1.5e: Exercises - Solve Equations with Rational Exponents

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A: Solve Equations with Rational Exponents I

Exercise \(\PageIndex{A}\)

\( \bigstar \) Solve.

\(x ^ { 1 / 2 } - 10 = 0\)
\(x ^ { 1 / 2 } - 6 = 0\)
\(x ^ { 1 / 3 } + 2 = 0\)
\(x ^ { 1 / 3 } + 4 = 0\)
\(( x - 1 ) ^ { 1 / 2 } - 3 = 0\)

\(( x + 2 ) ^ { 1 / 2 } - 6 = 0\)
\(( 2 x - 1 ) ^ { 1 / 3 } + 3 = 0\)
\(( 3 x - 1 ) ^ { 1 / 3 } - 2 = 0\)
\(( 4 x + 15 ) ^ { 1 / 2 } - 2 x = 0\)
\(( 3 x + 2 ) ^ { 1 / 2 } - 3 x = 0\)

\(( 2 x + 12 ) ^ { 1 / 2 } - x = 6\)
\(( 4 x + 36 ) ^ { 1 / 2 } - x = 9\)
\(2 ( 5 x + 26 ) ^ { 1 / 2 } = x + 10\)
\(3 ( x - 1 ) ^ { 1 / 2 } = x + 1\)

\(x ^ { 1 / 2 } + ( 3 x - 2 ) ^ { 1 / 2 } = 2\)
\(( 6 x + 1 ) ^ { 1 / 2 } - ( 3 x ) ^ { 1 / 2 } = 1\)
\(( 3 x + 7 ) ^ { 1 / 2 } + ( x + 3 ) ^ { 1 / 2 } - 2 = 0\)
\(( 3 x ) ^ { 1 / 2 } + ( x + 1 ) ^ { 1 / 2 } - 5 = 0\)

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

\( (x-6)^\frac{5}{7}=32\)
\( (x+3)^\frac{3}{7}+2=10\)
\( (x-1)^\frac{7}{3}=128\)
\( (x+2)^\frac{7}{5}+128=0\)
\( (x-2)^\frac{3}{5} - 4=4\)
\( (5x+7)^\frac{3}{5}=8 \)
\( 8x^\frac{5}{3}-24=0\)
\( (x+1)^\frac{5}{3}=32\)

\( 3x^\frac{1}{3}+5=17 \)
\( 7x^\frac{3}{7}+9=65 \)
\( x^\frac{4}{3}+11=92\)
\( x^\frac{2}{3}+1=65\)
\( (x-4)^\frac{2}{3}=16\)
\( (x+5)^\frac{2}{3}=4\)
\( 8(3x-1)^\frac{2}{3}=200\)
\( 5x^\frac{2}{3}=45 \)

\( (7x−3)^\frac{2}{5}=4 \)
\( (7x−8)^\frac{2}{3}=4(x−5)^\frac{2}{3} \)
\( 3x^\frac{5}{2}=96\)
\( x^\frac{3}{2}-2=6\)
\( 2x^\frac{3}{2}=54 \)
\( 6x^\frac{5}{2}-12=0\)
\( 6x^\frac{3}{2}-141=1917\)
\( 5x^\frac{3}{2}−3 = 4997 \)

\( -2(x-5)^\frac{3}{4}+48=-202\)
\( 2x^\frac{3}{4}=686 \)
\( (4x+5)^\frac{1}{2}=x−4 \)
\( x^3=(4x−3)^\frac{3}{2} \)
\( x^\frac{5}{2}=16x^\frac{1}{2} \)
\( (x^2-x-4)^\frac{3}{4}-2=6 \)
\( (x^2-3x+3)^\frac{3}{2}-1=0 \)

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. \( \{1, 2\} \) .

\( \star \)