By Chazelle B., Goodman J.E., Pollack R. (eds.)

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C) Condition 3 is violated. We multiply numerator and denominator by 12x; the effect is to multiply the expression by 1, so its value is unchanged, but the denominator is left free of radicals. 6 6 12x 612x 312x ϭ ؒ ϭ ϭ x 2x 12x 12x 12x (D) Condition 4 is violated. First we convert to rational exponent form. 8x4 81ր3x4ր3 ϭ B y y1ր3 y2ր3 3 Multiply by y2ր3 ؍1. ϭ 2x4ր3y2ր3 y x 4ր3 ؍xx 1ր3 ϭ 2xx1ր3y2ր3 y Write in radical form. ϭ 2x 2xy2 y 3 MATCHED PROBLEM 7 Write in simplified radical form.

A) Ϫ(5ր2 ϩ 7ր3) (C) 21 15 ؒ 20 14 Ϫ1 (B) Ϫ(8ր17) (D) 5 ؒ (1ր2 ϩ 1ր3) Rational numbers have decimal expansions that are repeating or terminating. 625 8 The number 6 repeats indefinitely. The block 142857 repeats indefinitely. Terminating expansion Conversely, any decimal expansion that is repeating or terminating represents a rational number (see Problems 49 and 50 in Exercise R-1). The number 12 is irrational because it cannot be written in the form a͞b, where a and b are integers, b 0 (for an explanation, see Problem 89 in Section R-3).

2x(u Ϫ 3v) ϩ 5y(u Ϫ 3v) 71. 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 2 3x 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.