Chapter 0
Problems 0.1 Problems 0.2
p
1. False; 13 is not a real number. 1. False, because 0 does not have a reciprocal.
66
2. True, because 2 and 7 are integers and 7 ¤ 0. 2. True; 6:6 D so the reciprocal of 6:6 is
10
10
D 0:151515 : : :.
3. False, because 3 is not positive. 66
0 3. False; the negative of 7 is 7 because
4. False, because 0 D : 7 C . 7/ D 0.
1
5. False, it is fairly easy to see (although the details 4. True; 1.x y/ D .1 x/.1 y/
are
p not relevant for a course using this text) that
n, for n an integer, is either an integer (if n is a 5. False; x C y D y C . x/ D y x.
perfect square) or irrational (if n is not a perfect
square). The perfect squares are 6. True; .x C 2/.4/ D .x/.4/ C .2/.4/ D 4x C 8.
16; 25; 36; : : : and since 3 is not among
1; 4; 9;p
these, 3 is not rational. x x 15 x C 15 xC3
7. false; C3D C D ¤ .
5 5 5 5 5
6. False; we cannot divide by 0.
b ab
8. True, because a D .
p c c
7. False, because 25 D 5, which is a positive
integer.
9. False; 2.x y/ D 2xy while
p .2x/ .2y/ D .2 2/ .x y/ D 4xy.
8. True; 2 is an irrational real number.
10. True; by the associative and commutative
9. False; we cannot divide by 0. properties, x.4y/ D .x 4/y D .4 x/y D 4xy.
10. False, we have 3 < < 4 so that is at best 11. distributive
rational and not an integer. It can be shown that
is irrational (but the details are not relevant for 12. the associative property of addition
our purposes).
13. associative
11. True; see Figure 0.1 in the text.
14. definition of division and commutative property
12. False, since the integer 0 is neither positive nor
15. commutative and distributive
negative.
16. associative
13. True; we can regard a terminating sequence as
being the same as the sequence obtained from it 17. the definition of division
by appending infinitely many zeros.
p 18. commutative
14. False; 13 is not a real number.
1
Copyright © 2017 Pearson Education, Inc. Publishing as Prentice Hall.
,Chapter 0: Review of Algebra ISM: Introductory Mathematical Analysis
19. distributive and commutative 37. . 1:6/. 0:5/ D .1:6/.0:5/ D 0:8
20. distributive 38. 19. 1/ D . 1/19 D .1 19/ D 19
1 a
21. 2x.y 7/ D .2x/y .2x/7 D 2xy .7/.2x/
39. 1
D 1 Da
D 2xy .7 2/x D 2xy 14x a
1
x 1 1 1 z 40. . 6 C x/ D . 6/ xD6 x
22. z D x zDx z Dx z D x where
y y y y y
we have used, in order, the definition of division, 41. 7.x/ D .7x/ D 7x
the associativity of multiplication, the
commutativity of multiplication, and the 42. 3.a b/ D 3a C 3b D 3.b a/
definition of division
43. Π6 C . y/ D . 6/ . y/ D 6 C y
23. (x C y/.2/ D 2.x C y/ D 2x C 2y
3 13 1
44. 3 .3a/ D D D
24. aŒb C .c C d/ D aŒ.b C c/ C d 3a 3a a
D aŒd C .b C c/ D aŒ.d C b/ C c 9 9 91 1
45. 9 . 27/ D D D D
27 27 93 3
25. xŒ.2y C 1/ C 3 D xŒ2y C .1 C 3/ D xŒ2y C 4
D x.2y/ C x.4/ D .x 2/y C 4x D .2x/y C 4x a a
46. . a/ . b/ D D
b b
D 2xy C 4x
9
26. .1 C a/.b C c/ D 1.b C c/ C a.b C c/ 47. 3 C .3 1 9/ D 3 C D3C3D6
3
D 1.b/ C 1.c/ C a.b/ C a.c/ D b C c C ab C ac
48. 3Œ 2.3/ C 6.2/ D 3Œ 6 C 12 D 3Œ6 D 18
27. .x y C z/w D ..x C . y// C z/w
D .x C . y//w C zw D xw C . y/w C zw 49. ( a/. b/. 1/ D ab. 1/ D ab
D xw C . yw/ C zw D xw yw C zw
50. . 12/. 12/ D .12/.12/ D 144
28. 2 C . 4/ D 6
51. X.1/ D X
29. aCbDb a
52. 71.x 2/ D 71x C 142 D 142 71x
30. 6 C . 4/ D 2
53. 4.5 C x/ D 4.5/ C 4.x/ D 20 C 4x
31. 7 2D5
54. .x y/ D xCyDy x
3
3 1 3 2 6
32. 1
D 1
D D D6 55. 0. x/ D 0
2 1 1 1
2
1 81 8
33. 5 . 13/ D 5 C 13 D 8 56. 8 D D
11 11 11
34. . a/ C . b/ D a b X
57. DX
1
35. . 2/.9/ D .2 9/ D 18
14x 27x 2x
36. 7. 9/ D .7 9/ D 63 58. D D
21y 37y 3y
2
Copyright © 2017 Pearson Education, Inc. Publishing as Prentice Hall.
,ISM: Introductory Mathematical Analysis Section 0.3
2x 2x 7
59. D D x 76. is not defined (we cannot divide by 0).
2 12 0
2 1 21 2 0
60. D D 77. D 0; for X ¤ 0
3 x 3x 3x X
a a.3b/ 3ab 0
61. .3b/ D D 78. is not defined (we cannot divide by 0).
c c c 0
62. 5a C .7 5a/ D 7 Problems 0.3
aby a by by 1. .23 /.22 / D 23C2 D 25 .D 32/
63. D D
ax ax x
2. x6 x9 D x6C9 D x15
a 1 a1 a
64. D D
b c bc bc 3. 175 172 D 175C2 D 177
2 5 25 10 4. z3 zz2 D z3C1C2 D z6
65. D D
x y xy xy
x3 x5 x3C5 x8
5. D D
1 1 3 2 3C2 5 y9 y5 y9C5 y14
66. C D C D D
2 3 6 6 6 6
6. .x12 /4 D x124 D x48
x y x 3y x C 3y
67. C D C D
3a a 3a 3a 3a .a3 /7 a37 a21
7. D D
.b4 /5 b45 b20
3 7 9 14 9 14 5
68. D D D !2
10 15 30 30 30 30
51 1 1314
D D 8. D .1314 1 2
/ D .131 /2
56 6 13
a c aCc D 1312 D 132 D 169
69. C D
b b b
9. .2x2 y3 /3 D 23 .x2 /3 .y3 /3 D 8x23 y33 D 8x6 y9
X Y X Y !2
70. p p D p
5 5 5 w2 s3 .w2 s3 /2 .w2 /2 .s3 /2
10. D D
y2 .y2 /2 y22
3 1 1 18 3 2
71. C D C w22 s32 w4 s6
2 4 6 12 12 12 D 4
D 4
y y
18 3 C 2 17
D D
12 12 x9
11. D x9 5 D x4
3 5 39 75 27 35 8 x5
72. D D D !6
7 9 63 63 63
2a4 .2a4 /6
12. D
6 x y 6y 7b5 .7b5 /6
73. x D6 D6 D
y y x x
26 .a4 /6
D
l 76 .b5 /6
w l m l 1 l
74. D D D 64a46
m w 1 w m wm D
117; 649b56
x
y2 x z x xy x2 64a24
75. z D D D D
xy y2 xy y2 z yz 117; 649b30
3
Copyright © 2017 Pearson Education, Inc. Publishing as Prentice Hall.
, Chapter 0: Review of Algebra ISM: Introductory Mathematical Analysis
0 12
.y3 /4 y34 y12 y12 2=5 5 !1=5
13. D D D D y12 8
D y4 243 3
.y2 /3 y2 y23 y2 y6 y2 y6C2 28. D@ A
1024 4
.x2 /3 .x3 /2 x23 x32 x6 x6 x12 2
14. D D 12 D 12 3 9
3
.x / 4 x34 x x D D
4 16
x12 12 D x0 D 1 p p p p p
29. 50 D 25 2 D 25 2 D 5 2
p
15. 25 D 5 p p p p p
30. 3
54 D 3
27 2 D 3
27 3 2 D 3 3 2
p
4 p p p p
16. 81 D 3 31.
3
2x3 D 3 3
2 x3 D x 3 2
p
7
p p p p
17. 128 D 2 32. 4x D 4 xD2 x
!1 p
1 33. 49u8 D 7u4
p
5 243 5 35 5
18. 0:00243 D D r p p
100000 105 x 4
x 4
x
34. 4
Dp 4
D
3 16 16 2
D D 0:3 p p p
10 35. 2 8 5 27 C 3 128
p p p
r p4
D 2 4p 2 5 9p 3 C 3p64 2
1 1 1
19. 4
Dp D D 2p2 2 p 5 3 3pC43 2
4
16 16 2 D 4 2 15 3 C 4 2 3
r r p
3 r r r p p
3 8 3 8 8 2 2 3 3 13 39 39 39
20. D D p
3
D D 36. D D D p D
27 27 27 3 3 13 13 13 132 13 2 13
p p p p p
21. .49/1=2 D 49 D 7 37. .9z4 /1=2 D 9z4 D 32 .z2 /2 D 32 .z2 /2
p D 3z2
22. .64/1=3 D 3
64 D 4
38. .729x6 /3=2 D .27x3 /3 D 19683x9
1 3
23. 813=4 D .81/ 4 D 33 D 27
!2=3 3 !2=3 2
27t3 3t 3t 9t2
1 1 1 1 39. D D D
24. .9/ 5=2 D D p 5 D 5 D 8 2 2 4
.9/5=2 9
3 243
3=4 4 ! 3=4 3
256 4 4
40. D D 3
2=5 1 1 1 1 x12 x3 x
25. .32/ D D p 2 D 2
D
.32/2=5 5
32
.2/ 4 4 3 4 3 x9 x9
D 3 3
D 9
D D
.x / x 43 64
1 1 1
26. .0:09/ 1=2 D Dp D a5 b 3 1 5 1 1 a5
.0:09/1=2 0:09 0:3 41. D a5 b 3
D a D
c2 c2 b3 c2 b3 c2
1 10
D 3
D
3 p x2=5 y3=5
10
42. 5
x2 y3 z 10 D x2=5 y3=5 z 10=5 D
z2
4=5 r !4 4
1 5 1 1 1 3
27. D D D 43. 3a 1b 2c 3 D
32 32 2 16
ab2 c3
4
Copyright © 2017 Pearson Education, Inc. Publishing as Prentice Hall.
Problems 0.1 Problems 0.2
p
1. False; 13 is not a real number. 1. False, because 0 does not have a reciprocal.
66
2. True, because 2 and 7 are integers and 7 ¤ 0. 2. True; 6:6 D so the reciprocal of 6:6 is
10
10
D 0:151515 : : :.
3. False, because 3 is not positive. 66
0 3. False; the negative of 7 is 7 because
4. False, because 0 D : 7 C . 7/ D 0.
1
5. False, it is fairly easy to see (although the details 4. True; 1.x y/ D .1 x/.1 y/
are
p not relevant for a course using this text) that
n, for n an integer, is either an integer (if n is a 5. False; x C y D y C . x/ D y x.
perfect square) or irrational (if n is not a perfect
square). The perfect squares are 6. True; .x C 2/.4/ D .x/.4/ C .2/.4/ D 4x C 8.
16; 25; 36; : : : and since 3 is not among
1; 4; 9;p
these, 3 is not rational. x x 15 x C 15 xC3
7. false; C3D C D ¤ .
5 5 5 5 5
6. False; we cannot divide by 0.
b ab
8. True, because a D .
p c c
7. False, because 25 D 5, which is a positive
integer.
9. False; 2.x y/ D 2xy while
p .2x/ .2y/ D .2 2/ .x y/ D 4xy.
8. True; 2 is an irrational real number.
10. True; by the associative and commutative
9. False; we cannot divide by 0. properties, x.4y/ D .x 4/y D .4 x/y D 4xy.
10. False, we have 3 < < 4 so that is at best 11. distributive
rational and not an integer. It can be shown that
is irrational (but the details are not relevant for 12. the associative property of addition
our purposes).
13. associative
11. True; see Figure 0.1 in the text.
14. definition of division and commutative property
12. False, since the integer 0 is neither positive nor
15. commutative and distributive
negative.
16. associative
13. True; we can regard a terminating sequence as
being the same as the sequence obtained from it 17. the definition of division
by appending infinitely many zeros.
p 18. commutative
14. False; 13 is not a real number.
1
Copyright © 2017 Pearson Education, Inc. Publishing as Prentice Hall.
,Chapter 0: Review of Algebra ISM: Introductory Mathematical Analysis
19. distributive and commutative 37. . 1:6/. 0:5/ D .1:6/.0:5/ D 0:8
20. distributive 38. 19. 1/ D . 1/19 D .1 19/ D 19
1 a
21. 2x.y 7/ D .2x/y .2x/7 D 2xy .7/.2x/
39. 1
D 1 Da
D 2xy .7 2/x D 2xy 14x a
1
x 1 1 1 z 40. . 6 C x/ D . 6/ xD6 x
22. z D x zDx z Dx z D x where
y y y y y
we have used, in order, the definition of division, 41. 7.x/ D .7x/ D 7x
the associativity of multiplication, the
commutativity of multiplication, and the 42. 3.a b/ D 3a C 3b D 3.b a/
definition of division
43. Π6 C . y/ D . 6/ . y/ D 6 C y
23. (x C y/.2/ D 2.x C y/ D 2x C 2y
3 13 1
44. 3 .3a/ D D D
24. aŒb C .c C d/ D aŒ.b C c/ C d 3a 3a a
D aŒd C .b C c/ D aŒ.d C b/ C c 9 9 91 1
45. 9 . 27/ D D D D
27 27 93 3
25. xŒ.2y C 1/ C 3 D xŒ2y C .1 C 3/ D xŒ2y C 4
D x.2y/ C x.4/ D .x 2/y C 4x D .2x/y C 4x a a
46. . a/ . b/ D D
b b
D 2xy C 4x
9
26. .1 C a/.b C c/ D 1.b C c/ C a.b C c/ 47. 3 C .3 1 9/ D 3 C D3C3D6
3
D 1.b/ C 1.c/ C a.b/ C a.c/ D b C c C ab C ac
48. 3Œ 2.3/ C 6.2/ D 3Œ 6 C 12 D 3Œ6 D 18
27. .x y C z/w D ..x C . y// C z/w
D .x C . y//w C zw D xw C . y/w C zw 49. ( a/. b/. 1/ D ab. 1/ D ab
D xw C . yw/ C zw D xw yw C zw
50. . 12/. 12/ D .12/.12/ D 144
28. 2 C . 4/ D 6
51. X.1/ D X
29. aCbDb a
52. 71.x 2/ D 71x C 142 D 142 71x
30. 6 C . 4/ D 2
53. 4.5 C x/ D 4.5/ C 4.x/ D 20 C 4x
31. 7 2D5
54. .x y/ D xCyDy x
3
3 1 3 2 6
32. 1
D 1
D D D6 55. 0. x/ D 0
2 1 1 1
2
1 81 8
33. 5 . 13/ D 5 C 13 D 8 56. 8 D D
11 11 11
34. . a/ C . b/ D a b X
57. DX
1
35. . 2/.9/ D .2 9/ D 18
14x 27x 2x
36. 7. 9/ D .7 9/ D 63 58. D D
21y 37y 3y
2
Copyright © 2017 Pearson Education, Inc. Publishing as Prentice Hall.
,ISM: Introductory Mathematical Analysis Section 0.3
2x 2x 7
59. D D x 76. is not defined (we cannot divide by 0).
2 12 0
2 1 21 2 0
60. D D 77. D 0; for X ¤ 0
3 x 3x 3x X
a a.3b/ 3ab 0
61. .3b/ D D 78. is not defined (we cannot divide by 0).
c c c 0
62. 5a C .7 5a/ D 7 Problems 0.3
aby a by by 1. .23 /.22 / D 23C2 D 25 .D 32/
63. D D
ax ax x
2. x6 x9 D x6C9 D x15
a 1 a1 a
64. D D
b c bc bc 3. 175 172 D 175C2 D 177
2 5 25 10 4. z3 zz2 D z3C1C2 D z6
65. D D
x y xy xy
x3 x5 x3C5 x8
5. D D
1 1 3 2 3C2 5 y9 y5 y9C5 y14
66. C D C D D
2 3 6 6 6 6
6. .x12 /4 D x124 D x48
x y x 3y x C 3y
67. C D C D
3a a 3a 3a 3a .a3 /7 a37 a21
7. D D
.b4 /5 b45 b20
3 7 9 14 9 14 5
68. D D D !2
10 15 30 30 30 30
51 1 1314
D D 8. D .1314 1 2
/ D .131 /2
56 6 13
a c aCc D 1312 D 132 D 169
69. C D
b b b
9. .2x2 y3 /3 D 23 .x2 /3 .y3 /3 D 8x23 y33 D 8x6 y9
X Y X Y !2
70. p p D p
5 5 5 w2 s3 .w2 s3 /2 .w2 /2 .s3 /2
10. D D
y2 .y2 /2 y22
3 1 1 18 3 2
71. C D C w22 s32 w4 s6
2 4 6 12 12 12 D 4
D 4
y y
18 3 C 2 17
D D
12 12 x9
11. D x9 5 D x4
3 5 39 75 27 35 8 x5
72. D D D !6
7 9 63 63 63
2a4 .2a4 /6
12. D
6 x y 6y 7b5 .7b5 /6
73. x D6 D6 D
y y x x
26 .a4 /6
D
l 76 .b5 /6
w l m l 1 l
74. D D D 64a46
m w 1 w m wm D
117; 649b56
x
y2 x z x xy x2 64a24
75. z D D D D
xy y2 xy y2 z yz 117; 649b30
3
Copyright © 2017 Pearson Education, Inc. Publishing as Prentice Hall.
, Chapter 0: Review of Algebra ISM: Introductory Mathematical Analysis
0 12
.y3 /4 y34 y12 y12 2=5 5 !1=5
13. D D D D y12 8
D y4 243 3
.y2 /3 y2 y23 y2 y6 y2 y6C2 28. D@ A
1024 4
.x2 /3 .x3 /2 x23 x32 x6 x6 x12 2
14. D D 12 D 12 3 9
3
.x / 4 x34 x x D D
4 16
x12 12 D x0 D 1 p p p p p
29. 50 D 25 2 D 25 2 D 5 2
p
15. 25 D 5 p p p p p
30. 3
54 D 3
27 2 D 3
27 3 2 D 3 3 2
p
4 p p p p
16. 81 D 3 31.
3
2x3 D 3 3
2 x3 D x 3 2
p
7
p p p p
17. 128 D 2 32. 4x D 4 xD2 x
!1 p
1 33. 49u8 D 7u4
p
5 243 5 35 5
18. 0:00243 D D r p p
100000 105 x 4
x 4
x
34. 4
Dp 4
D
3 16 16 2
D D 0:3 p p p
10 35. 2 8 5 27 C 3 128
p p p
r p4
D 2 4p 2 5 9p 3 C 3p64 2
1 1 1
19. 4
Dp D D 2p2 2 p 5 3 3pC43 2
4
16 16 2 D 4 2 15 3 C 4 2 3
r r p
3 r r r p p
3 8 3 8 8 2 2 3 3 13 39 39 39
20. D D p
3
D D 36. D D D p D
27 27 27 3 3 13 13 13 132 13 2 13
p p p p p
21. .49/1=2 D 49 D 7 37. .9z4 /1=2 D 9z4 D 32 .z2 /2 D 32 .z2 /2
p D 3z2
22. .64/1=3 D 3
64 D 4
38. .729x6 /3=2 D .27x3 /3 D 19683x9
1 3
23. 813=4 D .81/ 4 D 33 D 27
!2=3 3 !2=3 2
27t3 3t 3t 9t2
1 1 1 1 39. D D D
24. .9/ 5=2 D D p 5 D 5 D 8 2 2 4
.9/5=2 9
3 243
3=4 4 ! 3=4 3
256 4 4
40. D D 3
2=5 1 1 1 1 x12 x3 x
25. .32/ D D p 2 D 2
D
.32/2=5 5
32
.2/ 4 4 3 4 3 x9 x9
D 3 3
D 9
D D
.x / x 43 64
1 1 1
26. .0:09/ 1=2 D Dp D a5 b 3 1 5 1 1 a5
.0:09/1=2 0:09 0:3 41. D a5 b 3
D a D
c2 c2 b3 c2 b3 c2
1 10
D 3
D
3 p x2=5 y3=5
10
42. 5
x2 y3 z 10 D x2=5 y3=5 z 10=5 D
z2
4=5 r !4 4
1 5 1 1 1 3
27. D D D 43. 3a 1b 2c 3 D
32 32 2 16
ab2 c3
4
Copyright © 2017 Pearson Education, Inc. Publishing as Prentice Hall.
