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Additional info for How to keep the clock right by observations of the fixed stars with a small fixed telescope
Simplify. ϭ 2x3y Ϫ 2xy It should be clear that free use has been made of the real number properties discussed earlier. The steps done in the dashed box are usually done mentally, and the process is quickly mechanized as follows: Like terms in a polynomial are combined by adding their numerical coefficients. bar67388_chR_036-074 11/2/06 17:56 Page 39 S E C T I O N R–4 EXAMPLE 2 Polynomials: Basic Operations 39 Simplifying Polynomials Remove parentheses and combine like terms: (A) 2(3x2 Ϫ 2x ϩ 5) ϩ (x2 ϩ 3x Ϫ 7) ϭ 2(3x2 Ϫ 2x ϩ 5) ϩ 1(x2 ϩ 3x Ϫ 7) ϭ 6x2 Ϫ 4x ϩ 10 ϩ x2 ϩ 3x Ϫ 7 ϭ 7x2 Ϫ x ϩ 3 Distribute, remove parentheses.
A) u1ր5 (B) (6x2y5)2ր9 (C) (3xy)Ϫ3ր5 Change from radical form to rational exponent form. 4 (D) 2 9u 7 (E) Ϫ2 (2x)4 3 3 (F) 2 x ϩ y3 Z Properties of Radicals The exponent properties considered earlier imply the following properties of radicals. Z THEOREM 1 Properties of Radicals For n a natural number greater than 1, and x and y positive real numbers: n 1. 2xn ϭ x n 3 3 2 x ϭx n n 2. 2xy ϭ 2x2y n 3. EXAMPLE 2 x 2x ϭ n By 2y 5 5 5 2 xy ϭ 2 x2 y 4 1 x 4 x ϭ 4 Ay 1y n Simplifying Radicals Simplify: 5 (A) 2 (3x2y)5 ϭ 3x2y (B) 11015 ϭ 150 ϭ 125 ؒ 2 ϭ 12512 ϭ 512 (C) 3 3 x 1 x 1 x ϭ 3 ϭ A 27 3 1 27 3 or MATCHED PROBLEM 1 3 1x 3 2 Simplify: 7 (A) 2 (u2 ϩ v2)7 (B) 1612 (C) x2 B8 3 bar67388_chR_014-035 30 11/02/06 CHAPTER R 00:52 Page 30 BASIC ALGEBRAIC OPERATIONS ZZZ CAUTION ZZZ In general, properties of radicals can be used to simplify terms raised to powers, not sums of terms raised to powers.
2 144 40. 41. 6x 13x 42. 44. 4 16 Ϫ 2 45. 12y 2 16y 12 16 ϩ 2 1 17 3 3 254 3 43. 2 12 Ϫ 1 46. 12 110 Ϫ 2 In Problems 47–54, write in simplified form. 5 6 7 11 47. x 2 3xy 50. 5 2 32u12v8 uv 4 2 32m7n9 2mn 3 48. 2a2 8a8b13 49. 3 4 9 3 ab 51. 2 1 52. 21 x8 y6 6 bar67388_chR_014-035 11/2/06 17:10 Page 35 S E C T I O N R–3 3 3 53. 2 2x2y4 2 3x5y 4 4 54. 2 4m5n2 6m3n4 81. In Problems 55–60, rationalize denominators and write in simplified form. 55. 58. 12m15 120m 56. 5 1x 3 Ϫ 21x 59. 1618c 118c 57. 2 15 ϩ 3 12 5 15 ϩ 2 12 60.