 235 8 2 4 y
5818 5188
19. 37 37 27 7 2 xx
20. 6453 4356 71 1 7 LL
21. 2(34)5 685 658 8 qq qq qq q
22. 3(4) 123 312 212 12 2 6 nn nn
23. (4)62 462 22 32 2 3 rr
24. (2)55 552 53 63 31 62 xx xx xx x x
25. 8(5)2 8402 8240 1040 40 10 4 yy yy yy y y y
26. 4(7)7 2847 4728 3535 44 F F F F
27. 0.10.5(2)2 5(2)2(10) 51020 42010 105 42 xx xx xx x x
28. 1.50.3(4)6 153(4)6(10) 1531260 126012 48 12 4 xx xx xx x x x
29. 43(12)72 43672 7627 477 0 4 0 pp pp pp p p p
30. 36(23)5
31. 42(4)8
32. 25(73)2 4 4(2)35152
36. 71 71or71 1717 68 6or8 x xx xx xx xx
37. 5.80.3(6.0)0.5 0.55.80.31.8 0.50.37.6 0.87.6 7.6 0.8 9.5 xx xx xx x x
38. 1.90.5(4.0)0.8 1.92.00.50.8 1.90.51.2 2.41.2 1.2 2.4 0.50 tt tt tt
39. 0.24(0.50)0.63 0.240.120.63 0.240.630.12 0.240.51
40. 27.5(5.171.44)73.4 142.17539.673.4 39.673.4142.175 39.668.775 68.775 39.6 1.736742424 1.74 x x x x x x x
41. 17 2.06.0 17 2.0 6.0 5.6666666...
42. 3.0 7.042 3.0 42 7.0 18 R R R = = =
43. 16513 22315 151651513 132231315 2475 2899 0.85374267 0.85 V V V V V = = = = =
44. 2761360 17.046.4 1360 27617 46.4 498.2758621 276 1.805347326 1.81 x x x x x =
45. (a) 2332 2323 xx xx +=+ +=+ Isanidentity,sinceitistrueforallvaluesof x. (b) 2332 46 63 42 xx x x −=− = == Isconditionalas x hasoneansweronly.
46. Therearenovaluesof a thatresultinaconditionalequation.If0 a = ,thentheidentity22xx = results.If0 a ≠ , thenacontradictionresults.
47. 73(68) 03687 0415 3.75 xxx xxx x x
48. 0.05950.5258.85(0.0316)0 0.5950.5258.850.279660 9.3750.315340 0.033636266 0.0336 ii ii i i i
49. 0.030.06(2000)96 0.031200.0696
50. 15(5.5)24(5.5) 82.51513224 152413282.5 3949.5 49.5 39 1.269230769km/h 1.3km/h vv vv
51. 1.1(76) 40 40(1.1)76 4476 4476 120C T
52. 1.120.67(10.5)0 1.127.0350.670
53. 0.140.06(2000)0.09(2000)
54. 210(3)55.338.5(8.253)
63055.3317.625115.5
63055.3115.5317.625
690.2317.625 317.625 690.2 0.4601927m 0.460m
55. 30kWh 350mi107mi 30kWh 350mi 107mi 98kWh x x x = =×
56. 20min 250cal400cal 20min 400cal 250cal 32min x x x
1.11 Formulas and Literal Equations
3. 00 00 0000 0000 00 0 0 [1()] [1] VVbTT VVbTbT
4. 00 00 0 0 VVVT VVVT VV VT β β
EIR
pVnRT
7. 21 12 12 rLgg grLg ggrL =− += =−
8. d d d WSTQ QWST QSTW =− += =−
9. 12 12 12 nTWL B BnTWL B n TWL = = = 10. 2 2 PTf P T
11. a a a ppdgh ppdgh hpp dg =+
21. 112 21 21211 212211 221121 1121 2 2 () Kmm Km KmmKm KmKmKm KmKmKm KmKm m K + = += += =− =
23. 2 2 (2)2 22 22 22 mg a Mm aMmgm aMamgm aMgmam gmam M a = + += += =− =
24. ()VmM v m mvmVMV MVmvmV mvmV M V + = =+ =− =
26. 1
(1)AAM
27. ()NrAs NArrs rsNAr rsArN ArN s r
28. 21 21 12 12 2 1 3() 33 33 33 3 3 TTT
29. 21 21 21 12 100 100100 100100 100100 h TT TTh hTT
30. 2111 2111 21 11 (1) (1) (1) pprpp pprpp pp r pp =+− −=− =
31. 121 121 211 11 2 ()QPQQ QPQPQ PQQPQ QQPQ P =− =− =+ +
32. () 21 21 12 12 a a a a ppdgyy pp yydg pp yy dg pp yydg
33. 12 12 22 1 (1) (1) NNTNT NTNNT NNNT N T =−− =+− +− =
34. (1) acm acmm amcm mcma cma m ttht tttht thttt htttt ttt h t =+− =+− +=+ =+− +− =
35. 1212 1212 1212 212 1 ()22 22 22 22 Lrrxx Lrrxx rLrxx
36. () 21 12 12211 11221 1221 1 1 VRVR I RR IRRVRVRVR VRIRRVRVR IRRVRVR VR μ μ μ μ ++ = =++ =−+ −+
37. 121 2 121 2 121 2 1 2 1 ()VVV P gJ gJPVVV VVVgJP VVgJP V =
38. () 12 12 21 1 2 WTSSQ WQTSTS TSTSWQ STSWQ T
39. 12 12 1212 12 12 2 () ()2 () 2 eAkk C dkk CdkkeAkk Cdkk e Akk = + += + =
40. 23 23 23 3 2 3 6 63 36 6 3 LPxPx d EI dEILPxPx LPxdEIPx dEIPx L Px = =− =+ + =
41. 1 n VC N =− Cn VC N Cn VC N Cn CV N CnCNNV CNNV n C
42. () pAI PBAI pBAIAIP pBAIpAIP pBAIPAIp A BIPAIp p = +
43. () pCnnA −+= 1 1 1 1 (1) (1) 1 13.0L0.25(15.0L) 10.25 13.0L3.75L 0.75 9.25L 0.75 12.333333L 12L pCpnnA pnApC pnApC ApC n p n T T T T
44. 2 (10.500)2 tc PPm π =+ 2 2 10.500 680W
10.500(0.925) 680W
10.500(0.855625) 680W
10.4278125 680W
1.4278125 476.253009W 476W
VBLbL
2(38.6ft)(2.63ft)(16.1ft) 16.1ft
77.2ft42.343ft 16.1ft
34.857ft 16.1ft 2.16503106ft 2.16ft
(12.0V)(3.56)(3.56) 6.30V 6.7809523813.56 3.220952381 3.22
48. 1 (1)qpq
[ ] (1)1 (1)1 (1)1 1 (1) 1(0.66)(0.83) 0.66(10.83) 10.5478 0.66(0.17) 0.4522 0.1122 4.03030303 4processors qpq qpq pqq q p q p p p p p
49.
(2h)(4h) (4h)(2h) (4h)(2h)
15
1.12 Applied Word Problems
1. Let x =thenumberof25Wlights. Let31 x =thenumberof40Wlights. 2540(31)1000 251240401000 1510001240 15240 16 xx
Thereare16ofthe25Wlightsand(31–16)=15ofthe40Wlights.
Check: 251640(3116)1000 40040(15)1000 4006001000 10001000 ⋅+−=
2. Letx =thenumberofslideswith5mg. Let x 3=thenumberofslideswith6mg. (5mg)(6mg)(3) (5mg)(6mg)18mg 18 18slides xx xx x x =−
Thereare18slideswith5mgand(183)=15slideswith6mg.
Check:
5mg(18)6mg(15) 90mg90mg = =
3. Let t =thetimefortheshuttletoreachthesatellite. (29500km/h)6000km(27100km/h) (2400km/h)6000km 6000km 2400km/h 2.500h tt t
Itwilltaketheshuttle2.500htoreachthesatellite.
Check: (29500km/h)(2.500h)6000km(27100km/h)(2.500h) 73750km6000km67750km 73750km73750km =+ =+ =
4. Let x =thenumberoflitresof50%methanolblendthatmustbeadded. 0.0600(7250L)0.500()0.100(7250L) 435L0.500()725L0.100 0.400()290L 290L 0.400 725L xx xx x x x +=+ +=+ = = = 725Lofthe50%methanolblendmustbeadded.
Check:
0.0600(7250L)0.500(725L)0.100(7250L725L) 435L362.5L0.1(7975L)
797.5L797.5L
5. Let x =thecostofthecar6yearsago.
Let x +$5000=thecostofthecarmodeltoday. ($5000)$49000 2$44000 $44000 2 $22000 xx x
Thecostofthecar6yearsagowas$22000,andthecostofthetoday’smodelis($22000+5000)=$27000.
Check:
$22000($22000$5000)$49000
$22000+$27000$49000
$49000$49000
6. Let x =theflowfromthefirststreaminm3/s.
Let x –1700ft3/s=theflowfromthesecondstreaminm3/s.
Thefirststreamflows33600ft/sandthesecondstreamflows3600ft3/s–1700ft3/s=1900ft3/s.
Check:
36001.9810ft ft/s(3600ft/s1700ft/s)3600s
7200ft/s1700ft/s5500ft/s 5500ft/s5500ft/s
7. Let x =thenumberofcarsrecycledthefirstyear.
Let x+500000=thenumberofcarsrecycledthesecondyear. (500000cars)6900000cars 2500000cars6900000cars 26400000cars 6400000cars 2 3200000cars xx x x x x
Thefirstyear,3.2×106carswererecycled,andthesecondyear(3200000+500000)=3.7×106carswererecycled.
Check: 3200000cars(3200000cars500000cars)6900000cars 3200000cars3700000cars6900000cars 6900000cars6900000cars ++=
8. Let x =thenumberofhitstothewebsiteonthefirstday. Let1/2 x =thenumberofhitsonthesecondday. 1/2495000hits
330000hits xx x x x += = = =
3/2495000hits
495000hits 3/2
Thefirstdaytherewere330000hits,theseconddaytherewere1/2(33000hits=165000hits.
Check: 330000hits1/2(330000hits)495000hits
330000hits165000hits495000hits
495000hits495000hits += += =
9. Let x =thenumberacresoflandleasedfor$200peracre. Let140– x =thenumberofacresoflandleasedfor$300peracre. $200/acre$300/acre(140acre)$37000
$100/acre()$5000
$5000
$100/acre
50acres
Thereare50acresleasedat$200peracreand(140acres–50acres)=90hectaresleasedfor$300perhectare.
Check:
$200/acre(50acres)$300/acre(140acres50acres)$37000
$10000+$27000$37000 $37000$37000
10. Let x =thefirstdoseinmg. Let x +660mg=theseconddoseinmg. 660mg2000mg 21340mg 1340mg 2 670mg xx x x x
Thefirstdoseis670mg,andtheseconddoseis(670mg+660mg)=1130mg.
Check: 670mg670mg660mg2000mg 670mg+1330mg2000mg 2000mg2000mg
11. Let x =theamountofpollutantaftermodificationinppm/h. (5h)(3h)150ppm/h 450ppm 5h 90ppm/h x x x = = =
Theamountofpollutantaftermodificationis90ppm/h.Thedevicereducedemissionsby (150ppm/h–90ppm/h)=60ppm/h.
Check: (5h)90ppm/h(3h)150ppm/h 450ppm450ppm = =
12. Let x –13=thenumberofteeththatthefirstmeshedspurhas. Let x =thenumberofteeththatthesecondmeshedspurhas. Let x +15=thenumberofteeththatthethirdmeshedspurhas. 13teeth15teeth107teeth
32107teeth 3105teeth 105teeth 3 35teeth xxx
Thefirstspurhas(35–13)=22teeth,thesecondspurhas35teeth,andthethirdspurhas(35+15)=50teeth.
Check: 35teeth13teeth35teeth35teeth15teeth107teeth 107teeth107teeth −+++= =
13. Let x =amountpaidpermonthforfirstsixmonths.
Let x +10=amountpaidpermonthforfinalfourmonths. (6mo)(4mo)($10/mo)$890 (10mo)$40$890 (10mo)$850
$850 10mo $85/mo
Thebillwas$85/moforthefirstsixmonthsand$95/moforthenextfourmonths.
Check:
(6mo)$85/mo(4mo)($85/mo$10/mo)$890
$510(4mo)($95/mo)$890
$510$380$890
$890$890
14. Let x =amountpaidpermonthforfirstyear.
Let x +30=amountpaidpermonthfornexttwoyears.
Let(x +30)+20= x +50=amountpaidpermonthforfinaltwoyears. (12mo)(24mo)($30/mo)(24mo)($50/mo)$7320 (12mo)(24mo)$720(24mo)$1200$7320 (60mo)$1920$7320 (60mo)$5400
$5400
60mo
$90/mo
Forthefirstyear,thebillwas$90/mo,duringyears2and3,thebillwas$120/mo,andduringyears4and5,thebill was$140/mo.
Check:
(12mo)($90/mo)(24mo)($90/mo$30/mo)(24mo)($90/mo$50/mo)$7320
$1080(24mo)($120/mo)(24mo)($140/mo)$7320
$1080$2880$3360$7320
$7320$7320
15. Let x =thefirstcurrentinA μ
Let2x =thesecondcurrentinA μ
Let x +9.2A μ =thethirdcurrentinA μ
29.2A0A 49.2A 9.2A 4 2.3A xxx x x x μμ μ μ μ +++= =− = =−
Thefirstcurrentis2.3A μ ,thesecondcurrentis2(2.3A μ )=4.6A μ ,andthethirdcurrentis (2.3A μ +9.2A μ )=6.9A μ
Check:
2.3A2(2.3A)(-2.3)A9.2A0A 2.3A4.6A2.3A9.2A0A
16. Let x =thenumberoftrucksinthefirstfleet. Let x +5=thenumberoftrucksinthesecondfleet. (8h)(6h)(5)198h (8h)(6h)30h198h (14h)168h 168h 14h 12trucks xx xx x x x ++= ++= = = =
Thereare12trucksinthefirstfleetand(12trucks+5trucks)=17trucksinthesecondfleet.
Check: (8h)(12)(6h)(125)198h
96h(6h)(17)198h 96h102h198h 198h198h ++= += += =
17. Let x =thelengthofthefirstpipelineinkm. Let x +2.6km=thelengthofthe3otherpipelines. 3(2.6km)35.4km 37.8km35.4km 427.6km 27.6km 4 6.9km xx xx x x x ++= ++= = = =
Thefirstpipelineis6.9kmlong,andtheotherthreepipelinesareeach(6.9km+2.6km)=9.5kmlong.
Check: 6.9km3(6.9km2.6km)35.4km
6.9km3(9.5km)35.4km 6.9km28.5km35.4km 35.4km35.4km ++= += += =
18. Let x =thepowerofthefirstgeneratorinMW. Let750MW x =thepowerofthesecondgeneratorinMW.
0.650.75(750MW)530MW
0.65562.5MW0.75530MW
0.132.5MW
32.5MW 0.1 325MW xx xx x x x +−= +−= −=− = =
Thefirstgeneratorproduces325MWofpower,andthesecondgeneratorproduces(750MW–325MW)=425MW ofpower.
Check:
0.65(325MW)0.75(750MW(325MW))530MW
211.25MW0.75(425MW)530MW
211.25MW318.75MW530MW 530MW530MW +−= += += =
19. Let x =thenumberofdeluxesystems. Let2x =thenumbereconomysystems. Let x+75=thenumberofecono-plussystems.
$140$40(2)$80(75)$42000
$140$80$80$6000$42000
$300$36000 120systems xxx xxx x x +++= +++= = =
Thereare120deluxesystems,240economysystems,and195econo-plussystemssold.
Check:
$140(120)$40(240)$80(195)$42000
$16800$9600$15600$42000
$42000$42000 ++= ++= =
20. Theamountoflotterywinningsaftertaxesis$20000×(10.25)=$15000. Let x =theamountofmoneyinvestedata40%gain.
Let$15000 x =theamountofmoneyinvestedata10%loss.
0.400.10($15000)$2000
0.40$15000.10$2000 0.50$3500 $3500 0.50
$7000 xx xx x x x −−= −+= = = =
The40%gaininvestmenthad$7000invested,andthe10%lossinvestmenthad($15000$7000)=$8000invested.
Check:
0.40($7000)0.10($15000$7000)$2000
$2800$15000.10($7000)$2000
$2800$1500$700$2000 $2000$2000 −−= −+= −+= =
21. Let x =theamountoftimeinsecondsbetweenwhenthestartofthetrainspasseachothertowhentheendofthetrains passeachother.
Thetotaldistancetheendsmusttravelinthistimeis960feet.Wefirstconvertmi/hrintoft/sec.
5280ft22ft22 1mi/hrft/s
3600s15s15 ===
Therefore,trainAtravelsat60(22/15)=88ft/sandtrainBtravelsat40(22/15)=176/3ft/s. (88ft/s)(176/3ft/s)960ft
(440/3ft/s)960ft
960ft
440/3ft/s 72 s 11 xx x x x += = = =
Thetrainscompletelypasseachotherinabout6.55seconds.
Check:
7272
(88ft/s)s(176/3ft/s)s960ft 1111
576ft+384ft960ft
960ft960ft += = =
22. Let x =themortgagepaymentand x/0.23=themonthlyincome.
/0.23$3850 11$3850 0.23
10.23$3850 0.230.23 0.77$3850
Themortgagepaymentis$1150andthemonthlyincomeis$5000.
Check:
$1150/0.23$1150$3850
$5000$1150$3850
$3850$3850
23. Let x =theamountoftimetheskierspendsontheskiliftinminutes. Let24minutes x =theamountoftimetheskierspendsskiingdownthehillinminutes. (50m/min)(150m/min)(24min) (50m/min)3600m(150m/min) (200m/min)3600m 3600m 200m/min 18min xx xx x x x
Thelengthoftheslopeis18minutes×50m/minute=900m.
Check:
(50m/min)18min(150m/min)(24min18min)
900m3600m(150m/min)(18min)
900m3600m2700m
900m900m
24. Let x =thespeedofsound.
Let x –120mi/h=speedtravelledfor1h. Let x +410mi/h=thespeedtravelledfor3h. 1h(120mi/h)3h(410mi/h)=3990mi (1h)(1h)(120mi/h)(3h)(3h)(410mi/h)=3990mi
(1h)120mi(3h)1230mi=3990mi (4h)=2880mi 2880mi 4h 720mi/h
Thespeedofsoundis720mi/h.
Check:
1h(720mi/h120mi/h)3h(720mi/h410mi/h)=3990mi (1h)(600mi/h)(3h)(1130mi/h)=3990mi
600mi3390mi=3990mi
3990mi=3990mi
25. Let x =thespeedthetrainleavingEnglandinkm/h.
Let x +8km/h=speedofthetrainleavingFranceinkm/h. Thedistancetravelledbyeachtrainisspeed×time. 17min17min 60min/h(8km/h)=50km 60min/h
(0.28333h)(8km/h)(0.28333h)=50km (0.28333h)(0.28333h)2.26667km=50km
(0.56666h)=47.73333km
47.73333km
0.56666h 84.2352942
84.2km/h
ThetrainleavingEnglandwastravellingat84.2km/h,andthetrainleavingFrancewastravellingat (84.2km/h+8km/h)=92.2km/h.
Check: 84.2352942117min17minkm/h(84.23529421km/h8km/h)=50km 60min/h60min/h
23.8666617minkm(92.23529421km/h)=50km 60min/h
23.86666km26.13333km=50km 50km50km
26. Let x =timeleftuntiltheappointment.
Let x–10.0min=timetakentogettotheappointmenttravellingat60.0mi/h. Let x–5.0min=timetakentogettotheappointmenttravellingat45.0mi/h. Thedistancetravelledbytheexecutiveineachscenarioisthesame.Distance=speed×time 10.0min5.0min 60.0mi/h45mi/h 60min/h60min/h
(60.010.0min5.0min mi/h)60mi/h(45mi/h)45mi/h60min/h60min/h
(60.0mi/h)10mi(45.0mi/h)3.75mi (15.0mi/h)6.
15.0mi/h
0.416666667h
0.416666667hx60min/h 25min x x x x
Thereis25minutesleftuntiltheexecutive’sappointment. Check:
60.010.0min5.0min mi/h0.41667h45mi/h0.41667h 60min/h60min/h
60.0mi/h(0.25h)45mi/h(0.33333h) 15mi=15mi
27. Let x –30.0s=timesincethefirstcarstartedmovingintheraceinseconds. Let x=timesincethesecondcarstartedtheraceinseconds.
Thedistancetravelledbyeachcarwillbethesameatthepointwherethefirstcarovertakesthesecondcar. Distance=speed×time.
260.0ft/s(30.0s)240.0ft/s()
(260.0ft/s)(260.0ft/s)(30.0s)(240.0ft/s)
(260.0ft/s)7800ft(240.0ft/s)
(20.0ft/s)7800ft
7800ft
20.0ft/s 390s
Thefirstcarwillovertakethesecondcarafter390s.Thefirstcartravels260ft/s×(390s–30s)=93600ftbythis point.8lapsaroundthetrackis2.5mi/lap.8laps×5280ft/mi=105,600ft,sothefirstcarwillalreadybeintheleadat theendofthe8thlap.
Check:
260.0ft/s(390.0s30.0s)240.0ft/s(390s) 260.0ft/s(360.0s)240.0ft/s(390s) 93,600ft93,600ft
28. Let x =thenumberofthefirstchipsthatisdefective0.50%.
Let6100 x =thenumberofthesecondchipsthatisdefective0.80%. 0.0050()0.0080(6100chips)38chips (0.0050)48.8chips(0.0080)38chips (0.0030)10.8chips 10.8chips 0.0030 3600chips
Thereare3600chipsthatare0.50%defectiveand(6100chips–3600chips)=2500chipsthataredefective0.80%.
Check:
0.0050(3600chips)0.0080(6100chips3600chips)38chips
18chips0.0080(2500chips)38chips
18chips20chips38chips
38chips=38chips
Assumingthatthecustomerislocatedbetweenthetwogasolinedistributors: Let x =thedistanceinkmtothefirstgasolinedistributorthatcosts$2.90/gal. Let228mi x=thedistanceinkmtothesecondgasolinedistributorthatcosts$2.70/gal.
$2.90$0.002()$2.70$0.002(228)
$2.90$0.002()$2.70$0.456$0.002()
$0.004()$0.256
$0.256
$0.004
Thecustomeris64miawayfromthefirstgasdistributor($2.90/gal)and(228mi–64mi)=164miawayfromthe secondgasdistributor($2.70).
Check:
$2.90$0.002(64)$2.70$0.002(22864)
$2.90$0.128$2.70$0.002(164)
$3.028$2.70$0.328
$3.028$3.028
30. ?L75%Gas
(100%Gas)8.0Lgascan(needstobefullof93.75%gas/oilmixture)
A15:1gas/oilmixtureis15/16gasoline=93.75%.
Let x =theamountof100%gasolineaddedinL.
Let8.0L– x =theamountof75%gasolinemixtureinL.
1.00()0.75(8.0L)0.9375(8.0L)
1.00()6.0L0.75()7.5L
6.0Lof100%gasolinemustbeaddedtothe75%gas/oilmixturetomake8Lof15:1gasoline/oil.
Check:
1.00(6.0L)0.75(8.0L6.0L)0.9375(8.0L)
6L0.75(2.0L)7.5L
6L1.5L7.5L
7.5L7.5L
100%Antifreeze12.0Lradiator(needstobefilledwith50%mixture)
Let x =theamountinLof25%antifreezeleftinradiator
Let12.0L– x =theamountof100%antifreezeaddedinL.
0.25()1.00(12.0L)0.5(12.0L)
0.25()12.0L1.00()6.0L
0.75()6.0L 6.0L 0.75 8.0L xx xx x x x +−= +−= −=− = =
Thereneedstobe8Lof25%antifreezeleftinradiator,so(12.0L–8.0L)=4.0Lmustbedrained.
Check:
0.25(8.0L)1.00(12.0L8.0L)0.5(12.0L)
2.0L1.00(4.0L)6.0L 2.0L4.0L6.0L 6.0L6.0L +−= += += =
32. (x)lb Sand 250lbCement (22%SandMixture)
Let x =theamountofsandadded. Let250lb+ x =theamountinlbofthefinal25%sandmixture.
Copyright©2018PearsonEducation,Inc.
(x)Lof25%antifreeze
1.00()0.22(250lb)0.25(250lb) 1.00()55lb62.5lb0.25() 0.75()7.5lb 7.5lb 0.75 10lb xx xx x x x +=+ +=+ =
Check:
1.00(10lb)0.22(250lb)0.25(250lb10lb) 10lb55lb62.5lb2.5lb 65lb65lb +=+
33. (x)km/hm 70km/h
5.0m20.0m
Let x =thespeedthecarneedstotravelinkm/htopassthesemiin10s. Speed=distance/time.10sis10s/3600s/h=0.002777777h. () distanceneededtopasstruck+distancetravelledbytruckin10s 10
79km/h x s x x x x = + = + = = =
0.025km70km/h0.0027777h 0.0027777h
0.025km0.19444km 0.0027777h 2.19444km 0.0027777h
Thecarneedstotravelataspeedof79km/htopassthesemitrailerin10s.
Check:
() 0.025km70km/h0.0027777h 79km/h 0.0027777h
790.025km0.19444km km/h 0.0027777h 79km/h79km/h +
34. 5km/s8km/s
Seismic Station (?)km
Let x =thetimethefirstwavetakestotraveltotheseismicstationins. Let x +120s=thetimethefirstwavetakestotraveltotheseismicstationins. Distance=speed×time.Thedistancestravelledbybothwavestotheseismicstationarethesame.2.0minis (2.0min×60s/min)=120s.
8.0km/s()=5.0km/s(120s)
8.0km/s()=5.0km/s()(5km/s)(120s)
3.0km/s()=600km
600km
3.0km/s
200s xx xx x x x + + = =
Thedistancetotheseismicstationis(200s×8.0km/s)=1600km.
Check:
8.0km/s(200s)=5.0km/s(200s120s)
1600km=5.0km/s(320s)
1600km1600km + =
Review Exercises
1. False,because00 = whichisnotapositivevalue.
2. True.Theorderofoperationsdictatesperformingthedivisionfirst,thenthesubtraction.
3. False.Thereportedanswershouldhaveonlytwosignificantdigits.
4. False.Hadtheproblembeengivenas33 (2)8aa = ,thenitwouldbetrue.
5. True.
6. False.Theleft-handside,4,isnotarealnumber,infact.
7. False.Theleft-handsidesimplifiesto4(23)42323 xxxxx −+=−−=−
8. True.
9. False.Theleft-handsidesimplifiesto626231 222 xx x + =+=+
10. True.
11. False.Solvingfor c yields abcd bcda da c b da c b −= −=− = =−
12. False.Itislikelythatoneshouldsetupaphrasesuchas‘let x bethenumberofgearsofthefirsttype…’
13. (2)(5)325310 −+−−=−−−=−
14. 68(4)6842 −−−=−+=
15. (5)(6)(4)(20)(6) (2)(3) = (6) 20=−
16. (9)(12)(4)108(4)43218 242424 ===−
17. 15 52(6)512(5)512522 3 −−−+=−−−+−=−−−=−
18. 12 3532355(3)35(5)362519 4 −−−−=−−−−=−+=−=−
19. 2 1818 (4)(4)(4)91625352 −−=−−−=−−=−
20. 288827431 (3)(3)(3)9(2)4(2)46333 −−−=−−−−=−−=−−=−
21. 1664(4)(4)(8)(8)484 −=−=−=−
22. 81144225(5)(5)(3)(3)(3)(5)15 −+=−=−=−=−
23. 233 (7)8(7)(7)(2)(2)(2)725 −=−=−=
24. 424 16(6)(2)(2)(2)(2)(6)(6)264 −+=−+=−+=
25. 22222224(2)(2)4 x rtrtrt−=−=
26. 02333236 6 27 (3)(3)(1)27(1) abbb b ×− ===
27. 534135421 2 24 3(8)(3)(8)24 t mntmnmntmnt mn −−+−− −= −= −=
28. 4241 5 153 5 pqrpr pqr = 52 qr 3 3 3 p q =
29. 222121133 01 16()888 2(1)(1) NNTNTNTT NTN + +− ===
30. 12111232 111 35()77 5 xyxyyxyx xyx −+ + −+ == 2 x 73 y =−
31. 45(5)(3)(3)35 ==
32. 93645(5)(3)(3)35 +===
33. 8000has1significantdigit.Roundedto2significantdigits,itis8000.
34. 21490has4significantdigits.Roundedto2significantdigits,itis21000.
35. 9.050has4significantdigits.Roundedto2significantdigits,itis9.0.
36. 0.7000has4significantdigits.Roundedto2significantdigits,itis0.70.
37. 2 37.316.92(1.067)37.316.92(1.138489) 37.319.26323388 18.03676612 −=− =− = whichroundsto18.0.
38. 1212 13 8.896108.89610 3.59546.04499.6403 9.22792859110 ×× = =−× whichroundsto-9.228×10-13
39. 2 0.19582.8443.0398 3.142(65)3.142(4225) 1.743502223 13274.95 0.000131337 + = = = whichroundsto1.3×10-4
40. 2 13746637466 28.02690583 0.03568877.9369 29.63 28.0269058342.67504874 70.70195457 +=+ =+ = whichroundsto70.70,assumingthatthe1isexact.
41. 778.2ftlb1.356J 875Btu875Btu 1Btu1ftlb 923,334.3J =×× = whichroundsto923,000J.
42. 2.54cm1m 18.4in18.4in 1in100cm 0.46736m =×× = whichroundsto0.467m.
43. kmkm1hr0.6214mi5280ft 6565 hh3600s1km1mi ft 59.2401333 s =××× = whichroundsto59ft/s.
44. 3 3 12.25gg28.32L1kg2.205lb 12.25 LL1000g1kg 1ft lb 0.7649586 ft =××× = whichroundsto0.7650lb/ft3
45. 550ftlb/s1.356J60s 225hp225hp1hp1ftlb1min J 10068300 min ⋅ =××× = whichroundsto7 101000001.0110 =× J/min.
46. 2 22 2 lblb4.448N1in 89.789.7 inin1lb2.54cm N 61.8428917 cm =×× = whichroundsto61.8N/cm2
47. 322 aabaababa −−+=−−
48. 5436 xyyyxyxyy −−−=−−
49. 6(3)6373 LCLCLCLCLC −−=−+=−
50. (2)3(5)231516 xbxbxbxbbx −−−−−=−+++=+
51. 2 2 (21)(5)(2)(5)(2)()(1)(5)(1)() 1025 295 xxxxxx xxx xx −+=++−+− =+−− =+−
52. 22 22 (4)(2)()()()(2)(4)()(4)(2) 248 294 CDDCCDCCDDDC CDCDCD CCDD −−=+−+−+−− =−−+ =−+−
53. 2 2 2 (8)(8)(8) ()()()(8)(8)()(8)(8) 8864 1664 xxx xxxx xxx xx +=++ =+++ =+++ =++
54. 2 22 22 (29)(29)(29) (2)(2)(2)(9)(9)(2)(9)(9) 4181881 43681 rsrsrs rrrssrss rrsrss rrss −=−− =+−+−+−− =−−+ =−+
55. 32453245 222 32214251 24 2626 222 3 3 hkhkhkhk hkhkhk hkhk hkhk =− =− =−+
56. 234234 222 2132 4848 222 4 2 axaxaxax axaxax a ax =− =−+ 42 x a 2 42xax=−
57. [ ] [ ] [] 42(34)4234 463 463 76 RrRrRrRr RrR RrR Rr −−−=−−+ =−− =−+ =−
58. [ ] [ ] [] 33(3)44333 4323 4323 26 baabaabaab abab abab ab −−−−+=−−−+ =−−+ =−−− =−
59. [ ] [ ] [] 2{35(76)}2{3576)} 2{3117} 2{3117} 2{1011} 21011 1310 xyzxyzxyxyzxyzxy xyzxyz xyzxyz xyzxy xyzxy xyz −−−−=−−−+ =−−− =−−+ =−− =−+ =−
60. [ ] [ ] [] 22 2 2 2 3()3(2)3633) 3523 3523 223 xbbybyzxbbybyz xbbyz xbbyz xbyz ++−−−+=++−−+− =++−+− =+−+− =−+−
61. 222 322 32 (21)(3)(2)()(2)()(2)(3)(1)()(1)()(1)(3) 2263 273 xxxxxxxxxx xxxxx xxx +−−=+−+−++−+− =−−+−− =−−−
62. 222 322 32 (3)(213)()(2)()(1)()(3)(3)(2)(3)(1)(3)(3) 23639 29103 xxxxxxxxxx xxxxx xxx −+−=++−+−+−+−− =+−−−+ =−+−
63. 2 22 22 223 3(4)3(4)(4) 3[()()()(4)(4)()(4)(4)] 3[4416] 3[816] 32448 yxyyxyxy yxxxyyxyy yxxyxyy yxxyy xyxyy −−=−−− =−+−+−+−− =−−−+ =−−+ =−+−
64. 2 22 22 322 (43)(43)(43) [(4)(4)(4)(3)(3)(4)(3)(3)] [1612129] [16249] 16249 sstsstst ssssttstt ssststt ssstt sstst −−=−−− =−+−+−+−− =−−−+ =−−+ =−+−
65. 22 3[()2(13)]3[26] 3[36] 1893 pqppqpqpppq pqppq pqppq −−−=−−+ =−+ =−+
66. 2 3[24(2)]3[248] 3[274] 21126 xyrxrxyrxr xyrx rxxxy −−−=−−+ =+− =−+
67. 32453245 4444 214 43 12461246 2222 62 pqpqpqpqpqpq pqpqpqpq qpq
68. 32423242 2222 42 3221 2718927189 9999 2 3 stststststst stststst st st −+ =−+ =−+ t 92st 92st 2 231 sst =−−
69. 2 2 25 62730 212 530 530 0 x xxx xx x x ++− +
70.
74. 2 32 32 2 2 462 2380143 812 1214 1218 43 46 3 xx xxxx xx xx xx x x −+ ++−+ + + + 3 814323 462 2323 xx xx xx −+ =−+− ++
75. 3{()2[(32)(2)]}3{2[322)]}
3{2[3]} 3{62]}
3{5} 1533 rstrstsrstrsts rstrt rstrt rst rst −+−−−−−=−+−−−−+ =−+−−− =−+−−+ =−−++ =−−
76. 22 22 22 2 (12)(3)(4)(43) [(1)()(1)(3)(2)()(2)(3)][()(4)()(3)(4)(4)(4)(3) [326][431612] [273][3816] 2733816 1519 xxxx xxxxxxxx xxxxxx xxxx xxxx xx −−−+− =+−+−+−−−+−++− =−−+−+−++− =−+−−−−+ =−+−++− =+−
77. 2 32 32 2 2 51 212975 21 107 105 25 21 4 yy yyyy yy yy yy y y +− −+−+ −+ −+ 32 297524 51 2121 yyy yy yy +−+ =+−+
78. 22 2 2 2 34 2654 63 84 84 0 xy xyxxyy xxy xyy xyy + −+−
79. 318 29 9 2 xx x x +=−
80. 4357 10 10 yy y y −=+
81. 53 72
2(5)3(7) 1021 21 10 x x x x = = = =
82. 2(4)5 34 285 34 4(28)3(5) 83215 847 47 8 N N N N N
83. 653(4) 65312 37 7 3 xx xx x
84. 2(4)3 823 8 yy yy y
85. 24(3)6 21246 26 6 2 3 ss ss s s s +−=
87. 32(7)5(21) 3142105 514105 519 19 5 ttt
88. (8)2(2) 842 834 24 4 2 2 xxx
89. 2.72.0(2.13.4)0.1 2.74.26.80.1 4.24.10.1 4.24.2 4.2 4.2 1.0 x x x x x
90. 0.250(6.7212.44)2.08 1.680250.6102.08 0.6100.39975 0.39975 0.610 0.655327868 0.655
91. 60,000,000,000,000bytes=6×1013bytes
92. 25,000mi/h=2.5×104mi/h
93. 15,400,000,000km=1.54×1010km
94. 1.02×109Hz=1,020,000,000Hz
95. 2.53×1013mi=25,300,000,000,000mi
96. 107ft2=10,000,000ft2
97. 10-12W/m2=0.000000000001W/m2
98. 0.00000015m=1.5×10-7 m
99. 1.5×10-1Bq/L=0.15Bq/L
100. 0.00000018m=1.8×10-7 m
108. () mumMv mumvMv mumvMv mumv M v =+ =+
109. 1233 1323 2133 133 2 () NTNNN NNNTNT NTNNNT NNNT N T =−+
110. 21 21 21 12 12 () () kAtTT QL QLkAtTT
111. 21 21 21 1 2 ()ATT R H HRATAT ATHRAT HRAT T A
+
112. 2 2 2 2 2 22 2 2 1 2 2 2 2() 22 Zk a Z Zk a Z Zk a aZkZ aZak Z λ λ λ
113. 2 2 223 223 23 2 [3()] [33] 33 33 3 3 dkxabx dkxabx dakxbkxkx akxdbkxkx dbkxkx a kx =+−
114. 021
115. 13 8 4 5.2510bytes8.20312510 6.410bytes × =× × whichroundsto8.2×108.Thenewercomputer’smemoryis8.2×108larger.
116. 0.25662.0310096s t == whichroundsto2.0s.Itwouldtaketheperson2.0stofall66ft.
117. 0.553km 1.25113122 0.442km = whichroundsto1.25.TheCNToweris1.25timestallerthantheSearstower.
118. 32 4.8102cells(1.8113207)3.280882876s2650 t
whichroundsto3.28s.Itwouldtakethecomputer3.28stocheck4800memorycells.
119. 12 12 2 (0.0275)(0.0590) 0.02750.0590 0.0016225 0.0865 0.018757225 RR RR ΩΩ = +Ω+Ω Ω = Ω =Ω whichroundsto0.0188 Ω .Thecombinedelectricresistanceis0.0188 Ω
120.
1.5101.5105.9810kg 1.9910kg
1111
1.5100.000003005 1.510(0.0017335) 260025124.4m m M × ×=× × =× =× = whichroundsto2.6×108m.Thedistancethespacecraftwillbefromtheearthis2.6×108 m.
121. (2)3ft/yd(2)233(2) 236 44 xaxaxaxa xaxa
Thesumoftheirlengthis4x +4a ft.
122. 2 2 22 322 ()(1)()(1)(1) ()[(1)(1)(1)()()(1)()()] ()[21] ()()()(2)()(1)()()()(2)()(1) 22 AiRiAiRii
123. 2 22 22 22 4()2()442()() 442[()()()()()()()()] 442[2] 44242 22444 ththththth thttthhthh ththth ththth thhtth +−+=+−++
124. 22222222 2222 2 22 krhkhrvkrhkhrv krkrkrkr kr −+ =−+
125. 318(96)54(3)18 318965496660 ×÷−=÷= ×÷−=÷−=−= Yes,theremovaloftheparenthesesdoesaffecttheanswer.
126. (318)965496660 318965496660 ×÷−=÷−=−= ×÷−=÷−=−= No,theremovaloftheparenthesesdoesnotaffecttheanswer.
127. (3)23 323 2323 xxx xxx xx −−=− −+=− −=−
Theequationisvalidforallvaluesoftheunknown,sotheequationisanidentity.
128. 7(2)2 722 52 52 xx xx xx −−=+ −+=+ +=+ =
Theequationhasnovaluesoftheunknownforwhichitisvalid,sotheequationisacontradiction.
129. (a) 222448 4 −=− =− (b) 22(4)26 12 −−= =
130. For a<0, |a|=–a.
131. Given30, 372 (3)72 372 4 x xx xx xx x −= −+= −−+= −++= =
Thisisconsistentwith30,so4. xx −==
132. 463 436 436or(4)36 22436 1104 andsotheonlypossiblesolutionsare 1or5/2. xx xx xxxx xxx xx xx
Thefirstpossibility,1 x = ,yields363or93, −+== whichisfalse. Thesecondpossibility,5/2 x = ,yields3/2615/2or15/215/2, −+== whichistrue,and sotheonlysolutionis5/2 x =
133. ()()()() () () () () () () ()()() () 3 3 xyxyxyxy yxyxyx yxyxyx yx −=−−−
134. Generally,()() Wedemonstratethisusing8,4,2: (84)2221 8(42)824 abcabc abc ÷÷≠÷÷ === ÷÷=÷=
Divisionisnotassociative.
135. 3 7 4 810 410 210 × =× ×
136. (2)(2)(10 436210 10 422 + ===
137. 746.0W1kW 250hp250hp1hp1000W 186.5kW =×× = Thisisroundedto190kW.
138. 22 22 2 lblb4.448N1in100cm 3232 inin1lb2.54cm1m N 220,621.241 m =××× = Thisisroundedto220,000N/m2
139. 1lb1ft 110Nm110Nm 4.448N0.3048m 81.1358787ftlb ⋅=⋅×× =⋅ Thisisroundedto81footpounds.
140. 2 66 22 5 2 AA1000mA1m 1.2101.210 mm1A100cm mA 1.210 cm ×=××× =×
141. Let x= thecostofthefirstcomputerprogram.
Let x +$72=thecostofthesecondcomputerprogram.
($72)$190
2$72$190
2$118 $118 2 $59 xx x x x x ++= += = = =
Thecostofthefirstcomputerprogramis$59,andtheotherprogramcosts($59+$72)=$131.
Check:$59+$131=190
142. Let x= thecosttorunthecommercialonthefirststation.
Let x +$1100=thecosttorunthecommercialonthesecondstation.
($1100)$9500
2$1100$9500
2$8400 $8400 2 $4200 xx x x x x ++= += = = =
Thecostoftherunthecommercialonthefirststationis$4200,andthecostfortheotherstationis ($4200+$1100)=$5300.
Check:$4200+$5300=$9500
143. Let2x =theamountofoxygenproducedincm3bythefirstreaction. Let x =theamountofoxygenproducedincm3bythesecondreaction. Let4x =theamountofoxygenproducedincm3bythethirdreaction. 3 3 3 3 24560cm 7560cm 560cm 7 80cm xxx x x x ++= = = =
Thefirstreactionproduces(2×80cm3)=160cm3ofoxygen,thesecondreactionproduces80cm3ofoxygen,andthe thirdreactionproduces(4×80cm3)=320cm3ofoxygen.
Check:160cm3+80cm3+320cm3=560cm3*
144. Let x =thespeedthattheriverisflowinginmi/h. Let x +5.5mi/h=thespeedthattheboattravelsdownstream. Let x +5.5mi/h=thespeedthattheboattravelsupstream. Thedistancethattheboattravelledisthesameinbothexperiments.Distance=speed×time. (5.5mi/h)(5.0h)(5.5mi/h)(8.0h) (5.0h)()(5.5mi/h)(5.0h)(8.0h)()(5.5mi/h)(8.0h) (5.0h)()(27.5mi)(8.0h)()(44mi) (13.0h)16.5mi 16.5mi 13h
1.269230769mi/h
whichroundsto1.3mi/h.Thepollutedstreamisflowingat1.3mi/h. Check: (1.269230769mi/h5.5mi/h)(5.0h)(1.269230769mi/h5.5mi/h)(8.0h) (6.769230769mi/h)(5.0h)(4.2mi/h)(8.0h) (33.8mi)(33.8mi)
145. Let x =theresistanceinthefirstresistorin Ω
Let x +1200 Ω =theresistanceinthesecondresistorin Ω
Voltage=current×resistance.2.4 Aμ =2.3×10-6A.12mV=0.0120V 66 666 6 6 6 (2.410A)()(2.410A)(1200)0.0120V (2.410A)()(2.410A)()(2.410A)(1200)0.0120V (4.810A)()(0.00288V)0.0120V (4.010A)()0.00912V 0.00912V 4.810A 1900 xx xx x x x x ×+×+Ω= ×+×+×Ω=
Thefirstresistor’sresistanceis1900 Ω andthesecondresistor’sis(1900 Ω +1200 Ω )=3100 Ω
Check: 66 (2.410A)(1900)(2.410A)(19001200)0.0120V 0.00456V0.00744V0.0120V 0.0120V0.0120V ×Ω+×Ω +Ω= += =
146. Let x =theconcentrationofthefirstpollutantinppm. Let4x =theconcentrationofthesecondpollutantinppm. 44.0ppm 54.0ppm 4.0ppm 5 0.8ppm xx x x x += = = =
Theconcentrationofthefirstpollutantis0.8ppm,andtheconcentrationofthesecondis(4×0.8ppm)=3.2ppm.
Check: 0.8ppm4(0.8ppm)4.0ppm 0.8ppm3.2ppm4.0ppm 4.0ppm4.0ppm += += =
147. Let x =thetimetakeninhoursforthecrewtobuild250mofroad. Thecrewworksatarateof450m/12h,whichis37.5m/h.Time=distance/speed. 250m
37.5m/h 6.666666667h x x = = whichroundsto6.7h.
148. Let x =theamountofoilinLinthemixture. Let15x =theamountofgasinLinthemixture. 156.6L
166.6L 6.6L 16 0.4125L xx x x x += = = = whichroundsto0.41L.Thereis0.41Lofoilinthemixtureand(15×0.41L)=6.2Lofgas. Check:
149.
0.4125L15(0.4125L)6.6L
0.4125L6.1875L6.6L 6.6L6.6L += += =
634km
Let x =thetimetakenbythesecondshipinhours. Let x +2h=theamounttimetakenbythefirstshipinhours. Thedistancetravelledaddsupto634km.Distance=speed×time.
21.8km/h()17.4km/h(2h)634km
21.8km/h()17.4km/h()17.4km/h(2h)634km
39.2km/h()34.8km634km
39.2km/h()599.2km
599.2km
39.2km/h 15.2857h
whichroundsto15.2h.Theshipswillpass15.2hafterthesecondshipentersthecanal. Check:
21.8km/h(15.2857h)17.4km/h(15.2857h2h)634km
333.23km300.77km634km 634km634km
150. Let x =thetimetakeinhforthehelicoptertotravelfromthepondtothefire. Let0.5h x =thetimetakeinhforthehelicoptertotravelfromthefiretothepond. 30min/60min/h=0.5h.Thedistancetravelledbythehelicopteristhesameforbothtrips.Distance=speed×time.
105mi/h(0.5h)70mi/h()
52.5mi105mi/h()70mi/h()
52.5mi175mi/h()
52.5mi
175mi/h 0.3h
whichisreportedas0.30htotwosignificantdigits.Itwilltakethehelicopter0.30htoflyfromthepondtothefire.
Check:
105mi/h(0.5h0.3h)70mi/h(0.3h)
105mi/h(0.2h)70mi/h(0.3h)
21mi21mi
151. Let x =thenumberoflitresof0.50%gradeoilused. Let1000L x thenumberoflitresof0.75%gradeoilused. 0.005()0.0075(1000L)0.0065(1000L)
0.005()7.5L0.0075()6.5L 0.0025()1.0L
Itwilltake400Lofthe0.50%gradeoiland(1000L–400L)=600Lofthe0.75%gradeoiltomake1000Lof0.65% gradeoil.
Check:
0.005(400L)0.0075(1000L400L)0.0065(1000L)
2L4.5L6.5L 6.5L6.5L
152. Let x =theamountofrockcontaining72L/Mgofoil.
Let18000– x =theremainingamountofrockcontaining150L/Mgofoil. (72L/Mg)()(150L/Mg)(18000Mg)(120L/Mg)(18000Mg) 72L/Mg()2700000L150L/Mg()2160000L 78L/Mg()540000L 540000L 78L/Mg 6923.07692Mg
whichroundsto6900Mg.Itwilltake6900Mgof72L/Mgrockand11100Mgof150L/Mgrocktomakethe 18000Mgof120L/Mgrock.
Check:
(72L/Mg)(6923.07692Mg)(150L/Mg)(18000Mg6923.07692Mg)(120L/Mg)(18000Mg) 498461.538L2700000L1038461.538L2160000L 2160000L2160000L
153. Let x =theareaofspaceinft2inthekitchenandbath.
ftoftileinthehouse 0.25 ftinthehouse
0.15(2200ft)0.25 (2200ft)
330ft0.25()(0.25)(2200ft)
330ft0.25()550ft
0.75220ft
whichroundsto290ft2.Thekitchenandbathareais290ft2
Check:
623.3333333ft
154. Let x =thenumberofgramsof9-karatgold.
Let200g– x =thenumberofgramsof18-karatgold.9-karatgoldis9/24gold=0.375,18-karatgoldis 18/24gold=0.75,and14-karatgoldis14/24gold=0.583333333. 0.375()0.75(200g)0.583333333(200g)
0.375()150g0.75()116.6666666g
0.375()33.3333334g 33.3333334g 0.375 88.88888907g
whichroundsto89g.Thereis89gof9-karatgoldand(200g–89g)=111gof18-karatgoldneededtomake 200gof14-karatgold.
Check:
0.375(88.88888907g)0.75(200g88.88888907g)0.583333333(200g) 33.3333334g83.3333332g116.6666666g 116.6666666g116.6666666g
155. 00
$6250(4.000years)
$1375 25000 0.055
Therateisequalto5.500%.
Onthecalculatortype: (76256250)/(62504.000) −×
