# 1.5e: Exercises - Solve Equations with Rational Exponents


### A: Solve Equations with Rational Exponents I

Exercise $$\PageIndex{A}$$

$$\bigstar$$ Solve.

 $$x ^ { 1 / 2 } - 10 = 0$$ $$x ^ { 1 / 2 } - 6 = 0$$ $$x ^ { 1 / 3 } + 2 = 0$$ $$x ^ { 1 / 3 } + 4 = 0$$ $$( x - 1 ) ^ { 1 / 2 } - 3 = 0$$ $$( x + 2 ) ^ { 1 / 2 } - 6 = 0$$ $$( 2 x - 1 ) ^ { 1 / 3 } + 3 = 0$$ $$( 3 x - 1 ) ^ { 1 / 3 } - 2 = 0$$ $$( 4 x + 15 ) ^ { 1 / 2 } - 2 x = 0$$ $$( 3 x + 2 ) ^ { 1 / 2 } - 3 x = 0$$ $$( 2 x + 12 ) ^ { 1 / 2 } - x = 6$$ $$( 4 x + 36 ) ^ { 1 / 2 } - x = 9$$ $$2 ( 5 x + 26 ) ^ { 1 / 2 } = x + 10$$ $$3 ( x - 1 ) ^ { 1 / 2 } = x + 1$$ $$x ^ { 1 / 2 } + ( 3 x - 2 ) ^ { 1 / 2 } = 2$$ $$( 6 x + 1 ) ^ { 1 / 2 } - ( 3 x ) ^ { 1 / 2 } = 1$$ $$( 3 x + 7 ) ^ { 1 / 2 } + ( x + 3 ) ^ { 1 / 2 } - 2 = 0$$ $$( 3 x ) ^ { 1 / 2 } + ( x + 1 ) ^ { 1 / 2 } - 5 = 0$$

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

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