0.02e: Exercises - Whole number exponents
- Page ID
- 44370
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A: Zero and Negative Exponents
Exercise \(\PageIndex{1}\)
\( \bigstar \) Simplify. (Assume all variables represent nonzero numbers.)
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1. \(- 5 x ^ { 0 } \\[6pt]\) 2. \(3 x ^ { 2 } y ^ { 0 } \\[6pt]\) |
3. \(- 2 x ^ { - 3 } \\[6pt]\) 4. \(( - 2 x ) ^ { - 2 } \\[6pt]\) |
5. \(10 x ^ { - 3 } y ^ { 2 } \\[6pt]\) 6. \(- 3 x ^ { - 5 } y ^ { - 2 } \\[6pt]\) |
7. \(3 x ^ { - 2 } y ^ { 2 } z ^ { - 1 } \\[6pt]\) 8. \(- 5 x ^ { - 4 } y ^ { - 2 } z ^ { 2 } \\[6pt] \) |
- Answers to odd exercises.
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1. \(-5 \\[6pt]\) 3. \(- \displaystyle \frac { 2 } { x ^ { 3 } } \\[6pt]\) 5. \( \displaystyle\frac { 10 y ^ { 2 } } { x ^ { 3 } } \\[6pt]\) 7. \( \displaystyle \frac { 3 y ^ 2 } { x ^ { 2 } z } \\[6pt]\)
B: Product Rule
Exercise \(\PageIndex{2}\)
\( \bigstar \) Simplify. (Assume all variables represent nonzero numbers.)
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11. \(10 ^ { 4 } \cdot 10 ^ { 7 } \\[6pt] \\[6pt]\) 12. \(7 ^ { 3 } \cdot 7 ^ { 2 } \\[6pt]\) 13. \(x ^ { 3 } \cdot x ^ { 2 } \\[6pt]\) 14. \(y ^ { 5 } \cdot y ^ { 3 } \\[6pt]\) |
15. \(a ^ { 4 } \cdot a ^ { - 5 } \cdot a ^ { 2 } \\[6pt]\) 16. \(b ^ { - 8 } \cdot b ^ { 3 } \cdot b ^ { 4 } \\[6pt]\) 17. \(5 x ^ { 2 } y \cdot 3 x y ^ { 2 } \\[6pt]\) 18. \(- 10 x ^ { 3 } y ^ { 2 } \cdot 2 x y \\[6pt]\) |
19. \(- 6 x ^ { 2 } y z ^ { 3 } \cdot 3 x y z ^ { 4 } \\[6pt]\) 20. \(2 x y z ^ { 2 } \left( - 4 x ^ { 2 } y ^ { 2 } z \right) \\[6pt]\) 21. \(3 x ^ { n } y ^ { 2 n } \cdot 5 x ^ { 2 } y \\[6pt]\) 22. \(8 x ^ { 5 n } y ^ { n } \cdot 2 x ^ { 2 n } y \\[6pt]\) |
23. \(( 2 x + 3 ) ^ { 4 } ( 2 x + 3 ) ^ { 9 } \\[6pt]\) 24. \(( 3 y - 1 ) ^ { 7 } ( 3 y - 1 ) ^ { 2 } \\[6pt]\) 25. \(( a + b ) ^ { 3 } ( a + b ) ^ { 5 } \\[6pt]\) 26. \(( x - 2 y ) ^ { 7 } ( x - 2 y ) ^ { 3 } \\[6pt]\) |
- Answers to odd exercises.
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11. \(10^{11} \\[6pt]\)
13. \(x^{5} \\[6pt]\)
15. \(a \\[6pt]\)
17. \(15x^{3}y^{3} \\[6pt]\)
19. \(−18x^{3}y^{2}z^{7} \\[6pt]\)
21. \(15x^{n+2}y^{2n+1} \\[6pt]\)
23. \((2x + 3)^{13} \\[6pt]\)
25. \((a + b)^{8} \\[6pt]\)
C: Quotient Rule
Exercise \(\PageIndex{3}\)
\( \bigstar \) Simplify. (Assume all variables represent nonzero numbers.)
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31. \( \displaystyle \frac { 10 ^ { 2 } \cdot 10 ^ { 4 } } { 10 ^ { 5 } } \\[6pt]\) 32. \( \displaystyle \frac { 7 ^ { 5 } \cdot 7 ^ { 9 } } { 7 ^ { 2 } } \\[6pt]\) 33. \( \displaystyle \frac { a ^ { 8 } \cdot a ^ { 6 } } { a ^ { 5 } } \\[6pt]\) 34. \( \displaystyle \frac { b ^ { 4 } \cdot b ^ { 10 } } { b ^ { 8 } } \\[6pt]\) |
35. \( \displaystyle\frac { a ^ { 8 } \cdot a ^ { - 3 } } { a ^ { - 6 } } \\[6pt]\) 36. \( \displaystyle\frac { b ^ { - 10 } \cdot b ^ { 4 } } { b ^ { - 2 } } \\[6pt]\) 37. \( \displaystyle\frac { 25 x ^ { - 3 } y ^ { 2 } } { 5 x ^ { - 1 } y ^ { - 3 } } \\[6pt]\) 38. \( \displaystyle\frac { - 9 x ^ { - 1 } y ^ { 3 } z ^ { - 5 } } { 3 x ^ { - 2 } y ^ { 2 } z ^ { - 1 } } \\[6pt]\) |
39. \( \displaystyle \frac { 40 x ^ { 5 } y ^ { 3 } z } { 4 x ^ { 2 } y ^ { 2 } z } \\[6pt]\) 40. \( \displaystyle \frac { 8 x ^ { 2 } y ^ { 5 } z ^ { 3 } } { 16 x ^ { 2 } y z } \\[6pt]\) 41. \( \displaystyle \frac { 24 a ^ { 8 } b ^ { 3 } ( a - 5 b ) ^ { 10 } } { 8 a ^ { 5 } b ^ { 3 } ( a - 5 b ) ^ { 2 } } \\[6pt]\) 42. \( \displaystyle \frac { 175 m ^ { 9 } n ^ { 5 } ( m + n ) ^ { 7 } } { 25 m ^ { 8 } n ( m + n ) ^ { 3 } } \\[6pt]\) |
43. \( \displaystyle \frac { x ^ { 2 n } \cdot x ^ { 3 n } } { x ^ { n } } \\[6pt]\) 44. \( \displaystyle \frac { x ^ { n } \cdot x ^ { 8 n } } { x ^ { 3 n } } \\[6pt]\) |
- Answers to odd exercises.
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31. \(10 \\[6pt]\)
33. \(a^{9} \)
35. \(a^{11} \\[6pt]\)
37. \( \displaystyle\frac { 5 y ^ { 5 } } { x ^ { 2 } } \)
39. \(10x^{3}y \\[6pt]\)
41. \(3a^{3}(a − 5b)^{8} \)
43. \(x^{4n} \)
D: Power Rule for Products
Exercise \(\PageIndex{4}\)
\( \bigstar \) Simplify. (Assume all variables represent nonzero numbers.)
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51. \(\left( x ^ { 5 } \right) ^ { 3 } \\[6pt]\) 52. \(\left( y ^ { 4 } \right) ^ { 3 } \\[6pt]\) 53. \(\left( x ^ { 4 } y ^ { 5 } \right) ^ { 3 } \\[6pt]\) 54. \(\left( x ^ { 7 } y \right) ^ { 5 } \\[6pt]\) 55. \(( - 5 x ) ^ { 0 } \\[6pt]\) 56. \(\left( 3 x ^ { 2 } y \right) ^ { 0 } \) |
57. \(\left( x ^ { 2 } y ^ { 3 } z ^ { 4 } \right) ^ { 4 } \\[6pt]\) 58. \(\left( x y ^ { 2 } z ^ { 3 } \right) ^ { 2 } \\[6pt]\) 59. \(\left( - 5 x ^ { 2 } y z ^ { 3 } \right) ^ { 2 } \\[6pt]\) 60. \(\left( - 2 x y ^ { 3 } z ^ { 4 } \right) ^ { 5 } \\[6pt]\) 61. \(\left( x \cdot x ^ { 3 } \cdot x ^ { 2 } \right) ^ { 3 } \\[6pt]\) 62. \(\left( y ^ { 2 } \cdot y ^ { 5 } \cdot y \right) ^ { 2 } \) |
63. \(\left( - 2 x ^ { 4 } y ^ { 2 } z \right) ^ { 6 } \\[6pt]\) 64. \(\left( - 3 x y ^ { 4 } z ^ { 7 } \right) ^ { 5 } \\[6pt]\) 65. \(\left( - 2 a ^ { 2 } b ^ { 0 } c ^ { 3 } \right) ^ { 5 } \\[6pt]\) 66. \(\left( - 3 a ^ { 4 } b ^ { 2 } c ^ { 0 } \right) ^ { 4 } \\[6pt]\) 67. \(\left( - 5 x ^ { - 3 } y ^ { 2 } z \right) ^ { - 3 } \\[6pt]\) 68. \(\left( - 7 x ^ { 2 } y ^ { - 5 } z ^ { - 2 } \right) ^ { - 2 } \\[6pt]\) 69. \(\left( x ^ { 2 } y z ^ { 5 } \right) ^ { n } \) |
70. \(\left( x y ^ { 2 } z ^ { 3 } \right) ^ { 2 n } \\[6pt]\) 71. \( \displaystyle \frac { a ^ { 2 } \cdot \left( a ^ { 4 } \right) ^ { 2 } } { a ^ { 3 } } \\[6pt]\) 72. \( \displaystyle \frac { a \cdot a ^ { 3 } \cdot a ^ { 2 } } { \left( a ^ { 2 } \right) ^ { 3 } } \\[6pt]\) 73. \( \displaystyle \frac { \left( 9 x ^ { 3 } y ^ { 2 } z ^ { 0 } \right) ^ { 2 } } { 3 x y ^ { 2 } } \\[6pt]\) 74. \( \displaystyle \frac { \left( - 5 x ^ { 0 } y ^ { 5 } z \right) ^ { 3 } } { 25 y ^ { 2 } z ^ { 0 } } \) |
- Answers to odd exercises.
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51. \(x^{15} \\[6pt]\)
53. \(x^{12}y^{15} \\[6pt]\)
55. \(1 \)
57. \(x^{8}y^{12}z^{16} \\[6pt]\)
59. \(25x^{4}y^{2}z^{6} \\[6pt]\)
61. \(x^{18} \)
63. \(64x^{24}y^{12}z^{6} \)
65. \(- 32 a ^ { 10 } c ^ { 15 } \)
67. \(- \displaystyle\frac { x ^ { 9 } } { 125 y ^ { 6 } z ^ { 3 } } \)
69. \(x^{2n}y^{n}z^{5n} \\[6pt]\)
71. \(a^{7} \\[6pt]\)
73. \(27 x ^ { 5 } y ^ { 2 } \)
E: Power Rule for Quotients of Products
Exercise \(\PageIndex{5} \)
\( \bigstar \) Simplify. (Assume all variables represent nonzero numbers.)
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81. \(\left( \displaystyle \frac { - 3 a b ^ { 2 } } { 2 c ^ { 3 } } \right) ^ {3 } \\[6pt]\) 82. \(\left( \displaystyle \frac { - 10 a ^ { 3 } b } { 3 c ^ { 2 } } \right) ^ {2 } \\[6pt]\) 83. \(\left( \displaystyle \frac { - 2 x y ^ { 4 } } { z ^ { 3 } } \right) ^ {4 } \\[6pt]\) 84. \(\left( \displaystyle \frac { - 7 x ^ { 9 } y } { z ^ { 4 } } \right) ^ {3 } \) |
85. \(\left( \displaystyle \frac { 12 x ^ { 3 } y ^ { 2 } z } { 2 x ^ { 7 } y z ^ { 8 } } \right) ^ {3 } \\[6pt]\) 86. \(\left( \displaystyle \frac { 150 x y ^ { 8 } z ^ { 2 } } { 90 x ^ { 7 } y ^ { 2 } z } \right) ^ {2 } \\[6pt]\) 87. \(\left( \displaystyle \frac { 2 x ^ { - 3 } z } { y ^ { 2 } } \right) ^ {- 5 } \\[6pt]\) 88. \(\left( \displaystyle \frac { 5 x ^ { 5 } z ^ { - 2 } } { 2 y ^ { - 3 } } \right) ^ {- 3 } \) |
89. \(\left( \displaystyle \frac { x y ^ { 2 } } { z ^ { 3 } } \right) ^ {n } \\[6pt]\) 90. \(\left( \displaystyle \frac { 2 x ^ { 2 } y ^ { 3 } } { z } \right) ^ {n } \\[6pt]\) 91. \(\left( \displaystyle \frac { - 9 a ^ { - 3 } b ^ { 4 } c ^ { - 2 } } { 3 a ^ { 3 } b ^ { 5 } c ^ { - 7 } } \right) ^ {- 4 } \\[6pt]\) 92. \(\left( \displaystyle \frac { - 15 a ^ { 7 } b ^ { 5 } c ^ { - 8 } } { 3 a ^ { - 6 } b ^ { 2 } c ^ { 3 } } \right) ^ {- 3 } \) |
- Answers to odd exercises:
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81. \(- \displaystyle\frac { 27 a ^ { 3 } b ^ { 6 } } { 8 c ^ { 9 } } \\[6pt]\)
83. \( \displaystyle\frac { 16 x ^ { 4 } y ^ { 16 } } { z ^ { 12 } } \)
85. \( \displaystyle\frac { 216 y ^ { 3 } } { x ^ { 12 } z ^ { 21 } } \\[6pt]\)
87. \( \displaystyle\frac { x ^ { 15 } y ^ { 10 } } { 32 z ^ { 5 } } \)
89. \( \displaystyle\frac { x ^ { n } y ^ { 2 n } } { z ^ { 3 n } } \\[6pt]\)
91. \( \displaystyle\frac { a ^ { 24 } b ^ { 4 } } { 81 c ^ { 20 } } \)