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By Raymond A. Barnett

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321ր5 28. 6253ր4 29. 4xϪ1ր2 30. 32yϪ2ր5 31. x1ր3 Ϫ y1ր3 32. (x Ϫ y)1ր3 87. What is the result of entering 23 on a calculator? 2 88. Refer to Problem 87. What is the difference between 2(3 ) and 3 2 32 (2 ) ? Which agrees with the value of 2 obtained with a calculator? APPLICATIONS In Problems 33–38, change to rational exponent form. Do not simplify. 33. 1361 3 34. 2 172 5 3 35. 4x 2 y 4 36. 2 7x3y2 3 2 37. 2 x ϩ y2 3 2 3 2 38. 2 x ϩ2 y In Problems 39–50, evaluate each expression that represents a real number.

6(3x Ϫ 5)(2x Ϫ 3)2 ϩ 4(3x Ϫ 5)2(2x Ϫ 3) In Problems 29–34, factor completely, relative to the integers. 2 2 29. x ϩ 4x ϩ x ϩ 4 2 30. 2y Ϫ 6y ϩ 5y Ϫ 15 2 31. x Ϫ xy ϩ 3xy Ϫ 3y 32. 3a2 Ϫ 12ab Ϫ 2ab ϩ 8b2 33. 8ac ϩ 3bd Ϫ 6bc Ϫ 4ad In Problems 35–42, perform the indicated operations and simplify. 35. 2x Ϫ 35x ϩ 23 x Ϫ (x ϩ 5) 4 ϩ 16 78. 15ac Ϫ 20ad ϩ 3bc Ϫ 4bd 38. (x2 Ϫ 3xy ϩ y2)(x2 ϩ 3xy ϩ y2) 79. 3x2 Ϫ 2xy Ϫ 4y2 2 39. (3u Ϫ 2v) Ϫ (2u Ϫ 3v)(2u ϩ 3v) 80. 5u2 ϩ 4uv Ϫ v2 40. (2a Ϫ b)2 Ϫ (a ϩ 2b)2 3 42.

In other words, the empty set is a subset of every set. Two sets are equal if they have exactly the same elements. The order in which the elements of a set are listed does not matter. For example, {1, 2, 3, 4} ϭ {3, 1, 4, 2} Z The Real Number Line A one-to-one correspondence exists between the set of real numbers and the set of points on a line. That is, each real number corresponds to exactly one point, and each point to exactly one real number. A line with a real number associated with each point, and vice versa, as in Figure 1, is called a real number line, or simply a real line.

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