By Earl Swokowski, Jeffery A. Cole
The most recent 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 apparent, the time-tested workout units characteristic quite a few purposes, its uncluttered structure is attractive, and the trouble point of difficulties is suitable and constant. Mathematically sound, ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY, vintage variation, 12E, successfully prepares scholars for additional classes in arithmetic via its first-class, time-tested challenge units.
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Extra info for Algebra and Trigonometry with Analytic Geometry, Classic 12th Edition
Since am ϭ a и a и a и и и и и a, with a occurring as a factor m times, and since the number of such groups of m factors is n, the total number of factors of a is m и n. Thus, ͑am͒n ϭ amn. The cases m Յ 0 and n Յ 0 can be proved using the definition of nonpositive exponents. The remaining three laws can be established in similar fashion by counting factors. In laws 4 and 5 we assume that denominators are not 0. Laws of Exponents for Real Numbers a and b and Integers m and n Law Illustration (1) a a ϭ a (2) ͑am͒n ϭ amn (3) ͑ab͒n ϭ anbn a n an ϭ n (4) b b am (5) (a) n ϭ amϪn a am 1 (b) n ϭ nϪm a a m n ͩͪ mϩn 2 и 2 ϭ 2 ϭ 27 ϭ 128 ͑23͒4 ϭ 23и4 ϭ 212 ϭ 4096 ͑20͒3 ϭ ͑2 и 10͒3 ϭ 23 и 103 ϭ 8 и 1000 ϭ 8000 8 2 3 23 ϭ 3ϭ 5 5 125 25 ϭ 25Ϫ3 ϭ 22 ϭ 4 23 1 1 1 23 5 ϭ 5Ϫ3 ϭ 2 ϭ 2 2 2 4 3 4 3ϩ4 ͩͪ We usually use 5(a) if m Ͼ n and 5(b) if m Ͻ n.
For example, to find ͑4,500,000͒2 on a scientific calculator, we could enter the integer 4,500,000 and press the x 2 (or squaring) key, obtaining a display similar to one of those in Figure 6. 025 ϫ 1013. Thus, ͑4,500,000͒2 ϭ 20,250,000,000,000. Calculators may also use scientific form in the entry of numbers. The user’s manual for your calculator should give specific details. Before we conclude this section, we should briefly consider the issue of rounding off results. 1 Re a l N u m b e r s 13 tained by various types of measurements and, hence, are approximations to exact values.
If we begin with any collection of variables and real numbers, then an algebraic expression is the result obtained by applying additions, subtractions, multiplications, divisions, powers, or the taking of roots to this collection. If specific numbers are substituted for the variables in an algebraic expression, the resulting number is called the value of the expression for these numbers. The domain of an algebraic expression consists of all real numbers that may represent the variables. Thus, unless otherwise specified, we assume that the domain consists of the real numbers that, when substituted for the variables, do not make the expression meaningless, in the sense that denominators cannot equal zero and roots always exist.