Articles

P.4E: Exercises - Rational Exponents


Bookshelves/Algebra/Book:_Advanced_Algebra_(Redden)/05:_Radical_Functions_and_Equations/5.05:_Rational_Exponents
Bookshelves/Algebra/Book:_Intermediate_Algebra_(OpenStax_Marecek)/08:_Roots_and_Radicals/8.4E:_Simplify_Rational_Exponents

A: Radical to Exponential Notation

Exercise (PageIndex{A} [5pt]): Radical to Exponential Notation

Express using rational exponents.

  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])
Answers 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])
( igstar )

B: Exponential to Radical Notation.

Exercise (PageIndex{B} [5pt]): Exponential to Radical Notation

Express in radical form.

  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 } ight) ^ { - 1 / 3 } [5pt])
  1. (left( frac { 1 } { x } ight) ^ { - 3 / 5 } [5pt])
  2. (( 2 x + 1 ) ^ { 2 / 3 } [5pt])
  3. (( 5 x - 1 ) ^ { 1 / 2 } [5pt])
Answers 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])
( igstar )

C: Exponential to Radical Form; then Simplify.

Exercise (PageIndex{C} [5pt]): Exponential to Radical Form then Simplify

Write as a radical and then simplify.

  1. (64 ^ { 1 / 2 } [5pt])
  2. (49 ^ { 1 / 2 } [5pt])
  3. (left( frac { 1 } { 4 } ight) ^ { 1 / 2 } [5pt])
  4. (left( frac { 4 } { 9 } ight) ^ { 1 / 2 } [5pt])
  5. (4 ^ { - 1 / 2 } [5pt])
  6. (9 ^ { - 1 / 2 } [5pt])
  7. (left( frac { 1 } { 4 } ight) ^ { - 1 / 2 } [5pt])
  8. (left( frac { 1 } { 16 } ight) ^ { - 1 / 2 } [5pt])
  9. (8 ^ { 1 / 3 } [5pt])
  1. (125 ^ { 1 / 3 } [5pt])
  2. (left( frac { 1 } { 27 } ight) ^ { 1 / 3 } [5pt])
  3. (left( frac { 8 } { 125 } ight) ^ { 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 } ight) ^ { 1 / 5 } [5pt])
  4. (left( frac { 1 } { 243 } ight) ^ { 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 } ight) ^ { 3 / 4 } [5pt])
  4. (left( frac { 1 } { 81 } ight) ^ { 3 / 4 } [5pt])
  5. (( - 27 ) ^ { 2 / 3 } [5pt])
  6. (( - 27 ) ^ { 4 / 3 } [5pt])
  7. (( - 32 ) ^ { 3 / 5 } [5pt])
  8. (( - 32 ) ^ { 4 / 5 } [5pt])
Answers: 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)
( igstar )

D: Exponential Operations. PRODUCTS

Exercise (PageIndex{D} [5pt]): Exponential Operations

Perform the operations and simplify. Leave answers in exponential form.

  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})^{ frac{1}{6}} [5pt]) ​​​​​​
  8. ((r^{9}s^{12})^{ frac{1}{3}} [5pt])
  9. (left( 8 ^ { 1 / 2 } ight) ^ { 2 / 3 } [5pt])
  10. (left( 3 ^ { 6 } ight) ^ { 2 / 3 } [5pt])
  1. (left( x ^ { 2 / 3 } ight) ^ { 1 / 2 } [5pt])
  2. (left( y ^ { 3 / 4 } ight) ^ { 4 / 5 } [5pt])
  3. (left( y ^ { 8 } ight) ^ { - 1 / 2 } [5pt])
  4. (left( y ^ { 6 } ight) ^ { - 2 / 3 } [5pt])
  5. (left( 4 x ^ { 2 } y ^ { 4 } ight) ^ { 1 / 2 } [5pt])
  6. (left( 9 x ^ { 6 } y ^ { 2 } ight) ^ { 1 / 2 } [5pt])
  7. (left( 2 x ^ { 1 / 3 } y ^ { 2 / 3 } ight) ^ { 3 } [5pt])
  8. (left( 8 x ^ { 3 / 2 } y ^ { 1 / 2 } ight) ^ { 2 } [5pt])
  9. (left( 36 x ^ { 4 } y ^ { 2 } ight) ^ { - 1 / 2 } [5pt])
  10. (left( 8 x ^ { 3 } y ^ { 6 } z ^ { - 3 } ight) ^ { - 1 / 3 } [5pt])
  1. (left(27 q^{ frac{3}{2}} ight)^{ frac{4}{3}} [5pt])
  2. (left(64 s^{ frac{3}{7}} ight)^{ frac{1}{6}} [5pt])
  3. (left(a^{ frac{1}{3}} b^{ frac{2}{3}} ight)^{ frac{3}{2}} [5pt])
  4. (left(m^{ frac{4}{3}} n^{ frac{1}{2}} ight)^{ frac{3}{4}} [5pt])
  5. (left(16 u^{ frac{1}{3}} ight)^{ frac{3}{4}} [5pt])
  1. (left(625 n^{ frac{8}{3}} ight)^{ frac{3}{4}} [5pt])
  2. (left(4 p^{ frac{1}{3}} q^{ frac{1}{2}} ight)^{ frac{3}{2}} [5pt])
  3. (left(9 x^{ frac{2}{5}} y^{ frac{3}{5}} ight)^{ frac{5}{2}} [5pt])
  4. (left( 16 x ^ { 2 } y ^ { - 1 / 3 } z ^ { 2 / 3 } ight) ^ { - 3 / 2 } [5pt])
  5. (left( 81 x ^ { 8 } y ^ { - 4 / 3 } z ^ { - 4 } ight) ^ { - 3 / 4 } [5pt])
  6. (left( 100 a ^ { - 2 / 3 } b ^ { 4 } c ^ { - 3 / 2 } ight) ^ { - 1 / 2 } [5pt])
  7. (left( 125 a ^ { 9 } b ^ { - 3 / 4 } c ^ { - 1 } ight) ^ { - 1 / 3 } [5pt])
Answers 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^{ frac{1}{2}} b)
85. (8 u^{ frac{1}{4}} [5pt])
87. (8 p^{ frac{1}{2}} q^{ frac{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])
( igstar )

D: Exponential Operations. QUOTIENTS

Exercise (PageIndex{D} [5pt]): Exponential Operations

Perform the operations and simplify. Leave answers in exponential form.

  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^{ frac{5}{2}} cdot r^{- frac{1}{2}}}{r^{- frac{3}{2}}} [5pt])
  6. (dfrac{a^{ frac{3}{4}} cdot a^{- frac{1}{4}}}{a^{- frac{10}{4}}} [5pt])
  7. (dfrac{c^{ frac{5}{3}} cdot c^{- frac{1}{3}}}{c^{- frac{2}{3}}} [5pt])
  1. (dfrac{m^{ frac{7}{4}} cdot m^{- frac{5}{4}}}{m^{- frac{2}{4}}} [5pt])
  2. (left( dfrac { a ^ { 3 / 4 } } { a ^ { 1 / 2 } } ight) ^ { 4 / 3 } [5pt])
  3. (left( dfrac { b ^ { 4 / 5 } } { b ^ { 1 / 10 } } ight) ^ { 10 / 3 } [5pt])
  4. (left( dfrac { 4 x ^ { 2 / 3 } } { y ^ { 4 } } ight) ^ { 1 / 2 } [5pt])
  5. (left( dfrac { 27 x ^ { 3 / 4 } } { y ^ { 9 } } ight) ^ { 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^{ frac{1}{5}} t^{- frac{3}{2}}}{s^{- frac{9}{5}} t^{ frac{1}{2}}} ight)^{ frac{1}{2}} [5pt])
  6. (left(dfrac{27 b^{ frac{2}{3}} c^{- frac{5}{2}}}{b^{- frac{7}{3}} c^{ frac{1}{2}}} ight)^{ frac{1}{3}} [5pt])
  1. (left(dfrac{8 x^{ frac{5}{3}} y^{- frac{1}{2}}}{27 x^{- frac{4}{3}} y^{ frac{5}{2}}} ight)^{ frac{1}{3}} [5pt])
  2. (left(dfrac{16 m^{ frac{1}{5}} n^{ frac{3}{2}}}{81 m^{ frac{9}{5}} n^{- frac{1}{2}}} ight)^{ frac{1}{4}} [5pt])
  3. (dfrac { left( 9 x ^ { 2 / 3 } y ^ { 6 } ight) ^ { 3 / 2 } } { x ^ { 1 / 2 } y } [5pt])
  4. (dfrac { left( 125 x ^ { 3 } y ^ { 3 / 5 } ight) ^ { 2 / 3 } } { x y ^ { 1 / 3 } } [5pt])
  5. (dfrac { left( 27 a ^ { 1 / 4 } b ^ { 3 / 2 } ight) ^ { 2 / 3 } } { a ^ { 1 / 6 } b ^ { 1 / 2 } } [5pt])
  6. (dfrac { left( 25 a ^ { 2 / 3 } b ^ { 4 / 3 } ight) ^ { 3 / 2 } } { a ^ { 1 / 6 } b ^ { 1 / 3 } } [5pt])
Answers 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])
( igstar )

E: Radical to Exponential Form Operations.

Exercise (PageIndex{E} [5pt]): Radical to Exponential Form Operations

Rewrite in exponential form and then perform the operations.

  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])
Answers 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])
( igstar )

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Watch the video: Rewrite Using Rational Exponents (October 2021).