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P.4E: Exercícios - Expoentes Racionais


Estantes / Álgebra / Livro: _Advanced_Algebra_ (Redden) /05:_Radical_Functions_and_Equations/5.05:_Rational_Exponents
Estantes / Álgebra / Livro: _Intermediate_Algebra_ (OpenStax_Marecek) /08:_Roots_and_Radicals/8.4E:_Simplify_Rational_Exponents

A: Notação Radical para Exponencial

Exercício ( PageIndex {A} [5pt] ): Notação Radical para Exponencial

Expresse usando expoentes racionais.

  1. ( sqrt {10} [5pt] )
  2. ( sqrt {6} [5pt] )
  3. ( sqrt [3] {3} [5pt] )
  4. ( sqrt [4] {5} [5pt] )
  1. ( sqrt [3] {5 ^ {2}} [5pt] )
  2. ( sqrt [4] {2 ^ {3}} [5pt] )
  3. ( sqrt [3] {49} [5pt] )
  4. ( sqrt [3] {9} [5pt] )
  1. ( sqrt [5] {x} [5pt] )
  2. ( sqrt [6] {x} [5pt] )
  3. ( sqrt [6] {x ^ {7}} [5pt] )
  4. ( sqrt [5] {x ^ {4}} [5pt] )
  1. ( dfrac {1} { sqrt {x}} [5pt] )
  2. ( dfrac {1} { sqrt [3] {x ^ {2}}} [5pt] )
Respostas 1-13:
1. (10 ​​^ {1/2} [5pt] )
3. (3 ^ {1/3} [5pt] )
5. (5 ^ {2/3} [5pt] )
7. (7 ^ {2/3} [5pt] )
9. (x ^ {1/5} [5pt] )
11. (x ^ {7/6} [5pt] )
13. (x ^ {- 1/2} [5pt] )
( grande estrela )

B: Notação Exponencial para Radical.

Exercício ( PageIndex {B} [5pt] ): Exponencial para notação radical

Expresse de forma radical.

  1. (10 ​​^ {1/2} [5pt] )
  2. (11 ^ {1/3} [5pt] )
  3. (7 ^ {2/3} [5pt] )
  1. (2 ^ {3/5} [5pt] )
  2. (x ^ {3/4} [5pt] )
  3. (x ^ {5/6} [5pt] )
  1. (x ^ {- 1/2} [5pt] )
  2. (x ^ {- 3/4} [5pt] )
  3. ( left ( frac {1} {x} right) ^ {- 1/3} [5pt] )
  1. ( left ( frac {1} {x} right) ^ {- 3/5} [5pt] )
  2. ((2 x + 1) ^ {2/3} [5pt] )
  3. ((5 x - 1) ^ {1/2} [5pt] )
Respostas 15-25:
15. ( sqrt {10} [5pt] )
17. ( sqrt [3] {49} [5pt] )
19. ( sqrt [4] {x ^ {3}} [5pt] )
21. ( dfrac {1} { sqrt {x}} [5pt] )
23. ( sqrt [3] {x} [5pt] )
25. ( sqrt [3] {(2 x + 1) ^ {2}} [5pt] )
( grande estrela )

C: Forma Exponencial para Radical; então simplifique.

Exercício ( PageIndex {C} [5pt] ): Exponencial para a forma radical e depois simplificar

Escreva como um radical e depois simplifique.

  1. (64 ^ {1/2} [5pt] )
  2. (49 ^ {1/2} [5pt] )
  3. ( left ( frac {1} {4} right) ^ {1/2} [5pt] )
  4. ( left ( frac {4} {9} right) ^ {1/2} [5pt] )
  5. (4 ^ {- 1/2} [5pt] )
  6. (9 ^ {- 1/2} [5pt] )
  7. ( left ( frac {1} {4} right) ^ {- 1/2} [5pt] )
  8. ( left ( frac {1} {16} right) ^ {- 1/2} [5pt] )
  9. (8 ^ {1/3} [5pt] )
  1. (125 ^ {1/3} [5pt] )
  2. ( left ( frac {1} {27} right) ^ {1/3} [5pt] )
  3. ( left ( frac {8} {125} right) ^ {1/3} [5pt] )
  4. ((- 27) ^ {1/3} [5pt] )
  5. ((- 64) ^ {1/3} [5pt] )
  6. (16 ^ {1/4} [5pt] )
  7. (625 ^ {1/4} [5pt] )
  8. (81 ^ {- 1/4} [5pt] )
  9. (16 ^ {- 1/4} [5pt] )
  1. (100.000 ^ {1/5} [5pt] )
  2. ((- 32) ^ {1/5} [5pt] )
  3. ( left ( frac {1} {32} right) ^ {1/5} [5pt] )
  4. ( left ( frac {1} {243} right) ^ {1/5} [5pt] )
  5. (9 ^ {3/2} [5pt] )
  6. (4 ^ {3/2} [5pt] )
  7. (8 ^ {5/3} [5pt] )
  8. (27 ^ {2/3} [5pt] )
  1. (16 ^ {3/2} [5pt] )
  2. (32 ^ {2/5} [5pt] )
  3. ( left ( frac {1} {16} right) ^ {3/4} [5pt] )
  4. ( left ( frac {1} {81} right) ^ {3/4} [5pt] )
  5. ((- 27) ^ {2/3} [5pt] )
  6. ((- 27) ^ {4/3} [5pt] )
  7. ((- 32) ^ {3/5} [5pt] )
  8. ((- 32) ^ {4/5} [5pt] )
Respostas: 27-59
27. (8)
29. ( dfrac {1} {2} [5pt] )
31. ( dfrac {1} {2} [5pt] )
33. (2 [5pt] )
35. (2)
37. (8 [5pt] )
39. (- 3 [5pt] )
41. (2 [5pt] )
43. ( dfrac {1} {3} [5pt] )
45. (8 [5pt] )
47. ( dfrac {1} {2} [5pt] )
49. (27 [5pt] )
51. (32)
53. (64 [5pt] )
55. ( dfrac {1} {8} [5pt] )
57. (9 [5pt] )
59. (-8)
( grande estrela )

D: Operações exponenciais. PRODUTOS

Exercício ( PageIndex {D} [5pt] ): Operações Exponenciais

Execute as operações e simplifique. Deixe as respostas em formato exponencial.

  1. (5 ^ {3/2} cdot 5 ^ {1/2} [5pt] )
  2. (3 ^ {2/3} cdot 3 ^ {7/3} [5pt] )
  3. (5 ^ {1/2} cdot 5 ^ {1/3} [5pt] )
  4. (2 ^ {1/6} cdot 2 ^ {3/4} [5pt] )
  5. (y ^ {1/4} cdot y ^ {2/5} [5pt] )
  6. (x ^ {1/2} cdot x ^ {1/4} [5pt] )
  7. ((u ^ {12} v ^ {18}) ^ { tfrac {1} {6}} [5pt] )
  8. ((r ^ {9} s ^ {12}) ^ { tfrac {1} {3}} [5pt] )
  9. ( left (8 ^ {1/2} right) ^ {2/3} [5pt] )
  10. ( left (3 ^ {6} right) ^ {2/3} [5pt] )
  1. ( left (x ^ {2/3} right) ^ {1/2} [5pt] )
  2. ( left (y ^ {3/4} right) ^ {4/5} [5pt] )
  3. ( left (y ^ {8} right) ^ {- 1/2} [5pt] )
  4. ( left (y ^ {6} right) ^ {- 2/3} [5pt] )
  5. ( left (4 x ^ {2} y ^ {4} right) ^ {1/2} [5pt] )
  6. ( left (9 x ^ {6} y ^ {2} right) ^ {1/2} [5pt] )
  7. ( left (2 x ^ {1/3} y ^ {2/3} right) ^ {3} [5pt] )
  8. ( left (8 x ^ {3/2} y ^ {1/2} right) ^ {2} [5pt] )
  9. ( left (36 x ^ {4} y ^ {2} right) ^ {- 1/2} [5pt] )
  10. ( left (8 x ^ {3} y ^ {6} z ^ {- 3} right) ^ {- 1/3} [5pt] )
  1. ( left (27 q ^ { tfrac {3} {2}} right) ^ { tfrac {4} {3}} [5pt] )
  2. ( left (64 s ^ { tfrac {3} {7}} right) ^ { tfrac {1} {6}} [5pt] )
  3. ( left (a ^ { tfrac {1} {3}} b ^ { tfrac {2} {3}} right) ^ { tfrac {3} {2}} [5pt] )
  4. ( left (m ^ { tfrac {4} {3}} n ^ { tfrac {1} {2}} right) ^ { tfrac {3} {4}} [5pt] )
  5. ( left (16 u ^ { tfrac {1} {3}} right) ^ { tfrac {3} {4}} [5pt] )
  1. ( left (625 n ^ { tfrac {8} {3}} right) ^ { tfrac {3} {4}} [5pt] )
  2. ( left (4 p ^ { tfrac {1} {3}} q ^ { tfrac {1} {2}} right) ^ { tfrac {3} {2}} [5pt] )
  3. ( left (9 x ^ { tfrac {2} {5}} y ^ { tfrac {3} {5}} right) ^ { tfrac {5} {2}} [5pt] )
  4. ( left (16 x ^ {2} y ^ {- 1/3} z ^ {2/3} right) ^ {- 3/2} [5pt] )
  5. ( left (81 x ^ {8} y ^ {- 4/3} z ^ {- 4} right) ^ {- 3/4} [5pt] )
  6. ( left (100 a ^ {- 2/3} b ^ {4} c ^ {- 3/2} right) ^ {- 1/2} [5pt] )
  7. ( left (125 a ^ {9} b ^ {- 3/4} c ^ {- 1} right) ^ {- 1/3} [5pt] )
Respostas 61-91

61. (25 [5pt] )
63. (5 ^ {5/6} [5pt] )
65. (y ^ {13/20} [5pt] )
67. (U ^ {2} v ^ {3} [5pt] )

69. (2 [5pt] )
71. (x ^ {1/3} [5pt] )
73. ( dfrac {1} {y ^ {4}} [5pt] )
75. (2 x y ^ {2} [5pt] )
77. (8 x y ^ {2} [5pt] )
79. ( dfrac {1} {6 x ^ {2} y} [5pt] )
81. (81 q ^ {2} [5pt] )
83. (a ^ { tfrac {1} {2}} b )
85. (8 u ^ { tfrac {1} {4}} [5pt] )
87. (8 p ^ { tfrac {1} {2}} q ^ { tfrac {3} {4}} [5pt] )
89. ( dfrac {y ^ {1/2}} {64 x ^ {3} z} [5pt] )
91. ( dfrac {a ^ {1/3} b ^ {3/4}} {10 b ^ {2}} [5pt] )
( grande estrela )

D: Operações exponenciais. QUOCIENTES

Exercício ( PageIndex {D} [5pt] ): Operações Exponenciais

Execute as operações e simplifique. Deixe as respostas em formato exponencial.

  1. ( dfrac {5 ^ {11/3}} {5 ^ {2/3}} [5pt] )
  2. ( dfrac {2 ^ {9/2}} {2 ^ {1/2}} [5pt] )
  3. ( dfrac {2 a ^ {2/3}} {a ^ {1/6}} [5pt] )
  4. ( dfrac {3 b ^ {1/2}} {b ^ {1/3}} [5pt] )
  5. ( dfrac {r ^ { tfrac {5} {2}} cdot r ^ {- tfrac {1} {2}}} {r ^ {- tfrac {3} {2}}} [5pt] )
  6. ( dfrac {a ^ { tfrac {3} {4}} cdot a ^ {- tfrac {1} {4}}} {a ^ {- tfrac {10} {4}}} [5pt] )
  7. ( dfrac {c ^ { tfrac {5} {3}} cdot c ^ {- tfrac {1} {3}}} {c ^ {- tfrac {2} {3}}} [5pt] )
  1. ( dfrac {m ^ { tfrac {7} {4}} cdot m ^ {- tfrac {5} {4}}} {m ^ {- tfrac {2} {4}}} [5pt] )
  2. ( left ( dfrac {a ^ {3/4}} {a ^ {1/2}} right) ^ {4/3} [5pt] )
  3. ( left ( dfrac {b ^ {4/5}} {b ^ {1/10}} right) ^ {10/3} [5pt] )
  4. ( left ( dfrac {4 x ^ {2/3}} {y ^ {4}} right) ^ {1/2} [5pt] )
  5. ( left ( dfrac {27 x ^ {3/4}} {y ^ {9}} right) ^ {1/3} [5pt] )
  6. ( dfrac {y ^ {1/2} y ^ {2/3}} {y ^ {1/6}} [5pt] )
  7. ( dfrac {x ^ {2/5} x ^ {1/2}} {x ^ {1/10}} [5pt] )
  1. ( dfrac {x y} {x ^ {1/2} y ^ {1/3}} [5pt] )
  2. ( dfrac {x ^ {5/4} y} {x y ^ {2/5}} [5pt] )
  3. ( dfrac {49 a ^ {5/7} b ^ {3/2}} {7 a ^ {3/7} b ^ {1/4}} [5pt] )
  4. ( dfrac {16 a ^ {5/6} b ^ {5/4}} {8 a ^ {1/2} b ^ {2/3}} [5pt] )
  5. ( left ( dfrac {36 s ^ { tfrac {1} {5}} t ^ {- tfrac {3} {2}}} {s ^ {- tfrac {9} {5}} t ^ { tfrac {1} {2}}} right) ^ { tfrac {1} {2}} [5pt] )
  6. ( left ( dfrac {27 b ^ { tfrac {2} {3}} c ^ {- tfrac {5} {2}}} {b ^ {- tfrac {7} {3}} c ^ { tfrac {1} {2}}} right) ^ { tfrac {1} {3}} [5pt] )
  1. ( left ( dfrac {8 x ^ { tfrac {5} {3}} y ^ {- tfrac {1} {2}}} {27 x ^ {- tfrac {4} {3}} y ^ { tfrac {5} {2}}} right) ^ { tfrac {1} {3}} [5pt] )
  2. ( left ( dfrac {16 m ^ { tfrac {1} {5}} n ^ { tfrac {3} {2}}} {81 m ^ { tfrac {9} {5}} n ^ {- tfrac {1} {2}}} right) ^ { tfrac {1} {4}} [5pt] )
  3. ( dfrac { left (9 x ^ {2/3} y ^ {6} right) ^ {3/2}} {x ^ {1/2} y} [5pt] )
  4. ( dfrac { left (125 x ^ {3} y ^ {3/5} right) ^ {2/3}} {x y ^ {1/3}} [5pt] )
  5. ( dfrac { left (27 a ^ {1/4} b ^ {3/2} right) ^ {2/3}} {a ^ {1/6} b ^ {1/2}} [5pt] )
  6. ( dfrac { left (25 a ^ {2/3} b ^ {4/3} right) ^ {3/2}} {a ^ {1/6} b ^ {1/3}} [5pt] )
Respostas 101-125
101. (125 [5pt] )
103. (2 a ^ {1/2} [5pt] )
105.9a. (r ^ { frac {7} {2}} [5pt] )
107.1a. (c ^ {2} [5pt] )
109. (a ^ {1/3} [5pt] )
111. ( dfrac {2 x ^ {1/3}} {y ^ {2}} [5pt] )
113. (y )
115. (x ^ {1/2} y ^ {2/3} [5pt] )
117. (7 a ^ {2/7} b ^ {5/4} [5pt] )
119. ( dfrac {6 s} {t} [5pt] )
121. ( dfrac {2x} {3y} [5pt] )
123. (27 x ^ {1/2} y ^ {8} [5pt] )
125. (9 b ^ {1/2} [5pt] )
( grande estrela )

E: Operações de forma radical a exponencial.

Exercício ( PageIndex {E} [5pt] ): Operações de forma radical para exponencial

Reescreva de forma exponencial e, em seguida, execute as operações.

  1. ( sqrt [3] {9} cdot sqrt [5] {3} [5pt] )
  2. ( sqrt {5} cdot sqrt [5] {25} [5pt] )
  3. ( sqrt {x} cdot sqrt [3] {x} [5pt] )
  4. ( sqrt {y} cdot sqrt [4] {y} [5pt] )
  5. ( sqrt [3] {x ^ {2}} cdot sqrt [4] {x} [5pt] )
  6. ( sqrt [5] {x ^ {3}} cdot sqrt [3] {x} [5pt] )
  7. ( dfrac { sqrt [3] {100}} { sqrt {10}} [5pt] )
  1. ( dfrac { sqrt [5] {16}} { sqrt [3] {4}} [5pt] )
  2. ( dfrac { sqrt [3] {a ^ {2}}} { sqrt {a}} [5pt] )
  3. ( dfrac { sqrt [5] {b ^ {4}}} { sqrt [3] {b}} [5pt] )
  4. ( dfrac { sqrt [3] {x ^ {2}}} { sqrt [5] {x ^ {3}}} [5pt] )
  5. ( dfrac { sqrt [4] {x ^ {3}}} { sqrt [3] {x ^ {2}}} [5pt] )
  1. ( sqrt { sqrt [5] {16}} [5pt] )
  2. ( sqrt { sqrt [3] {9}} [5pt] )
  3. ( sqrt [3] { sqrt [5] {2}} [5pt] )
  4. ( sqrt [3] { sqrt [5] {5}} [5pt] )
  5. ( sqrt [3] { sqrt {7}} [5pt] )
  6. ( sqrt [3] { sqrt {3}} [5pt] )
Respostas 131-147:
131. ( sqrt [15] {3 ^ {13}} [5pt] )
133. ( sqrt [6] {x ^ {5}} [5pt] )
135. ( sqrt [12] {x ^ {11}} [5pt] )
137. ( sqrt [6] {10} [5pt] )
139. ( sqrt [6] {a} [5pt] )
141. ( sqrt [15] {x} [5pt] )
143. ( sqrt [5] {4} [5pt] )
145. ( sqrt [15] {2} [5pt] )
147. ( sqrt [6] {7} [5pt] )
( grande estrela )

.