By Earl Swokowski, Jeffery A. Cole
The newest version within the hugely revered Swokowski/Cole precalculus sequence keeps the weather that experience made it so well liked by teachers and scholars alike: its exposition is obvious, the time-tested workout units function numerous purposes, its uncluttered format is attractive, and the trouble point of difficulties is acceptable and constant. Mathematically sound, ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY, vintage variation, 12E, successfully prepares scholars for extra classes in arithmetic via its very good, time-tested challenge units.
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Extra resources for Algebra and Trigonometry with Analytic Geometry, Classic 12th Edition
N n n n This process is called rationalizing a denominator. Some special cases are listed in the following chart. 2 Exponents and Radicals 23 Rationalizing Denominators of Quotients (a > 0) Factor in denominator Multiply numerator and denominator by Resulting factor 2a 2a 2 a 2 a ϭ 2 a2 ϭ a 3 7 2 a 2 a2 ϭ 2 a3 ϭ a 3 2a 3 2 a2 7 2a 3 3 2 a 2 a ϭ 2 a7 ϭ a 7 2a 3 4 3 7 4 7 The next example illustrates this technique. EXAMPLE 4 Rationalizing denominators ͱ Rationalize each denominator: 1 1 (a) (b) 3 (c) 25 2x 2 3 (d) ͱ 5 x y2 SOLUTION (a) (b) (c) (d) 1 ͙5 1 3 x ͙ ͱ ͱ 5 1 ͙5 ͙5 ͙5 2 5 ͙5 ͙5 ͙5 3 2 3 2 3 2 1 ͙ x x x ͙ ͙ 3 3 2 3 3 x ͙x ͙x ͙x 2 ͙2 ͙2 ͙3 ͙2 и 3 ͙6 3 3 ͙3 ͙3 ͙3 ͙32 5 5 5 3 5 5 x x x ͙ y xy 3 ͙ xy 3 ͙ ͙ ͙ 2 5 2 5 2 5 3 5 5 y y ͙y ͙y ͙y ͙y L If we use a calculator to find decimal approximations of radicals, there is no advantage in rationalizing denominators, such as 1͞ 25 ϭ 25͞5 or 22͞3 ϭ 26͞3, as we did in Example 4(a) and (c).
The following chart contains two particular warnings concerning commonly made mistakes. Y Warning! Y If a 0 and b 0 Illustration (1) 2a2 ϩ b2 a ϩ b 2a ϩ 2b (2) 2a ϩ b 232 ϩ 42 ϭ 225 ϭ 5 3ϩ4ϭ7 24 ϩ 29 ϭ 5 24 ϩ 9 ϭ 213 If c is a real number and c n occurs as a factor in a radical of index n, then we can remove c from the radicand if the sign of c is taken into account. For example, if c Ͼ 0 or if c Ͻ 0 and n is odd, then n 2c nd ϭ 2c n 2 d ϭ c 2d, n n n n provided 2d exists. If c Ͻ 0 and n is even, then n n n 2c nd ϭ 2c n 2d ϭ ͉ c ͉2d, n n provided 2d exists.
Y If a 0 and b 0 Illustration (1) 2a2 ϩ b2 a ϩ b 2a ϩ 2b (2) 2a ϩ b 232 ϩ 42 ϭ 225 ϭ 5 3ϩ4ϭ7 24 ϩ 29 ϭ 5 24 ϩ 9 ϭ 213 If c is a real number and c n occurs as a factor in a radical of index n, then we can remove c from the radicand if the sign of c is taken into account. For example, if c Ͼ 0 or if c Ͻ 0 and n is odd, then n 2c nd ϭ 2c n 2 d ϭ c 2d, n n n n provided 2d exists. If c Ͻ 0 and n is even, then n n n 2c nd ϭ 2c n 2d ϭ ͉ c ͉2d, n n provided 2d exists. ILLUS TRATION n Removing nth Powers from 2 2x7 ϭ 2x5 и x2 ϭ 2x5 2x2 ϭ x 2x2 5 5 5 5 5 2 x7 ϭ 2 x 6 и x ϭ 2 ͑x 2͒3x ϭ 2 ͑x 2͒3 2 x ϭ x 2 2 x 3 3 3 3 3 3 2x 2y ϭ 2x 2 2y ϭ ͉ x ͉ 2y 2x 6 ϭ 2͑x 3͒2 ϭ ͉ x 3 ͉ 2 x 6y 3 ϭ 2 x 4 и x 2y 3 ϭ 2 x 4 2 x 2y 3 ϭ ͉ x ͉ 2 x 2y 3 4 4 4 4 4 Note: To avoid considering absolute values, in examples and exercises involving radicals in this chapter, we shall assume that all letters—a, b, c, d, x, y, 22 CHAPTER 1 FUNDAMENTAL CONCEPTS OF ALGEBRA and so on—that appear in radicands represent positive real numbers, unless otherwise specified.