By Alex Bellos

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**Sample text**

X − 3 = 1 3. −x + 5 = 0 4. 2x + 4 = 1 3 6. x + 1 = 0 4 8. 3x + 1 = x 5. 4x − 5 = 8 7. 7x + 55 = 98 9. x + 1 = 2x + 2 11. ax + b = c (a = 0) 10. x + 1 = 3x + 1 12. x − 1 = cx + d(c = 1) 27. x 4 − 10x 2 + 9 = 0 28. x 4 − 2x 2 + 1 = 0 29. x 4 + x 2 − 1 = 0 30. x 3 + 2x 2 + x = 0 Find all possible real solutions of each equation in Exercises 31–44. 31. x 3 + 6x 2 + 11x + 6 = 0 32. x 3 − 6x 2 + 12x − 8 = 0 33. x 3 + 4x 2 + 4x + 3 = 0 34. y 3 + 64 = 0 By any method, determine all possible real solutions of each equation in Exercises 13–30.

59. Convert the expressions in Exercises 65–80 to exponential form. √ √ √ 65. 3 66. 8 67. x 3 √ 3 68. x 2 69. 3 x y 2 70. x 2 y 19. −x 2 x −3 y 3 4x y 3 x2 y 3 a2 b2 Convert the expressions in Exercises 81–92 to radical form. 81. 22/3 82. 34/5 84. y 7/4 85. (x 1/2 y 1/3 ) 1/5 86. x −1/3 y 3/2 3 87. 1 11 −1/7 x 90. −4/3 − x 7 89. 2x −2/3 + 88. 91. 83. x 4/3 3 7x −1/2 3 4(1 − x) 5/2 92. 9 4(1 − x) −7/3 Simplify the expressions in Exercises 93–102. 93. 4−1/2 47/2 96. 21/3 2−1 22/3 2−1/3 99. 101. x 1/2 y 2 x −1/2 y x y 1/3 95.

X 2 + 1) (x + 1) 4 − 3 (x + 1) 7 = 0 13. (x + 1) (2x + 3) − (x + 1)(2x + 3) = 0 2 17. 18. 19. 21. (x + 1) 5 = 0 23. 2 14. (x 2 − 1) 2 (x + 2) 3 − (x 2 − 1) 3 (x + 2) 2 = 0 (x + 1) 2 (x + 2) 3 − (x + 1) 3 (x + 2) 2 =0 (x + 2) 6 25. 6x(x 2 + 1) 2 (x 2 + 2) − 8x(x 2 + 1) 3 (x 2 + 2) 3 =0 (x 2 + 2) 8 √ 4 2(x 2 − 1) x 2 + 1 − √ x2 x +1 =0 x2 + 1 √ 4 4x x 3 − 1 − √3x3 x −1 =0 x3 − 1 4 1 x− =0 20. 1 − 2 = 0 x x 1 1 9 1 − 3 =0 =0 22. 2 − x x x x +1 x 2x − 3 2x + 3 x −4 − =0 − =0 24. x +1 x −1 x −1 x +1 x +4 2x − 3 2x − 3 x +4 + =0 − =0 26.