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1.4e: Exercises - Radical Equations

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    A: Radical Equations (I)

    Exercise \(\PageIndex{A}\) 

    \( \bigstar \) Solve.

    1. \(\sqrt { x } = 7\)
    2. \(\sqrt { x } = 4\)
    3. \(\sqrt { x } + 8 = 9\)
    4. \(\sqrt { x } - 4 = 5\)
    5. \(\sqrt { x } + 7 = 4\)
    6. \(\sqrt { x } + 3 = 1\)
    7. \(5 \sqrt { x } - 1 = 0\)
    8. \(3 \sqrt { x } - 2 = 0\)
    1. \(\sqrt { 3 x + 1 } = 2\)
    2. \(\sqrt { 5 x - 4 } = 4\)
    3. \(\sqrt { 7 x + 4 } + 6 = 11\)
    4. \(\sqrt { 3 x - 5 } + 9 = 14\)
    5. \(2 \sqrt { x - 1 } - 3 = 0\)
    6. \(3 \sqrt { x + 1 } - 2 = 0\)
    7. \(\sqrt { x + 1 } = \sqrt { x } + 1\)
    8. \(\sqrt { 2 x - 1 } = \sqrt { 2 x } - 1\)
    1. \(\sqrt { 4 x - 1 } = 2 \sqrt { x } - 1\)
    2. \(\sqrt { 4 x - 11 } = 2 \sqrt { x } - 1\)
    3. \(\sqrt { x + 8 } = \sqrt { x } - 4\)
    4. \(\sqrt { 25 x - 1 } = 5 \sqrt { x } + 1\)
    5. \(\sqrt [ 3 ] { x } = 3\)
    6. \(\sqrt [ 3 ] { x } = - 4\)
    7. \(\sqrt [ 3 ] { 2 x + 9 } = 3\)
    1. \(\sqrt [ 3 ] { 4 x - 11 } = 1\)
    2. \(\sqrt [ 3 ] { 5 x + 7 } + 3 = 1\)
    3. \(\sqrt [ 3 ] { 3 x - 6 } + 5 = 2\)
    4. \(4 - 2 \sqrt [ 3 ] { x + 2 } = 0\)
    5. \(6 - 3 \sqrt [ 3 ] { 2 x - 3 } = 0\)
    6. \(\sqrt [ 5 ] { 3 ( x + 10 ) } = 2\)
    7. \(\sqrt [ 5 ] { 4 x + 3 } + 5 = 4\)
    Answers to odd exercises:

    1. \(49\)    3. \(1\)    5. \(Ø\)    7. \(\frac{1}{25}\)     9. \(1\)     11. \(3\)     13. \(\frac{13}{4}\)     15. \(0\)     17. \(\frac{1}{4}\)     19. \(Ø\)     21. \(27\)     23. \(9\)     25. \(−3\)     27. \(6\)     29. \(\frac{2}{3}\).

    B: Radical Equations (II)

    Exercise \(\PageIndex{B}\) 

    \( \bigstar \) Solve.

    1. \(\sqrt { 8 x + 11 } = 3 \sqrt { x + 1 }\)
    2. \(2 \sqrt { 3 x - 4 } = \sqrt { 2 ( 3 x + 1 ) }\)
    3. \(\sqrt { 2 ( x + 10 ) } = \sqrt { 7 x - 15 }\)
    1. \(\sqrt { 5 ( x - 4 ) } = \sqrt { x + 4 }\)
    2. \(\sqrt [ 3 ] { 5 x - 2 } = \sqrt [ 3 ] { 4 x }\)
    3. \(\sqrt [ 3 ] { 9 ( x - 1 ) } = \sqrt [ 3 ] { 3 ( x + 7 ) }\)
    1. \(\sqrt [ 3 ] { 3 x + 1 } = \sqrt [ 3 ] { 2 ( x - 1 ) }\)
    2. \(\sqrt [ 3 ] { 9 x } = \sqrt [ 3 ] { 3 ( x - 6 ) }\)
    1. \(\sqrt [ 5 ] { 3 x - 5 } = \sqrt [ 5 ] { 2 x + 8 }\)
    2. \(\sqrt [ 5 ] { x + 3 } = \sqrt [ 5 ] { 2 x + 5 }\)
    Answers to odd exercises:

    31. \(2\)     33. \(7\)     35. \(2\)     37. \(−3\)     39. \(13\).

    C: Radical Equations (III)

    Exercise \(\PageIndex{C}\) 

    \( \bigstar \) Solve.

    1. \(\sqrt { 4 x + 21 } = x\)
    2. \(\sqrt { 8 x + 9 } = x\)
    3. \(\sqrt { 4 ( 2 x - 3 ) } = x\)
    4. \(\sqrt { 3 ( 4 x - 9 ) } = x\)
    5. \(2 \sqrt { x - 1 } = x\)
    6. \(3 \sqrt { 2 x - 9 } = x\)
    7. \(\sqrt { 9 x + 9 } = x + 1\)
    8. \(\sqrt { 3 x + 10 } = x + 4\)
    9. \(\sqrt { x - 1 } = x - 3\)
    10. \(\sqrt { 2 x - 5 } = x - 4\)
    11. \(\sqrt { 16 - 3 x } = x - 6\)
    1. \(\sqrt { 7 - 3 x } = x - 3\)
    2. \(3 \sqrt { 2 x + 10 } = x + 9\)
    3. \(2 \sqrt { 2 x + 5 } = x + 4\)
    4. \(3 \sqrt { x - 1 } - 1 = x\)
    5. \(2 \sqrt { 2 x + 2 } - 1 = x\)
    6. \(\sqrt { 10 x + 41 } - 5 = x\)
    7. \(\sqrt { 6 ( x + 3 ) } - 3 = x\)
    8. \(\sqrt { 8 x ^ { 2 } - 4 x + 1 } = 2 x\)
    9. \(\sqrt { 18 x ^ { 2 } - 6 x + 1 } = 3 x\)
    10. \(5 \sqrt { x + 2 } = x + 8\)
    11. \(4 \sqrt { 2 ( x + 1 ) } = x + 7\)
    1. \(\sqrt { x ^ { 2 } - 25 } = x\)
    2. \(\sqrt { x ^ { 2 } + 9 } = x\)
    3. \(3 + \sqrt { 6 x - 11 } = x\)
    4. \(2 + \sqrt { 9 x - 8 } = x\)
    5. \(\sqrt { 4 x + 25 } - x = 7\)
    6. \(\sqrt { 8 x + 73 } - x = 10\)
    7. \(2 \sqrt { 4 x + 3 } - 3 = 2 x\)
    8. \(2 \sqrt { 6 x + 3 } - 3 = 3 x\)
    9. \(2 x - 4 = \sqrt { 14 - 10 x }\)
    10. \(3 x - 6 = \sqrt { 33 - 24 x }\)
    1. \(\sqrt [ 3 ] { x ^ { 2 } - 24 } = 1\)
    2. \(\sqrt [ 3 ] { x ^ { 2 } - 54 } = 3\)
    3. \(\sqrt [ 3 ] { x ^ { 2 } + 6 x } + 1 = 4\)
    4. \(\sqrt [ 3 ] { x ^ { 2 } + 2 x } + 5 = 7\)
    5. \(\sqrt [ 3 ] { 25 x ^ { 2 } - 10 x - 7 } = - 2\)
    6. \(\sqrt [ 3 ] { 9 x ^ { 2 } - 12 x - 23 } = - 3\)
    7. \(\sqrt [ 3 ] { 4 x ^ { 2 } - 1 } - 2 = 0\)
    8. \(4 \sqrt [ 3 ] { x ^ { 2 } } - 1 = 0\)
    9. \(\sqrt [ 5 ] { x ( 2 x + 1 ) } - 1 = 0\)
    10. \(\sqrt [ 5 ] { 3 x ^ { 2 } - 20 x } - 2 = 0\)
    Answers to odd exercises:

    41. \(7\)     43. \(2, 6\)     45. \(2\)     47. \(−1, 8\)     49. \(5\)     51. \(Ø\)     53. \(−3, 3\)     55. \(2, 5\)     57. \(−4, 4\)     59. \(\frac{1}{2}\)     61. \(2, 7\)     63. \(Ø\)     65. \(10\)     67. \(−6, −4\)     69. \(−\frac{1}{2}, \frac{3}{2}\)     71. \(Ø\)     73. \(−5, 5\)     75. \(−9, 3\)     77. \(\frac{1}{5}\)     79. \(− \frac{3}{2} ,\frac{ 3}{2}\)     81. \(−1, \frac{1}{2}\)

    D: Radical Equations (IV)

    Exercise \(\PageIndex{D}\) 

    \( \bigstar \) Solve.

    1. \(\sqrt { 2 x ^ { 2 } - 15 x + 25 } = \sqrt { ( x + 5 ) ( x - 5 ) }\)
    2. \(\sqrt { x ^ { 2 } - 4 x + 4 } = \sqrt { x ( 5 - x ) }\)
    3. \(\sqrt [ 3 ] { 2 \left( x ^ { 2 } + 3 x - 20 \right) } = \sqrt [ 3 ] { ( x + 3 ) ^ { 2 } }\)
    4. \(\sqrt [ 3 ] { 3 x ^ { 2 } + 3 x + 40 } = \sqrt [ 3 ] { ( x - 5 ) ^ { 2 } }\)
    5. \(\sqrt { 2 x - 5 } + \sqrt { 2 x } = 5\)
    6. \(\sqrt { 4 x + 13 } - 2 \sqrt { x } = 3\)
    1. \(\sqrt { 8 x + 17 } - 2 \sqrt { 2 - x } = 3\)
    2. \(\sqrt { 3 x - 6 } - \sqrt { 2 x - 3 } = 1\)
    3. \(\sqrt { 2 ( x - 2 ) } - \sqrt { x - 1 } = 1\)
    4. \(\sqrt { 2 x + 5 } - \sqrt { x + 3 } = 2\)
    5. \(\sqrt { 2 ( x + 1 ) } - \sqrt { 3 x + 4 } - 1 = 0\)
    1. \(\sqrt { 6 - 5 x } + \sqrt { 3 - 3 x } - 1 = 0\)
    2. \(\sqrt { x - 2 } - 1 = \sqrt { 2 ( x - 3 ) }\)
    3. \(\sqrt { 14 - 11 x } + \sqrt { 7 - 9 x } = 1\)
    4. \(\sqrt { x + 1 } = \sqrt { 3 } - \sqrt { 2 - x }\)
    5. \(\sqrt { 2 x + 9 } - \sqrt { x + 1 } = 2\)
    Answers to odd exercises:

    83. \(5, 10\)     85. \(−7, 7\)     87. \(\frac{9}{2}\)     89. \(1\)     91. \(10\)     93. \(Ø\)     95. \(3\)     97. \(-1, 2\).

    E: Radical Equations (V)

    Exercise \(\PageIndex{E}\) 

    \( \bigstar \) Solve.

    1. \(x ^ { 1 / 2 } - 10 = 0\)
    2. \(x ^ { 1 / 2 } - 6 = 0\)
    3. \(x ^ { 1 / 3 } + 2 = 0\)
    4. \(x ^ { 1 / 3 } + 4 = 0\)
    5. \(( x - 1 ) ^ { 1 / 2 } - 3 = 0\)
    1. \(( x + 2 ) ^ { 1 / 2 } - 6 = 0\)
    2. \(( 2 x - 1 ) ^ { 1 / 3 } + 3 = 0\)
    3. \(( 3 x - 1 ) ^ { 1 / 3 } - 2 = 0\)
    4. \(( 4 x + 15 ) ^ { 1 / 2 } - 2 x = 0\)
    5. \(( 3 x + 2 ) ^ { 1 / 2 } - 3 x = 0\)
    1. \(( 2 x + 12 ) ^ { 1 / 2 } - x = 6\)
    2. \(( 4 x + 36 ) ^ { 1 / 2 } - x = 9\)
    3. \(2 ( 5 x + 26 ) ^ { 1 / 2 } = x + 10\)
    4. \(3 ( x - 1 ) ^ { 1 / 2 } = x + 1\)
    5. \(x ^ { 1 / 2 } + ( 3 x - 2 ) ^ { 1 / 2 } = 2\)
    1. \(( 6 x + 1 ) ^ { 1 / 2 } - ( 3 x ) ^ { 1 / 2 } = 1\)
    2. \(( 3 x + 7 ) ^ { 1 / 2 } + ( x + 3 ) ^ { 1 / 2 } - 2 = 0\)
    3. \(( 3 x ) ^ { 1 / 2 } + ( x + 1 ) ^ { 1 / 2 } - 5 = 0\)
    4. \(\sqrt{3x+7} +\sqrt{x+1} = 2 \)
    5. \( \sqrt{2x+4} - \sqrt{x+3} =1\)
    6. \( \sqrt{2x}-\sqrt{x+1}=1 \)
    Answers to odd exercises:

    99. \(100\)     101. \(−8\)     103. \(10\)     105. \(−13\)     107. \(\frac{5}{2}\)     109. \(−6, −4\)     111. \(−2, 2\)     113. \(1\)     115. \(−2\)      117. {-1}     119. {8}

    \( \star\)

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