College algebra and trigonometry by Bernard Kolman; Arnold Shapiro

By Bernard Kolman; Arnold Shapiro

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X2(x 2) • missing in the denominator of the expression. x2(x + 2) · Step 3. Add the rational expressions. Do not multiply out the denominators since it may be possible Step 3. x+ l 2x2 - to cancel. x2(x + 2) PROGRESS CHECK Perform the indicated operations. (a) x-8 3 + - 4 x2 - 2x x2 ANSWERS (a) COMPLEX FRACTIONS x-3 x(x + 2)' x =l= 2 {b) 4r - 3 9r3 (b) _ 2r + 4r2 I + 1_ 3r 6r2 + 1r - 12 36r3 At the beginning of this section we said that we wanted fractions like I 1 -- x - to be able to simplify 36 THE FOUNDATIONS OF ALGEBRA This is an example of a complex fraction, which is a fractional form with frac­ tions in the numerator or denominator or both .

12. 16. 20. 24. 2S. 32. 36. 40. 44. 21x + 4YI 2x4 + x2 x2 + 2x - S x2 - 49 4b2 - a2 4a2 - b2 x2 - Sx - 20 2x2 + 7x + 6 4y2 - 9 9/ - 16x2 x1 2 - I 3. 7. 10. 13. 17. 21. 25. 29. 33. 37. 41. 45. -2x - Sy -3y2 - 4y5 9a3b3 + 12a2b - 15ab2 y2 - Sy + 15 y2 - -9I x2 - 5x - 14 x2 - 6x + 9 x2 + I I x + 24 3a2 - I la + 6 Sm2 - 6m - 9 6a2 - 5ab - 6b2 16 - 9x2y2 4. S. 14. IS. 22. 26. 30. 34. 3S. 42. 46. 5 47. 51. 55. 59. 63. 67. 71. 73. 75. 76. 48. 15 + 4x - 4x2 8n2 - 1 8n - 5 52. x4y4 x2y2 30x2 - 35x + 10 56.

We will assume that the previous rules for exponents apply to a0 and see if this leads us to a definition of a0 . For example, applying the rule a"'a" = a"'+" yields Dividing both sides by a"', we obtain a0 nonzero real number by =L We therefore define a0 for any a0 = l The same approach will lead us to a definition of negative exponents . For con­ sistency, we must have or (1) Division of both sides of Equation ( l ) by am suggests that we define a-m by a-"' = J_ " a" a*0 Dividing Equation ( 1 ) by a -m, we have Thus, a-m is the reciprocal of a"' , and a"' is the reciprocaJ of a-m.

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