By Bell E.T.
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Extra resources for Invariant Sequences
Example text
2. a. The associative law of addition states that a ϩ (b ϩ c) ϭ . b. The distributive law states that ab ϩ ac ϭ 3. What can you say about a and b if ab ϶ 0? How about a, b, and c if abc ϶ 0? 1 Exercises In Exercises 1–10, classify the number as to type. ) 1. Ϫ3 5. 211 ෆ2ෆ1ෆ 9. 4 2. Ϫ420 3 3. 8 4 4. Ϫ 125 6. Ϫ 25 p 7. 2 2 8. p 10. 71828. . In Exercises 11–16, indicate whether the statement is true or false. 11. Every integer is a whole number. 12. Every integer is a rational number. 13. Every natural number is an integer.
3)2 и (Ϫ3)3 12. (Ϫ2x)3(Ϫ2x)2 In Exercises 13–56, perform the indicated operations and simplify. 13. (2x ϩ 3) ϩ (4x Ϫ 6) 2 45. (2x ϩ 3y)2 Ϫ (2y ϩ 1)(3x Ϫ 2) ϩ 2(x Ϫ y) 4 6. Ϫ aϪ b 5 5. Ϫ43 42. (3r ϩ 4s)(3r Ϫ 4s) 43. (2x Ϫ 1) ϩ 3x Ϫ 2(x ϩ 1) ϩ 3 2 14. (Ϫ3x ϩ 2) Ϫ (4x Ϫ 3) 47. (t 2 Ϫ 2t ϩ 4)(2t 2 ϩ 1) 48. (3m 2 Ϫ 1)(2m 2 ϩ 3m Ϫ 4) 49. 2x Ϫ {3x Ϫ [x Ϫ (2x Ϫ 1)]} 50. 3m Ϫ 2{m Ϫ 3[2m Ϫ (m Ϫ 5)] ϩ 4} 51. x Ϫ {2x Ϫ [Ϫx Ϫ (1 Ϫ x)]} 52. 3x 2 Ϫ {x 2 ϩ 1 Ϫ x[x Ϫ (2x Ϫ 1)]} ϩ 2 53. (2x Ϫ 3)2 Ϫ 3(x ϩ 4)(x Ϫ 4) ϩ 2(x Ϫ 4) ϩ 1 54.
N ϭ a mϪn a 3. (am)n ϭ a mn m n Illustration x и x ϭ x2ϩ3 ϭ x 5 x7 ϭ x7Ϫ4 ϭ x 3 x4 (x 4)3 ϭ x 4и3 ϭ x12 mϩn 4. (ab)n ϭ an и bn a n an 5. a b ϭ n (b b b 2 0) 3 (2x)4 ϭ 24 и x 4 ϭ 16x 4 x 3 x3 x3 a b ϭ 3ϭ 2 8 2 It can be shown that these laws are valid for any real numbers a and b and any integers m and n. 28 1 FUNDAMENTALS OF ALGEBRA Simplifying Exponential Expressions The next two examples illustrate the use of the laws of exponents. EXAMPLE 4 Simplify the expression, writing your answer using positive exponents only.