Chapter P Fundamental Concepts of Algebra
Section P.1
Check Point Exercises
1. 22 86(3)86(133) x +−=+− 2 86(10) 86(100) 8600 608 =+ =+ =+ =
2. Since2010is10yearsafter2000,substitute10for x 2 2 172613257 17(10)261(10)3257 7567 Txx=++ =++ = Iftrendscontinue,thetuitionandfeeswillbe$7567
3. Theelementscommonto{3,4,5,6,7}and {3,7,8,9}are3and7.
{3,4,5,6,7}{3,7,8,9}{3,7} ∩=
4. Theunionisthesetcontainingalltheelementsof eitherset.
{3,4,5,6,7}{3,7,8,9}{3,4,5,6,7,8,9} ∪=
5. 9,1.3,0,0.3,,9,10 2 π ⎧⎫ ⎨⎬ ⎩⎭
a. Naturalnumbers:9because93 =
b. Wholenumbers:0,9
c. Integers:9,0,9
d. Rationalnumbers:9,1.3,0,0.3,9
e. Irrationalnumbers:,10 2 π
f. Realnumbers: 9,1.3,0,0.3,,9,10 2 π ⎧⎫ ⎨⎬ ⎩⎭
6. a. 12
Because21.4, ≈ thenumberinsidethe absolutevaluebarsisnegative.Theabsolute valueof x when x <0is–x. Thus, () 121221 −=−−=−
b. 3 π
Because3.14, π ≈ thenumberinsidethe absolutevaluebarsispositive.Theabsolute valueofapositivenumberisthenumberitself. Thus,
33.ππ−=−
c. x x
Because0,. xxx >= Thus,1 x x xx ==
7. 4(5)99 −−=−= Thedistancebetween–4and5is9.
8. 22 7(43)2(5) xxxx +++ 22 22 2 7(43)2(5) 2821102 3823 xxxx xxxx xx =+++ =+++ =+
9. 64[7(2)] x +−− 64[72)] 64[9] 6364 424 x x x x =+−+ =+− =+− =−
Exercise Set P.1
1. 75(10)75057 +=+=
2. () 86583038 +=+=
3. 6(3)818810 −=−=
4. () 83424420 −=−=
5. 2 83(8)642488 +=+=
6. () 2 656363066 +=+=
7. 2 76(7)3494237310 −+=−+=+=
8. () 2 8784645648412 −+=−+=+=
9. 33 45(97)45(2) 45(8)44044 +−=+ =+=+=
10. ()() () 336586652 658 64046 +−=+ =+ =+=
11. 2 83(82)643(6) 641846 −−=− =−=
12. ()() 2 84836445642044 −−=−=−=
13. 5(2)5(102) 2142(10)14 5(12) 6 52 10 x x ++ = = =⋅ =
14. 7(3)7(93)7(6)7321 2162(9)162 x x ===⋅=
15. 23 ;2,4 1 xy xy x + =−= + ()() 22344128 8 2111 −+ −+ ====− −+−−
16. 2 ;2and4 2 xy xy xyx + =−= () ()()() 224440 0 2422844 −+ −+ === −−−−+
17. 5(5032) 9 C =− 5 9(18)10== 10 ° Cisequivalentto50 ° F.
18. 555 (32)(8632)(54)30 999 CF=−=−== 30 ° Cisequivalentto86 ° F.
19. 22 46016460(2)16(2) 412016(4)412064 1246460 htt=+−=+− =+−=+− =−=
Twosecondsafteritiskicked,theball’sheightis 60feet.
20. 2 2 46016 460(3)16(3) 418016(9) 4180144 18414440 htt =+− =+− =+− =+− =−=
Threesecondsafteritiskicked,theball’sheightis 40feet.
21. { } { } { }1,2,3,42,4,52,4 = ∩
22. { } { } { }1,3,72,3,83 = ∩
23. { } { } { } ,,,,,, settesset = ∩
24. { } { } { } ,,,,,,,,, reallearreal = ∩
25. { } { } { } 1,3,5,72,4,6,8,10 = ∩ Theemptysetisalsodenotedby. ∅
26. { } { } { } 1,3,5,75,3,1−−−= ∩ or ∅
27. { } ,,, abcd ∅=∅ ∩
28. { } ,, wyz ∅=∅ ∩
29. { } { } { }1,2,3,42,4,51,2,3,4,5 = ∪
30. { } { } { }1,3,7,82,3,81,2,3,7,8 = ∪
31. { } { } {} 1,3,5,72,4,6,8,10 1,2,3,4,5,6,7,8,10 = ∪
32. { } { } { }0,1,3,52,4,60,1,2,3,4,5,6 = ∪
33. { } { } ,,,,,,,, aeiouaeiou ∅= ∪
34. { } { } ,,,,,,,, emptyempty ∅= ∪
35. a. 100 b. 0,100
c. 9,0,100
d. 4 9,,0,0.25,9.2,100 5
e. 3
f. 4 9,,0,0.25,3,9.2,100 5
36. a. 49
b. 0,49
c. 7,0,49
d. 7,0.6,0,49
e. 50
f. 7,0.6,0,49,50
37. a. 64
b. 0,64
c. 11,0,64
d. 5 11,,0,0.75,64 6
e. 5, π
f. 5 11,,0,0.75,5,,64 6 π
38. a. 4
b. 0,4
c. 5,0,4
d. 5,0.3,0,4
e. 2 f. 5,0.3,0,2,4
39. 0
40. Answersmayvary.Anexampleis 1 . 2
41. Answersmayvary.Anexampleis2.
42. Answersmayvary.Anexampleis2.
43. true;–13istotheleftof–2onthenumberline.
44. false;–6istotheleftof2onthenumberline.
45. true;4istotherightof–7onthenumberline.
46. true;–13istotheleftof–5onthenumberline.
47. true; −π=−π
48. true;–3istotherightof–13onthenumberline.
49. true;0istotherightof–6onthenumberline.
50. true;0istotherightof–13onthenumberline.
51. 300300 = 52. 203203−= 53. 1212−π=−π 54. 77−π=−π 55. 2552 −=− 56. 513135 −=− 57. 33 1 33 ==−
58. 77 1 77 ==−
59. 373744 −−−=−=−=
60. 51351388 −−−=−=−=
61. 2(5)33 xy+=+−=−=
62. 2(5)77 xy−=−−==
63. 25257 xy+=+−=+=
64. 25253 xy−=−−=−=−
65. 55 1 55 y y ===−
66. 2525 25251(1)0 xy xy +=+=+=+−=
67. Thedistanceis2171515 −=−=
68. Thedistanceis4151111 −=−=
69. Thedistanceis2577 −−=−=
70. Thedistanceis681414 −−=−=
71. Thedistanceis19(4)1941515 −−−=−+=−=
72. Thedistanceis26(3)2632323 −−−=−+=−=
73. Thedistanceis
3.6(1.4)3.61.42.22.2 −−−=−+=−=
74. Thedistanceis
5.4(1.2)5.41.24.24.2 −−−=−+=−= .
75. 6+(–4)=(–4)+6; commutativepropertyofaddition
76. 11(74)117114; ⋅+=⋅+⋅ distributivepropertyofmultiplicationoveraddition
77. 6+(2+7)=(6+2)+7; associativepropertyofaddition
78. 6(23)6(32); ⋅⋅=⋅⋅ commutativepropertyofmultiplication
79. (2+3)+(4+5)=(4+5)+(2+3); commutativepropertyofaddition
80. 7(118)(118)7; ⋅⋅=⋅⋅ commutativepropertyofmultiplication
81. 2(–8+6)=–16+12; distributivepropertyofmultiplicationoveraddition
82. 8(311)24(88) −+=−+− ; distributivepropertyofmultiplicationoveraddition
83. () 1 31;3 3 xx x +=≠− + , inversepropertyofmultiplication
84. ()()440;xx++−+= ⎡⎤ ⎣⎦ inversepropertyofaddition
85. 5(34)453544 15204 1516 xx x x +−=⋅+⋅− =+− =+
86. 2(54)325243 1083 105 xx x x +−=⋅+⋅− =+− =+
87. 5(32)12535212 151012 2710 xxxx xx x −+=⋅−⋅+ =−+ =−
88. 2(51)14252114 10214 242 xxxx xx x −+=⋅−⋅+ =−+ =−
89. 7(35)2(43) 73752423 213586 2929 yy yy yy y −++ =⋅−⋅+⋅+⋅ =−++ =−
90. 4(26)3(510) 424635310 8241530 236 yy yy yy y −++ =⋅−⋅+⋅+⋅ =−++ =+
91. ()() 53272151072 812 yyyy y −−+=−−− =−
92. 4(53)(63)201263 1415 yyyy y −−+=−−− =−
93. () [] 7434574345 yy −−−=−−+ ⎡⎤ ⎣⎦ []7484 73216 1625 y y y =−− =−+ =−
94. [] [] 65[8(24)]65824 65122 66010 1054 yy y y y −−−=−−+ =−− =−+ =−
⎡⎤+−−+⎣⎦
95. () () 22 22 22 22 22 22 184625 1846125 18467 18467 18647 186111211 xx xx xx xx xx xx
⎡⎤=+−−+⎣⎦
⎡⎤=+−−⎣⎦ =+−+ =−++ =−+=+
104. 2050 2050 2050 < Since2050,2050. <−<−
96. () () 22 22 22 22 22 2 2 145724 1457144 145710 145710 147510 14715 715 xx xx xx xx xx x x ⎡⎤+−−+⎣⎦
⎡⎤=+−−+⎣⎦
⎡⎤=+−−⎣⎦ =+−+ =−++ =−+ =+
97. –(–14x)=14x
98. ()1717yy−−=
99. –(2x –3y –6)=–2x +3y +6
100. () 51315131 xyxy −−−=−++
101. [] 1 3(3)(4)(4)0 xyyx x ++−=+ =
102. ()() 1 2770 2 yxxyy +−+=+= ⎡⎤ ⎣⎦
103. 63 63 63 > Since63,63. >−>−
105. 3 0.6 5 0.60.6 0.60.6 0.60.6 =
Since 3 0.60.6,0.6. 5 ==−
106. 5 2.5 2 2.52.5 .52.5 2.52.5 2 =
Since 5 2.52.5,2.5. 2 ==−
107. 3031415 4041514 303014 404015 15 14 01 01 < Since 3031415 01, 4041514 <−<⋅ .
108. 1718505 1817606 17 ⋅− 18 18 17 5050 6060 10 10 > Since 1718505 10,> 1817606 >⋅−
109. 88 1 1313 813 1 138 11 11 ÷− = Since 88 11,=1. 1313 =÷−
110. 44 2 1717 417 2 174 21 21 −÷ > Since 44 21,2>. 1717 >−÷
111. 22 8162436416443 64443 64163 483 45 −÷⋅−=−÷⋅−
112. 22 101005231001002523 100423 10083 923 89 −÷⋅−=−÷⋅− =−⋅− =−− =− =
113. 2 222 2 2 523529 [3(2)][9(2)] 109 [92] 109 11 1 121 ⋅−⋅− = = + = = 114. 22 2 10234512 (1232)(126) 17 6 17 36 ÷+⋅+ = = =
115. 83[2(25)4(86)]83[2(3)4(2)] 83[68] 83[2] 86 14 −−−−−=−−−−
116. 83[2(57)5(42)]83[2(2)5(2)] 83[410] 83[6] 818 26 −−−−−=−−−− =−− =−− =+ =
117. 2(2)4(3)412 583 8 3 8 3 −−−−+ = = =−
118. 6(4)5(3)2415 9101 9 1 9 −−−−+ = = =
119. 22 2 (56)237(1)24 893589325 12(4) 8975 18 14 7 14 1 2 = = = = =− 120. 22 2 123523123549 7367336 4513 1036 20(13) 26 260 26 10 ÷⋅+ ÷⋅+ = +− +− ⋅ = = = =−
121. ()444xxxx −+=−−=−
122. ()8828xxxxx −−=−+=−
123. () 6530xx−=−
124. () 10440xx−=−
125. 523xxx −=
126. () 62628 xxxxx −−=+=
127. () 83683656 xxxxx −+=−−=−
128. () 8368318310 xxx −+=−−=−−
129. a. () () () 7 220 10 7 22020 10 7 200 10 140 Ha H =− =− = =
Thelowerlimitoftheheartratefora20-yearoldwiththisexercisegoalis140beatsper minute.
b. () () () 4 220 5 4 22020 5 4 200 5 160 Ha H =− =− = =
Theupperlimitoftheheartratefora20-yearoldwiththisexercisegoalis160beatsper minute.
130. a. () () () 1 220 2 1 22030 2 1 190 2 95 Ha H =− =− = =
Thelowerlimitoftheheartratefora30-yearoldwiththisexercisegoalis95beatsper minute.
b. () () () 3 220 5 3 22030 5 3 190 5 114 Ha H =− =− = =
Theupperlimitoftheheartratefora30-yearoldwiththisexercisegoalis114beatsper minute.
131. a. 2 2 15,3959882 15,395988(7)2(7) 22,213 Txx =+− =+− = Theformulaestimatesthecosttohavebeen $22,213in2007.
b. Thisunderestimatesthevalueinthegraphby $5.
c. 2 2 15,3959882
15,395988(10)2(10) 25,075 Txx =+− =+− =
Theformulaprojectsthecosttobe$25,075in 2010.
132. a. 2 2 15,3959882
Txx =+− =+− =
15,395988(6)2(6) 21,251
Theformulaestimatesthecosttohavebeen $21,251in2006.
b. Thisunderestimatesthevalueinthegraphby $16.
c. 2 2 15,3959882
Txx =+− =+− =
15,395988(12)2(12) 26,963
Theformulaprojectsthecosttobe$26,963in 2012.
133. a. ()0.050.1210,000 0.0512000.12 12000.07 xx xx x +− =+− =−
b. 12000.0712000.07(6000) $780 x −=− =
134. a. 0.060.5(50)0.06250.5 250.44 tttt t +−=+− =−
b. 0.06(20)0.5(5020) 1.20.5(30) 1.215 16.2miles +− =+ =+ =
135. – 144. Answersmayvary.
145. doesnotmakesense;Explanationswillvary. Sampleexplanation:Modelsdonotalways accuratelypredictfuturevalues.
146. doesnotmakesense;Explanationswillvary. Sampleexplanation:Tousethemodel,substitute0 for x
147. makessense
148. doesnotmakesense;Explanationswillvary. Sampleexplanation:Thecommutativeproperty changesorderandtheassociativepropertychanges groupings.
149. false;Changestomakethestatementtruewillvary. Asamplechangeis:Somerationalnumbersarenot integers.
150. false;Changestomakethestatementtruewillvary. Asamplechangeis:Allwholenumbersareintegers
151. true
152. false;Changestomakethestatementtruewillvary. Asamplechangeis:Someirrationalnumbersare negative.
153. false;Changestomakethestatementtruewillvary. Asamplechangeis:Theterm x hasacoefficientof 1.
154. false;Changestomakethestatementtruewillvary. Asamplechangeis: 53(4)531237. xxx +−=+−=−
155. false;Changestomakethestatementtruewillvary. Asamplechangeis:2. xxx−−=−
156. true
157. 21.4 1.41.5
160. a. 437 ()() bbbbbbbbbb ⋅=⋅⋅⋅⋅⋅=
b. 5510 ()() bbbbbbbbbbbbb ⋅=⋅⋅⋅⋅⋅⋅⋅⋅=
c. addtheexponents
161. a. 7 4 3 bbbbbbbb b bbb b == ⋅⋅ b. 8 6 2 bbbbbbbbb b bb b == ⋅
c. subtracttheexponents
162. 3 6.2106.21010106200 ×=×××= Itmovesthedecimalpoint3placestotheright.
Section P.2
Check Point Exercises
1. a. 32325 3333or243 + ==
b. ()()34263246410410 xyxyxxyy =⋅⋅⋅⋅⋅ 3246 510 40 40 xy xy ++ =⋅ =⋅
2. a. 6 3 3 (3)(3)27 (3) =−=−
b. 148148 14385113 3535 2727 99 33 xyxy xyxy xyxy =⋅⋅==
3. a. 2 2 11 5 525 ==
b. 3 3 111 (3)2727 (3) −===−
c. 2 2 2 2 114 1416 411 4 ==⋅==
d. 4 644 66 13 33 y xyy xx =⋅⋅=
4. a. () 2 3326 333or729 ==
b. () 2 77(2)14 14 1 yyy y ===
c. () 4 33(4)12 bbb ==
5. 3333 (4)(4)()64 xxx−=−=−
6. a. () 55 55 2 232 yyy ⎛⎞ −==⎜⎟ ⎝⎠
b. () 33 5 515 3 327 3 x xx⎛⎞ == ⎜⎟ ⎝⎠
7. a. () () ()() 444 4 36361224 2216 xyxyxy ==
b. ()() () 253253 38 6363 18 xyxyxxyy xy −=−⋅⋅⋅⋅⋅ =−
c. 122122 164164 100100 2020 xyxy xyxy ⎛⎞⎛⎞ ⎛⎞ = ⎜⎟⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠⎝⎠ 12162(4) 46 6 4 5 5 5 xy xy y x = = =
d. ()() () 222 42 4 55 x x y y ⎛⎞ = ⎜⎟ ⎝⎠ ()() () 22 42 22 8 8 22 8 2 5 5 5 25 x y x y y x y x = = = =
8. a. 9 2.6102,600,000,000−×=−
b. 6 3.017100.000003017 ×=
9. a. 95,210,000,0005.2110 =×
b. 0.000000068936.893108 −=−×
10. () 727 410104.11010 ×=×× () 27 9 4.11010 4.110 =×× =×
11 a. ()() 577.110510 ×× () () 57 2 12 12 1 7.151010 35.510 3.551010 3.551010 3.5510 =⋅×⋅ =× =×× =×× =×
b. 66 33 1.2101.210 3 31010 × =⋅ × 6(3) 9 8 0.410 0.410 410 =× =× =×
12. 99 66 3 13101310 5.1105.110 2.510 2500 × =⋅ × ≈⋅ ≈ TheaveragePellgrantwas$2500in2006.
Exercise Set P.2
1. 2 52(55)225250 ⋅=⋅⋅=⋅=
2. 2 62(66)236272 ⋅=⋅⋅=⋅=
3. 6 (2)(2)(2)(2)(2)(2)(2)64 −=−−−−−−=
4. 4 (2)(2)(2)(2)(2)16 −=−−−−=
5. 6 222222264 −=−⋅⋅⋅⋅⋅=−
6. 4 2222216 −=−⋅⋅⋅=−
7. 0 (3)1 −=
8. 0 (9)1 −=
9. 0 31−=−
10. 0 91−=−
11. 3 3 111 4 444464 ===
12. 6 6 111 2 222222264 ===
13. 23235 22222222232 + ⋅===⋅⋅⋅⋅=
14. 32325 333333333243 + ⋅===⋅⋅⋅⋅=
15. 23236 (2)2222222264 ===⋅⋅⋅⋅⋅=
16. 32326 (3)33333333729 ===⋅⋅⋅⋅⋅=
17. 8 844 4 2 22222216 2 ===⋅⋅⋅=
18. 8 844 4 3 33333381 3 ===⋅⋅⋅=
19. 3312 2 111 3333 3339 −−+− ⋅=====
20. 3312 2 111 2222 2224 −−+− ⋅===== ⋅
21. 3 374 74 2111 22 22222216 =====
22. 4 473 73 3111 33 3333327 =====
23. 2 22 1 y xyy xx =⋅=
24. 3 33 1 x xyx yy =⋅=
25. 0555 1 xyyy =⋅=
26. 7077 1 xyxx ⋅=⋅=
27. 373710 xxxx + ⋅==
28. 11511516 xxxx + ⋅==
29. 5105105 xxxx −−+ ⋅==
30. 6126126 xxxx −−+ ⋅==
31. 373721()xxx ==
32. 11511555()xxx ==
33. 535315 15 1 ()xxx x ===
34. 646424 24 1 ()xxx x ===
35. 14 1477 7 x xx x ==
36. 30 301020 10 x xx x ==
37. 14 14(7)14721 7 x xxx x −−+ ===
38. 30 30(10)301040 10 x xxx x −−+ ===
39. 322322326 (8)8()864 xxxx ===
40. 422422428 (6)(6)()636 xxxx ===
41. 33 33 4(4)64 xxx ⎛⎞ −==−⎜⎟ ⎝⎠
42. 33 33 6(6)216 y yy ⎛⎞ −==−⎜⎟ ⎝⎠
43. 25222252 2252 410 (3)(3)()() 9 9 xyxy xy xy −=−⋅ = =
44. 46334363 4363 1218 (3)(3)()() 27 27 xyxy xy xy −=− =− =−
45. 47474711 (3)(2)3266 xxxxxx + =⋅⋅==
46. 51251251217 (11)(9)1199999 xxxxxx + =⋅==
47. 364364 3614 95 (9)(2)(9)(2) 18 18 xyxyxxyy xy xy ++ −−=−− = =
48. 47114711 47111 1112 (5)(6)(5)(6) 30 30 xyxyxxyy xy xy ++ −−=−− = =
49. 2020 20416 44 88 44 22 xx xx xx ⎛⎞ ⎛⎞ === ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠
50. 2424 24618 66 2020 22 1010 xx xx xx ⎛⎞ ⎛⎞ === ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠
51. 134134 2323 13243 11 2525 55 5 5 abab abab ab ab ⎛⎞⎛⎞ ⎛⎞ = ⎜⎟⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠⎝⎠
52. 146146 7373 14763 73 3535 77 5 5 abab abab ab ab ⎛⎞⎛⎞ ⎛⎞ = ⎜⎟⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠⎝⎠ =−
53. 77 7147 14147 14142 22 77 bb bb bbb ⎛⎞ ⎛⎞ ==⋅==
1010 2020 1020 10 10 2020 1010 2 2 2 bb bb b b
32232 26 26 6 (4)(4)() 4 1 4 1 16 xx x x x = = = =
2332(3) 36 36 6 (10)10 10 1 10 1 1000 xx x x x = = = =
35 375(9) 79 414 14 4 243 324 3 4 3 4 xy xy xy xy y x ⋅ = =
49 4129(3) 123 812 12 8 101 303 1 3 3 xy xy xy xy y x =
59. 2 3262 6 55 225 xxy y x y ⎛⎞ == ⎜⎟ ⎝⎠ 60. 433 4 3 343 3 12 3 3 3 27 xy yx y x y x ⎛⎞ ⎛⎞ = ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ = =
61. () 334223 103104 53 6 15 18 153 5 3 27 abb aba b a b a ⎛⎞ ⎛⎞ = ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎛⎞ = ⎜⎟ ⎝⎠ =
62. () 3314882 1721714 103 30 9 303 10 3 3 27 abb aba b a b a ⎛⎞ ⎛⎞ = ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎛⎞ = ⎜⎟ ⎝⎠ =
63. 520 34 3 1 12 ab ab ⎛⎞ = ⎜⎟ ⎝⎠
64. 530 35 4 1 12 ab ab ⎛⎞ = ⎜⎟ ⎝⎠
65. 2 3.810380 ×=
66. 2 9.210920 ×=
67. 4 6100.0006 ×=
68. 5 7100.00007 ×=
69. 6 7.16107,160,000 −×=−
70. 6 8.17108,170,000 −×=−
71. 1 7.9100.79 ×=
72. 1 6.8100.68 ×=
73. 3 4.15100.00415 −×=−
74. 3 3.14100.00314 −×=−
75. 10 6.000011060,000,100,000 −×=−
76. 10 7.000011070,000,100,000 −×=−
77. 4 32,0003.210 =×
78. 4 64,0006.410 =×
79. 17 638,000,000,000,000,000 6.3810=×
80. 17579,000,000,000,000,0005.7910 =×
81. 357165.71610 −=−×
82. 338293.82910 −=−×
83. 0.00272.7103 =×
84. 0.00838.3103 =×
85. 0.000000005045.04109 −=−×
86. 0.000000004054.05109 −=−×
87. ()() () ()4343 437 3102.11032.11010 6.3106.310 + ××=×× =×=×
88. ()()437 2104.1108.210 ××=×
89. ()() () () 15111511 15(11) 4 1.6104101.641010 6.410 6.410 +− ××=×× =× =×
90. ()() () () 15111511 15(11) 4 1.4103101.431010 4.210 4.210 +− ××=×× =× =×
91. ()() () () 8484 8(4) 12 11 6.1102106.121010 12.210 12.210 1.2210 −+− ××=×× =× =× =×
92. ()()8412 11 5.11031015.310 1.5310 ××=× =×
93. ()() () () 84 84 84 12 1313 4.3106.210 4.36.21010 26.6610 26.6610 2.666102.6710 + ×× =×× =× =× =×≈×
94. ()() 84 8412 1313 8.2104.610 37.721037.7210 3.772103.7710 + ×× =×=× =×≈×
95. 88 55 853 8.4108.410 4 41010 2.1102.110 × =× × =×=×
96. 8 853 5 6.910 2.3102.310 310 × =×=× ×
97. 44 22 4(2) 65 3.6103.610 9 91010 0.410 0.410410 × =× × =× =×=×
98. () 4 4(2)6 2 165 1.210 0.6100.610 210 61010610 × =×=× × =××=×
99. 22 66 268 4.8104.810 2.4 2.41010 210210 × =× × =×=×
100. 2 268 6 7.510 310310 2.510 × =×=× ×
101. 22 66 2(6) 43 2.4102.410 4.8 4.81010 0.510 0.510510 × =× × =× =×=×
102. 2 2(6) 6 43 1.510 0.510 510 0.510510 × =× × =×=×
105. ()() 43 4 43 3 4 0.000720.003 0.00024 7.210310 2.410 7.231010910 2.410 × ×× = × ×⋅ =×=×
106. ()() ()() () 43 33 1 16 6 7 660000.0016.610110 0.0030.002310210 6.610 1.110 610 1.110 ×× × = × ×× × ==× × =×
107. () () () 2363 21363 33 66001 xy xy xy xy xyxy = ===
108. () () () 2224 2363 7 43 2687 8 xy xy xy xy y xyxy x = ===
109. () () 5 363655 486 48686 2222 2 216 xyzxxyzx yy xyz xzxz =⋅⋅ ===
110. () () 3 474733 277 27777 3333 3 39 xyzxxyzx yy xyz xzxz =⋅⋅ ===
103. 11 4 11 4 11(4) 15
480,000,000,0004.810 0.000121.210 4.810 1.210 410 410 × = × =× =× =×
111. () 34522 6810 345 121620 121620 1 xyz xyz xyz xyz xyz ⎛⎞ = ⎜⎟ ⎝⎠ ==
112. () 4564 810124 456 324048 324048 1 xyz xyz xyz xyz xyz ⎛⎞ = ⎜⎟ ⎝⎠ ==
104. 11 3 11(3) 14
282,000,000,0002.8210 0.001411.4110 210 210 × = × =× =×
113. ()()() () ()() () () 220 1214333 352 222286 2610 186 2216 2 221 2 4 xyxyxy xy xyxy xy xy = =
114. ()()() () ()() () () 220 1316433 462 2622128 2812 266 229 2 221 2 4 xyxyxy xy xyxy xy xy = =
115. a. 122.5210 ×
b. 8310 ×
c. 1212 88 4 2.52102.5210 3 31010 0.8410 8400 × =× × =× = $8400perAmerican
116. a. 122.2710 × b. 82.9810 ×
c. 1212 88 4 2.27102.2710 2.98 2.981010 0.761710 7617 × =× × =× = $7617perAmerican
117. 912 12 12 1450106.601.45106.6 1.456.610 9.5710 ×⋅=×⋅ =⋅× =×
Box-officereceiptswere12 $9.5710 × in2006.
118. 912 12 12 1400106.401.4106.4 1.46.410 8.9610 ×⋅=×⋅ =⋅× =×
Box-officereceiptswere12 $8.9610 × in2005.
119. 23234 234 19 119 18 5.31020,0005.310210 5.321010 10.610 1.061010 1.0610 ×⋅=×⋅× =⋅×⋅ =× =×⋅ =×
Themassis18 1.0610 × gram.
120. 24244 244 20 120 19 1.671080,0001.6710810 1.6781010 13.3610 1.3361010 1.33610 ×⋅=×⋅× =⋅×⋅ =× =×⋅ =×
Themassis19 1.33610 × gram.
121. 772 72 9 3.2101273.2101.2710 3.21.271010 4.06410 ×⋅=×⋅× =⋅×⋅ =×
Americanseat9 4.06410 × chickensperyear.
122. 365daysequals36524or8760hours. ⋅ 8760hoursequals876060or525,600minutes. ⋅ 525,600min.equals 525,60060or31,536,000seconds. Thereare7 3.153610 × secondsinayear.
123. – 130. Answersmayvary.
131. doesnotmakesense;Explanationswillvary. Sampleexplanation:392712 36()36not36. xxx =
132. makessense
133. doesnotmakesense;Explanationswillvary. Sampleexplanation:12 4.610 × representsover4 trillion.Theentireworldpopulationismeasuredin billions(910).
134. makessense
135. false;Changestomakethestatementtruewillvary. Asamplechangeis:23 44. >
136. true
137. false;Changestomakethestatementtruewillvary. Asamplechangeis:441 (2)2because16. 16 −≠≠
138. false;Changestomakethestatementtruewillvary. Asamplechangeis:2255 5522. ⋅=⋅
139. false;Changestomakethestatementtruewillvary. Asamplechangeis:534.75347. ≠
140. false;Changestomakethestatementtruewillvary. Asamplechangeis: 30 30(5)35 5 810 410410. 210 × =×=× ×
141. false;Changestomakethestatementtruewillvary. Asamplechangeis: 53 (710)(210)700,000.002. ×+×=
142. true
143. Thedoctorhasgathered: 12112132222444 2 +=+=+= So, 31 1 44 −= isremaining.
144. ,, ACD bMNbMbN === bbbACD A CD = =+
145. 70bts min 60min hr 24hrs day 365days yr 80yrs 9 9
70602436580beats 2943360000beats 2.9433610beats 2.9410beats =⋅⋅⋅⋅ = =× ≈× Theheartbeatsapproximately9 2.9410 × timesovera lifetimeof80years.
146. Answersmayvary.
147. a. 164428 ⋅=⋅=
b. 164648 ⋅==
c. 164164 ⋅=⋅
148. a. 30017.32 ≈
b. 10317.32 ≈
c. 300103 =
149. a. 211031 xxx += b. 212102312 +=
Section P.3
Check Point Exercises
1. a. 819 =
b. 93−=− c. 11 255 = d. 366410010 +==
e. 36646814 +=+=
2. a. 7525325353 =⋅==
b. 2 2 2 510510 50 252 252 xxxx x x x ⋅=⋅ = =⋅ =⋅ 52 x =
3. a. 25255 164 16 ==
b. 33150150 22 xx x x = 2 2 75 253 53 x x x = =⋅ =
4. a. 813913(89)3 1713 +=+ = b. 172017 1172017 (120)17 1917 xx xx x x =− =− =−
5. a. 52712 + 59343 53323 15323 (152)3 173 =⋅+⋅ =⋅+ =+ =+ = b. 61848xx 692442 632422 18282 (188)2 102 xx xx xx x x =⋅−⋅ =⋅−⋅ =− =− =
6. a. Ifwemultiplynumeratoranddenominatorby 3,thedenominatorbecomes 3393. ⋅== Therefore,multiplyby1, choosing 3 3 for1. 5535353 33393 =⋅==
b. The smallest numberthatwillproduceaperfect squareinthedenominatorof 6 12 is3 because123⋅= 366. = Somultiplyby1, choosing 3 3 for1. 6636363 3 12123366 =⋅===
7. Multiplyby 45 45 8845 454545 8(45) 224(5) 8(45) 165 8(45)3285 or 1111 =⋅ ++− = = =
8. a. 33333 40858525 =⋅=⋅= b. 555555 886432222 ⋅==⋅= c. 3 3 3 1251255 273 27 == 9. 33 38143 33 33 33 3 3 327343 33343 9343 (94)3 53 =⋅− =⋅− =− =− =
10. a. 1 2 25255 == b. 1 33 882 ==
c. 1 44 81813 −=−=−
d. () 1 33 882 −=−=− e. 1 3 13 3 111 27 27273 ===
11. a. () 44 334 2727(3)81 === b. () 33 223 44(2)8 ===
c. () 2 5 222 5 5 1111 32 323224 ====
2555 −=−=−
5. 36,Thesquarerootofanegativenumberisnot real.
6. 25,Thesquarerootofanegativenumberisnot real.
7. 251693 −==
1442516913
2516541
1442512517
5025225252 =⋅== 14. 27939333 =⋅==
25. 49497 164 16 ==
26. 12112111 93 9 ==
27. 33 48482164 33 xx xx x x ===
28. 33 7272293 88 xx xx x x ===
29. 44 3 2 150150 33 50 252 52 xx x x x xx xx = = =⋅ =
30. 44 3 2 2424 33 8 42 22 xx x x x xx xx = = =⋅ =
31. () 33 11 31 4 4 200200 1010 20 20 45 2 25 xx x x x x x x = = = =⋅ =
32. () 33 31 11 442 50050050 1010 5025252 xx x x x xxx == ==⋅=
33. 7363(76)3133 +=+=
34. 85115(811)5195 +=+=
35. 617817(68)17217 xxxx −=−=−
36. 413613(46)13213 xxxx −=−=−
37. 8324232 2232 (23)2 52 +=⋅+ =+ =+ =
38. 20654565 2565 (26)5 85 +=⋅+ =+ =+ =
39. 50825242 5222 (52)2 32 xxxx xx x x −=⋅−⋅ =− =− =
40. 63289747 3727 (32)7 7 xxxx xx x x −=⋅−⋅ =− =− =
41. 3185503925252 332552 92252 (925)2 342 +=⋅+⋅ =⋅+⋅ =+ =+ =
42. 4122754432253 423253 83103 (810)3 23 −=⋅−⋅ =⋅−⋅ =− =− =−
43. 383237275 −+− 3421623362253 3224236253 624218253 20253 =⋅−⋅+⋅−⋅ =⋅−+⋅− =−+− =−
44. 35422496463 −−+ 396246166497 33622646437 964646127 6127 =⋅−⋅−⋅+⋅ =⋅−⋅−+⋅ =−−+ =+
45. 1177 7777 =⋅=
46. 221021010 101010105 =⋅==
47. 22510 5555 =⋅=
48. 77321 3333 =⋅=
49. 22 1313311 311311311 13(311) 3(11) 13(311) 911 13(311) 2 =⋅ ++− = = =
50. 22 3337 373737 3(37) 3(7) 3(37) 97 3(37) 2 =⋅ ++− = = =
51. 22 7752 525252 7(52) (5)2 7(52) 54 7(52) + =⋅ −−+ + = + = =+
52. 22 5531 313131 5(31) (3)1 5(31) 31 5(31) 2 + =⋅ −−+ + = + = + =
53. 22 6653 535353 6(53) (5)(3) 6(53) 53 6(53) 2 3(53) =⋅ ++− = = = =−
54. 22 111173 737373 11(73) (7)(3) 11(73) 73 11(73) 4 + =⋅ −−+ + = + = + =
55. 33312555 == 56. 333822 ==
57. 33 3 8(2)2 −=−=−
58. 33 3 125(5)5 −=−=−
59. 416isnotarealnumber.
60. 481isnotarealnumber.
61. 44(3)33 −=−=
62. 44(2)22 −=−=
63. 55(3)3 −=−
64. 55(2)2 −=−
65. 55 5 111 322 2 −=−=−
66. 6 6 66 111 642 2 ==
67. 33333 32848424 =⋅==⋅
68. 3150cannotbesimplifiedfurther.
69. 33433 xxxxx =⋅=⋅
70. 3335322 xxxxx ==
71. 3333333 965427227232 ⋅==⋅==
72. 33333 124488626 ⋅==⋅=
73. 566 555 5 6464 322 22 xx xx x x ===
74. 455 444 4 162162 813 22 xx xx x x ===
75. 555 423272 +=
76. 555 632383 +=
77. 3333 51654582272 +=⋅+⋅ 33 33 3 52232 10232 132 =⋅+ =+ =
78. 3333 3248183273 +=⋅+⋅ 33 33 3 32333 6333 93 =⋅+ =+ =
79. 33354128 xyyx
333 33 3 272642 3242 2 xyyx yxyx yx =⋅−⋅ =− =−
80. 3332481 xyyx
333 33 3 83273 2333 3 xyyx yxyx yx =⋅−⋅ =− =−
81. 3 2822 +=+
82. 3 315 + willnotsimplify
83. 1/2 36366 ==
84. 1/2 12112111 ==
85. 1/33 882 ==
86. 1/33 27273 ==
87. () 2 2/32 3 125125525 ===
88. () 2 2/33 884 ==
89. 4/5 4/54 111 32 32216 ===
90. 5/2 5/25 5 1111 16 1641024 (16) ====
91. ()()1/31/41/31/4
1/31/4 7/12 7272 14 14 xxxx x x + =⋅⋅ =⋅ =
92. 2/33/42/33/4 2/33/4 17/12 (3)(4)34 12 12 xxxx x x + =⋅⋅ =⋅ =
93. 1/21/2 1/41/4 1/21/4 1/4 2020 55 4 4 xx xx x x ⎛⎞ ⎛⎞ = ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ =⋅ =
94. 3/43/4 3/41/35/12 1/31/3 7272 88 99 xx xx xx ⎛⎞ ⎛⎞ ==⋅= ⎜⎟ ⎜⎟
95. ()3 2/32/332 xxx ==
96. 4/554/554()xxx ==
97. 461/21/241/261/223(25)255xyxyxy ==
98. 961/31/39/36/332(125)1255xyxyxy ==
99. 13 43 431 412 11 1212 3 27 27 y y y yy ⎛⎞ ⎜⎟ ⎝⎠ == 82 1232727yy==
100. () () 441/541/5 3/103/10 22yy yy = 4/5 4/53/101/2 3/10 16 1616 y yy y ===
101. 422/41/2 5555 ===
102. 422/41/2 7777 ===
103. 366/32 xxx ==
104. 41212/43 xxx ==
105. 66/2344/22 xxx ==
106. 99/3366/32 xxx ==
107. 6321 63293 9933 xyxyxyxy ===
108. 4812 12482 1212333 xyxyxyxy ===
109. 333 416625225273 +=+==
110. 333 3 3 33 16991000216 133106 1616 448 2 +++ =+++ =+ =+= =
()()()() () () 1/2 241/2 1/21/21/2241/2
21/241/21/2 1/2 121/21121/2 2 23/2 3/2 49 49 1 49 11 77 1 77 xyxy xyxy xyxy xyxyxy x xy y −+ −+ = = =⋅= ==
111. ()() () ()()()
112. ()() ()()()()()()()() () 1/36635/61/3 1/361/331/35/661/36 21522512 3 31 8 8 22 2 2 xyxy xyxy xyxyxy x xy y +− −−−+ = == ==
113. ()() () () ()()()() 5/41/366 5/43/41/3 3/4 2/41/362/461/36 3 32 2 xy xy x xyxy x xy y ⎛⎞ = ⎜⎟ ⎝⎠ == ==
114. ()() () () ()()()() 1/27/444 1/27/45/4 5/4 1/22/441/242/44 2 22 2 xy xy y xyxy y xy x ⎛⎞ = ⎜⎟ ⎝⎠ == ==
