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Mathematics LibreTexts

1.5e: Exercises - Solve Equations with Rational Exponents

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    45458
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    A: Solve Equations with Rational Exponents I

    Exercise \(\PageIndex{A}\) 

    \( \bigstar \) 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. \(3 ( x - 1 ) ^ { 1 / 2 } = x + 1\)
    1. \(x ^ { 1 / 2 } + ( 3 x - 2 ) ^ { 1 / 2 } = 2\)
    2. \(( 6 x + 1 ) ^ { 1 / 2 } - ( 3 x ) ^ { 1 / 2 } = 1\)
    3. \(( 3 x + 7 ) ^ { 1 / 2 } + ( x + 3 ) ^ { 1 / 2 } - 2 = 0\)
    4. \(( 3 x ) ^ { 1 / 2 } + ( x + 1 ) ^ { 1 / 2 } - 5 = 0\)
    Answers to odd exercises:

    1. \( \{100\} \)     3. \( \{−8\} \)     5. \( \{10\} \)     7. \( \{−13\} \)     9. \( \{\frac{5}{2}\} \)     11. \( \{−6, −4\} \)     13. \( \{−2, 2\} \)     15. \( \{1\} \)     17. \( \{−2\} \).

    B: Solve Equations with Rational Exponents II

    Exercise \(\PageIndex{B}\) 

    \( \bigstar \) Solve

    1. \( (x-6)^\frac{5}{7}=32\)
    2. \( (x+3)^\frac{3}{7}+2=10\)
    3. \( (x-1)^\frac{7}{3}=128\)
    4. \( (x+2)^\frac{7}{5}+128=0\)
    5. \( (x-2)^\frac{3}{5} - 4=4\)
    6. \( (5x+7)^\frac{3}{5}=8 \)
    7. \( 8x^\frac{5}{3}-24=0\)
    8. \( (x+1)^\frac{5}{3}=32\)
    1. \( 3x^\frac{1}{3}+5=17 \)
    2. \( 7x^\frac{3}{7}+9=65 \)
    3. \( x^\frac{4}{3}+11=92\)
    4. \( x^\frac{2}{3}+1=65\)
    5. \( (x-4)^\frac{2}{3}=16\)
    6. \( (x+5)^\frac{2}{3}=4\)
    7. \( 8(3x-1)^\frac{2}{3}=200\)
    8. \( 5x^\frac{2}{3}=45 \)
    1. \( (7x−3)^\frac{2}{5}=4 \)
    2. \( (7x−8)^\frac{2}{3}=4(x−5)^\frac{2}{3} \)
    3. \( 3x^\frac{5}{2}=96\)
    4. \( x^\frac{3}{2}-2=6\)
    5. \( 2x^\frac{3}{2}=54 \)
    6. \( 6x^\frac{5}{2}-12=0\)
    7. \( 6x^\frac{3}{2}-141=1917\)
    8. \( 5x^\frac{3}{2}−3 = 4997 \)
    1. \( -2(x-5)^\frac{3}{4}+48=-202\)
    2. \( 2x^\frac{3}{4}=686 \)
    3. \( (4x+5)^\frac{1}{2}=x−4 \)
    4. \( x^3=(4x−3)^\frac{3}{2} \)
    5. \( x^\frac{5}{2}=16x^\frac{1}{2} \)
    6. \( (x^2-x-4)^\frac{3}{4}-2=6 \)
    7. \( (x^2-3x+3)^\frac{3}{2}-1=0 \)
    Answers to odd exercises:

    21. \( \{134\} \)     23. \( \{9\} \)     25. \( \{34\} \)     27. \( \ \{3^{\frac{3}{5}}\} \)     29. \( \{ 64 \}  \)     31. \( \{27, -27\} \)     33. \( \{−60, 68\} \)     35. \( \{42, \; -\frac{124}{3}\} \)     37. \( \{5, \frac{-29}{7}\} \)     39. \( \{4 \} \)     41. \( \{ 9 \} \)     43. \( \{49\} \)     45. \( \{630\} \)     47. \( \{11\} \)     49. \( \{ 0, 4, -4\} \)     51. \( \{1, 2\} \) .

    \( \star \)

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