Appendix 1: Homework Answers for Chapter 0

Section 0.1 Answers

  1. 6
  2. 0
  3. \dfrac{2}{21}
  4. \dfrac{19}{24}
  5. -\dfrac{1}{3}
  6. -1
  7. \dfrac{3}{5}
  8. 8
  9. -\dfrac{7}{8}
  10. Undefined
  11. 0
  12. Undefined
  13. \dfrac{23}{9}
  14. -\dfrac{4}{99}
  15. -\dfrac{24}{7}
  16. 0
  17. \dfrac{243}{32}
  18. \dfrac{13}{48}
  19. \dfrac{9}{22}
  20. \dfrac{25}{4}
  21. 5
  22. -3\sqrt{3}
  23. \dfrac{107}{27}
  24. -\dfrac{3\sqrt[5]{3}}{8} = -\dfrac{3^{6/5}}{8}
  25. \sqrt{10}
  26. \dfrac{\sqrt{61}}{2}
  27. \sqrt{7}
  28. \dfrac{-4 + \sqrt{2}}{7}
  29. -1
  30. 2 + \sqrt{5}
  31. \dfrac{15}{16}
  32. 13
  33. -\dfrac{385}{12}
  34. 1.38 \times 10^{10237}

Section 0.2 Answers

  1. 3|x|
  2. 2t
  3. 5|y^3|\sqrt{2}
  4. |2t+1|
  5. |w-8|
  6. \sqrt{3x+1}
  7. \dfrac{\sqrt{c^2-v^2}}{|c|}
  8. \dfrac{2r \sqrt[3]{3 \pi r^2}}{L}
  9. \dfrac{2 \varepsilon^2 \sqrt[4]{2\pi}}{|\rho^3|}
  10. -\dfrac{1}{\sqrt{x}}
  11. \dfrac{3-6t^2}{\sqrt{1-t^2}}
  12. \dfrac{6-8z}{3 (\sqrt[3]{1-z})^2}
  13. \dfrac{4x-3}{(2x-1)\sqrt[3]{2x-1}}

Section 0.3 Answers

  1. 2x(1 - 5x)
  2. 4t^3(3t^2-2)
  3. 4xy(4y-3x)
  4. -(m+3)^2(4m+7)
  5. (2x-1)(x-1)
  6. (t-5)(t^2+1)
  7. (w-11)(w+11)
  8. (7-2t)(7+2t)
  9. (3t-2)(3t+2)(9t^2+4)
  10. (3z-8y^2)(3z+8y^2)
  11. -3(y - 3)(y+1)
  12. (x+h)(x+h-1)(x+h+1)
  13. (y-12)^2
  14. (5t+1)^2
  15. 3x(2x-3)^2
  16. (m^2+5)^2
  17. (3-2x)(9 + 6x + 4x^2)
  18. t^3(t+1)(t^2 - t + 1)
  19. (x-7)(x+2)
  20. (y-9)(y-3)
  21. (3t+1)(t+5)
  22. (2x-5)(3x-4)
  23. (7-m)(5+m)
  24. (-2w+1)(w-3)
  25. 3m(m-1)(m+4)
  26. (x-2)(x+2)(x^2+5)
  27. (2t-3)(2t+3)(t^2+1)
  28. (x-3)(x+3)(x-5)
  29. (t-3)(1-t)(1+t)
  30. (y^2-y+3)(y^2+y+3)

Section 0.4 Answers

  1. O is the odd natural numbers
  2. X = \{ 0, 1, 4, 9, 16, \ldots \}
  3. a. \dfrac{20}{10} = 2 and 117
    b. \sqrt{3} and 5.2020020002
    c. \left\{ -3, \dfrac{20}{10}, 117\right\}
    d. \left\{ -3, -1.02, -\dfrac{3}{5}, 0.57, 1.\overline{23},\dfrac{20}{10}, 117 \right \}
  4. Completed Table
    A chart of the answers to exercise 4. The first column of the table is the subsets of real numbers using set notation, the second column is the corresponding interval notation, and the last column is the region on the real number line corresponding to the subset of real numbers.
  5. (-1,5] \cap [0,8) = [0,5]
  6. (-1,1) \cup [0,6] = (-1,6]
  7. (-\infty,4]\cap (0,\infty) = (0,4]
  8. (-\infty,0) \cap [1,5] = \emptyset
  9. (-\infty, 0) \cup [1,5] = (-\infty,0) \cup [1,5]
  10. (-\infty, 5] \cap [5,8) = \left\{ 5\right\}
  11. (-\infty, 5) \cup (5, \infty)
  12. (-\infty, -1) \cup (-1, \infty)
  13. (-\infty, -3) \cup (-3, 4)\cup (4, \infty)
  14. (-\infty, 0) \cup (0, 2)\cup (2, \infty)
  15. (-\infty, -2) \cup (-2, 2)\cup (2, \infty)
  16. (-\infty, -4) \cup (-4, 0) \cup (0, 4) \cup (4, \infty)
  17. (-\infty, -1] \cup [1, \infty)
  18. [2, 3)
  19. (-\infty, -3] \cup (0, \infty)
  20. \emptyset
  21. \{-1\} \cup \{1\} \cup (2, \infty)
  22. (3,4) \cup (4, 13)
  23. A \cup C
    A three circle Venn Diagram with the outside box labeled with a U. The three circles are A, B, and C. All of circles A and B are shaded.
  24. B \cap C
    A three circle Venn Diagram with outside rectangle of U. The circles are labeled A, B, and C. The region where B and C overlap is shaded.
  25. (A \cup B) \cup C
    A three circle Venn Diagram with outside rectangle U. The circles are labeled A, B, and C. All three circles are shaded.
  26. (A \cap B) \cap C
    A three circle Venn Diagram with outside rectangle U. The circles are labeled A, B, and C. The mostly triangular region where all three circles overlap is shaded.
  27. A \cap (B \cup C)
    A three circle Venn Diagram with outside rectangle U. The circles are labeled A, B, and C. The region where A and B overlap is shaded as well as the region A and C overlap is shaded.
  28. (A \cap B) \cup (A \cap C)
    A three circle Venn Diagram with outside rectangle U. The circles are labeled A, B, and C. The region where A and B overlap is shaded as well as the region A and C overlap is shaded.
  29. Yes, A \cup (B \cap C) = (A \cup B) \cap (A \cup C)
    A three circle Venn Diagram with outside rectangle U. The circles are labeled A, B, and C. All of circle A is shaded as well as the region where circles B and C overlap is shaded.

Section 0.5 Answers

  1. x = \dfrac{18}{7}
  2. t = -\dfrac{1}{30}
  3. w = \dfrac{61}{33}
  4. y = 50000
  5. All real numbers
  6. No solution
  7. t = -\dfrac{5}{3\sqrt{7}} = -\dfrac{5\sqrt{7}}{21}
  8. y = \dfrac{6}{17\sqrt{2}} = \dfrac{3 \sqrt{2}}{17}
  9. x = \dfrac{27}{18+\sqrt{7}}
  10. y = \dfrac{4 - 3x}{2} or y = -\dfrac{3}{2}x + 2
  11. x = \dfrac{4 - 2y}{3} or x = -\dfrac{2}{3} y + \dfrac{4}{3}
  12. C = \dfrac{5}{9}(F - 32) or C = \dfrac{5}{9} F - \dfrac{160}{9}
  13. x = \dfrac{p - 15}{-2.5} = \dfrac{15-p}{2.5} or x = -\dfrac{2}{5} p + 6
  14. x = \dfrac{C - 1000}{200} or x = \dfrac{1}{200} C - 5
  15. y = \dfrac{x-7}{4} or y = \dfrac{1}{4} x - \dfrac{7}{4}
  16. w = \dfrac{3v+1}{v}, provided v \neq 0
  17. v = \dfrac{1}{w-3}, provided w \neq 3
  18. y = \dfrac{3x+1}{x-2}, provided x \neq 2
  19. \pi = \dfrac{C}{2r}, provided r \neq 0
  20. V = \dfrac{nRT}{P}, provided P \neq 0
  21. R = \dfrac{PV}{nT}, provided n \neq 0, T \neq 0
  22. g = \dfrac{E}{mh}, provided m \neq 0, h \neq 0
  23. m = \dfrac{2E}{v^2}, provided v^2 \neq 0 (so v \neq 0)
  24. V_{2} = \dfrac{P_{1}V_{1}}{P_{2}}, provided P_{2} \neq 0
  25. t = \dfrac{x - x_{0}}{a}, provided a \neq 0
  26. x = \dfrac{y-y_{0} + mx_{0}}{m} or x = x_{0} + \dfrac{y-y_{0}}{m}, provided m \neq 0
  27. T_{1} = \dfrac{mcT_{2} - q}{mc} or T_{1} = T_{2} - \dfrac{q}{mc}, provided m \neq 0, c \neq 0
  28. x = -6 or x=6
  29. t = -3 or t= \dfrac{11}{3}
  30. w = -3 or w= 11
  31. y = -1 or y= 1
  32. m=-\dfrac{1}{2} or m= \dfrac{1}{10}
  33. No solution
  34. x=-3 or x= 3
  35. w = -\dfrac{13}{8} or w= \dfrac{53}{8}
  36. t = \dfrac{\sqrt{2} \pm 2}{3}
  37. v = -1 or v = 0
  38. No solution
  39. y = \dfrac{3}{2}
  40. t = -1 or t = 9
  41. x = -\dfrac{1}{7} or x = 1
  42. y = 0 or y = \dfrac{2}{\sqrt{2} - 1}
  43. x=1
  44. z = -\dfrac{3}{10}
  45. w = \dfrac{\sqrt{3} \pm 2}{\sqrt{3} \mp 2} See footnote[1]
  46. x = -\dfrac{3}{7} or x = 5
  47. t = \dfrac{1}{2} or t = -4
  48. y = \dfrac{5}{3} or y = -2
  49. t = 0 or t = 4
  50. y = -1 or y = \dfrac{3}{2}
  51. x = \dfrac{3}{2} or x = \dfrac{7}{4}
  52. x = 0 or x = \pm \dfrac{3}{4}
  53. w = -\dfrac{5}{2} or w = \dfrac{2}{3}
  54. w=-5 or w = -\dfrac{1}{2}
  55. x=3 or x = \pm 4
  56. t = -1, t= -\dfrac{1}{2}, or t = 0
  57. a = \pm 1
  58. t = -\dfrac{3}{4} or t = \dfrac{3}{2}
  59. x = \dfrac{2}{3}
  60. y = \pm 3
  61. x = -\dfrac{3}{2}
  62. y = -1, 2
  63. t = -\dfrac{\sqrt[3]{3}}{2}
  64. x = 5
  65. t = \pm 3 \sqrt{7}
  66. x=3
  67. y=-3
  68. t = -\dfrac{1}{3}, \dfrac{2}{3}
  69. x = \dfrac{5 + \sqrt{57}}{8}
  70. w = \sqrt{3}
  71. x = 6
  72. x = 4
  73. h = \sqrt[3]{\dfrac{12I}{b}}
  74. a = \dfrac{2 \sqrt[4]{I_{0}}}{\sqrt[4]{5\sqrt{3}}}
  75. g = \dfrac{4 \pi^2 L}{T^2}
  76. v = \dfrac{c \sqrt{L_{0}^2 - L^2}}{L_{0}}
  77. x = \dfrac{3 \pm \sqrt{5}}{6}
  78. t = -\dfrac{4}{5}, -\dfrac{2}{5}
  79. y = \pm 1, \pm \sqrt{5}
  80. x = \dfrac{-1 \pm \sqrt{5}}{2}
  81. w = -1, \dfrac{2}{3}
  82. y = -2 \pm \sqrt{5}
  83. z = \dfrac{1 \pm \sqrt{65}}{16}
  84. v = -3, 1
  85. No real solution
  86. t = \dfrac{-5 \pm \sqrt{33}}{4}
  87. x = 0
  88. y = \dfrac{2 \pm \sqrt{10}}{6}
  89. w = \pm \sqrt{\dfrac{\sqrt{13} - 3}{2}}
  90. x = \pm 1\
  91. y = \dfrac{4 \pm \sqrt{6 + 2 \sqrt{13}}}{2}
  92. x = 0, \dfrac{5 \pm \sqrt{17}}{2}
  93. p = -\dfrac{1}{3}, \pm \sqrt{2}
  94. v = 0, \pm \sqrt{2}, \pm \sqrt{5}
  95. y = \dfrac{5\sqrt{2} \pm \sqrt{46}}{2}
  96. x = \dfrac{\sqrt{2} \pm \sqrt{10}}{2}
  97. v = -\dfrac{\sqrt{3}}{2}, 2\sqrt{3}
  98. b = \pm \dfrac{\sqrt{13271}}{50}
  99. r = \pm \sqrt{\dfrac{37}{\pi}}
  100. r = \dfrac{-4\sqrt{2} \pm \sqrt{54\pi + 32}}{\pi}
  101. t = \dfrac{500 \pm 10\sqrt{491}}{49}
  102. x = \dfrac{99 \pm 6 \sqrt{165}}{13}
  103. A = \dfrac{-107 \pm 7 \sqrt{70}}{330}
  104. x = 1, 2, \dfrac{3 \pm \sqrt{17}}{2}
  105. x = \pm 1, 2 \pm \sqrt{3}
  106. x = -\dfrac{1}{2}, 1, 7
  107. The discriminant is: D = p^2 - 4p^2 = -3p^2 < 0. because D < 0, there are no real solutions.
  108. t = \dfrac{v \pm \sqrt{v^2 + 2gh}}{g}
  109. 7i
  110. 3i
  111. -10
  112. 10
  113. -12
  114. 12
  115. 3
  116. -3i
  117. i^{5} = i^{4} \cdot i = 1 \cdot i = i
  118. i ^{6} = i^{4} \cdot i^{2} = 1 \cdot (-1) = -1
  119. i^{7} = i^{4} \cdot i^{3} = 1 \cdot (-i) = -i
  120. i^{8} = i^{4} \cdot i^{4} = \left(i^{4}\right)^{2} = (1)^{2} =1
  121. i^{15} = \left(i^{4}\right)^{3} \cdot i^{3} = 1 \cdot (-i) = -i
  122. i ^{26} = \left(i^{4}\right)^{6} \cdot i^{2} = 1\cdot (-1) = -1
  123. i^{117} = \left(i^{4}\right)^{29} \cdot i = 1 \cdot i = i
  124. i ^{304} = \left(i^{4}\right)^{76} = 1^{76} = 1
  125. For z = 2+3i and w = 4i
    • z+w = 2+7i
    • zw = -12+8i
    • z^2 = -5 + 12i
    • \frac{1}{z} = \frac{2}{13} - \frac{3}{13} \, i
    • \frac{z}{w} = \frac{3}{4} - \frac{1}{2} \, i
    • \frac{w}{z} = \frac{12}{13} + \frac{8}{13} \,i
  126. For z = 1+i and w = -i
    • z+w = 1
    • zw = 1-i
    • z^2 = 2i
    • \frac{1}{z} = \frac{1}{2} - \frac{1}{2} \, i
    • \frac{z}{w} = -1+i
    • \frac{w}{z} = -\frac{1}{2} - \frac{1}{2} \, i
  127. For z = i and w = -1+2i
    • z+w = -1+3i
    • zw = -2-i
    • z^2 = -1
    • \frac{1}{z} = -i
    • \frac{z}{w} = \frac{2}{5} - \frac{1}{5} \, i
    • \frac{w}{z} = 2+i
  128. For z = 4i and w = 2-2i
    • z+w = 2+2i
    • zw = 8+8i
    • z^2 = -16
    • \frac{1}{z} = -\frac{1}{4} \,i
    • \frac{z}{w} = -1+i
    • \frac{w}{z} = -\frac{1}{2} - \frac{1}{2} \,i
  129. For z = 3-5i and w = 2+7i
    • z+w = 5+2i
    • zw = 41+11i
    • z^2 = -16-30i
    • \frac{1}{z} = \frac{3}{34} + \frac{5}{34} \,i
    • \frac{z}{w} = -\frac{29}{53} - \frac{31}{53} \, i
    • \frac{w}{z} = -\frac{29}{34} + \frac{31}{34} \,i
  130. For z = -5+i and w = 4+2i
    • z+w = -1+3i
    • zw = -22-6i
    • z^2 = 24-10i
    • \frac{1}{z} = -\frac{5}{26} - \frac{1}{26} \,i
    • \frac{z}{w} = -\frac{9}{10} + \frac{7}{10} \, i
    • \frac{w}{z} = -\frac{9}{13} - \frac{7}{13} \,i
  131. For z = \sqrt{2} - i\sqrt{2} and w = \sqrt{2} + i\sqrt{2}
    • z+w = 2\sqrt{2}
    • zw = 4
    • z^2 = -4i
    • \frac{1}{z} = \frac{\sqrt{2}}{4} + \frac{\sqrt{2}}{4} \,i
    • \frac{z}{w} = -i
    • \frac{w}{z} = i
  132. For z = 1 - i\sqrt{3} and w = -1-i\sqrt{3}
    • z+w = -2i\sqrt{3}
    • zw = -4
    • z^2 = -2-2i\sqrt{3}
    • \frac{1}{z} = \frac{1}{4} + \frac{\sqrt{3}}{4} \,i
    • \frac{z}{w} = \frac{1}{2} + \frac{\sqrt{3}}{2} \,i
    • \frac{w}{z} = \frac{1}{2} - \frac{\sqrt{3}}{2} \,i
  133. For z = \frac{1}{2} + \frac{\sqrt{3}}{2} \, i and w = -\frac{1}{2} + \frac{\sqrt{3}}{2} \,i
    • z+w = i\sqrt{3}
    • zw = -1
    • z^2 = -\frac{1}{2} + \frac{\sqrt{3}}{2} \,i
    • \frac{1}{z} = \frac{1}{2} - \frac{\sqrt{3}}{2} \, i
    • \frac{z}{w} = \frac{1}{2} - \frac{\sqrt{3}}{2} \, i
    • \frac{w}{z} = \frac{1}{2} + \frac{\sqrt{3}}{2} \, i
  134. For z = -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \, i and w = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \, i
    • z + w = -\sqrt{2}
    • zw = 1
    • z^2 =-i
    • \frac{1}{z} = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \, i
    • \frac{z}{w} = -i
    • \frac{w}{z} = i
  135. x = \dfrac{2 \pm i\sqrt{14}}{3}
  136. t = 5, \pm \dfrac{i \sqrt{3}}{3}
  137. y = \pm 2, \pm i
  138. w = \dfrac{1 \pm i \sqrt{7}}{2}
  139. y = \pm \dfrac{3i \sqrt{2}}{2}
  140. x= 0, \dfrac{1 \pm i\sqrt{2}}{3}
  141. x = \dfrac{\sqrt{5} \pm i\sqrt{3}}{2}
  142. y = \pm i, \pm \dfrac{i\sqrt{2}}{2}
  143. z = \pm 2, \pm 2i

Section 0.6 Answers

  1. \left(-\infty, \dfrac{3}{4}\right]
  2. \left(-\infty, \dfrac{7}{6} \right)
  3. \left( -\infty, \dfrac{3}{13}\right]
  4. \left(-\dfrac{4}{3}, \infty\right)
  5. No solution
  6. (-\infty, \infty)
  7. (4, \infty)
  8. \left[ \dfrac{7}{2 - \sqrt[3]{18}}, \infty\right)
  9. [0, \infty)
  10. \left[ \dfrac{1}{2}, \dfrac{7}{10}\right]
  11. \left(-\dfrac{23}{6}, \dfrac{19}{2} \right]
  12. \left(-\dfrac{13}{10}, -\dfrac{7}{10} \right]
  13. (-4,1]
  14. \{1 \} = [1,1]
  15. \left[-6, \dfrac{18}{19} \right)
  16. (-\infty, -1] \cup [0, \infty)
  17. (-\infty, -7) \cup [4, \infty)
  18. (-\infty, \infty)
  19. \left[\dfrac{1}{3}, 3\right]
  20. \left(-\infty, -\dfrac{12}{7} \right) \cup \left(\dfrac{8}{7}, \infty\right)
  21. (-3,2)
  22. (-\infty,1] \cup [3,\infty)
  23. No solution
  24. (-\infty, \infty)
  25. (-\infty, -6-\sqrt{5}) \cup (6-\sqrt{5}, \infty)
  26. \left[ -\dfrac{3}{4}, \dfrac{3}{4}\right]
  27. No solution
  28. (-3,2] \cup [6,11)
  29. [3, 4) \cup (5, 6]
  30. \left(\dfrac{2 \sqrt{3} - 3}{2}, \dfrac{2 \sqrt{3} - 1}{2} \right) \cup \left(\dfrac{2 \sqrt{3} +1}{2}, \dfrac{2 \sqrt{3} +3}{2} \right)

 

  1. That is, w = \dfrac{\sqrt{3} + 2}{\sqrt{3} - 2} or w = \dfrac{\sqrt{3} - 2}{\sqrt{3} + 2} ↵

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