By James Stewart
This most sensible promoting writer staff explains strategies easily and obviously, with no glossing over tricky issues. challenge fixing and mathematical modeling are brought early and bolstered all through, in order that whilst scholars end the path, they've got a great beginning within the rules of mathematical considering. This entire, calmly paced publication presents whole assurance of the functionality thought and integrates gigantic graphing calculator fabrics that support scholars increase perception into mathematical rules. The authors' awareness to element and readability, as in James Stewart's market-leading Calculus textual content, is what makes this article the marketplace chief.
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Extra info for Algebra and trigonometry, 2nd Edition
1ab2 ϭ 1a # a # . . # a2 # 1b # b # . . # b2 ϭ a n b n 144424443 n factors 1442443 n factors 1442443 n factors Here we have used the Commutative and Associative Properties repeatedly. If n Յ 0, Law 4 can be proved using the definition of negative exponents. ■ You are asked to prove Laws 2 and 5 in Exercise 76. 24 CHAPTER P Prerequisites Example 3 Using Laws of Exponents (a) x 4x 7 ϭ x 4ϩ7 ϭ x 11 Law 1: a ma n ϭ a mϩn (b) y 4y Ϫ7 ϭ y 4Ϫ7 ϭ y Ϫ3 ϭ 1 y3 c9 ϭ c 9Ϫ5 ϭ c 4 c5 # (d) 1b 4 2 5 ϭ b 4 5 ϭ b 20 Law 2: a m/a n ϭ a mϪn (c) Law 3: (a m) n ϭ a mn (e) 13x2 3 ϭ 33x 3 ϭ 27x 3 Law 4: (ab)n ϭ a nb n x 5 x5 x5 (f) a b ϭ 5 ϭ 2 32 2 Example 4 Law 1: a ma n ϭ a mϩn Law 5: (a/b) n ϭ a n/b n ■ Simplifying Expressions with Exponents Simplify: x 3 y 2x 4 (b) a b a b z y (a) 12a 3b 2 2 13ab 4 2 3 Solution (a) 12a 3b 2 2 13ab 4 2 3 ϭ 12a 3b 2 2 333a 3 1b 4 2 3 4 ϭ 12a 3b 2 2 127a 3b 12 2 ϭ 122 1272a 3a 3b 2b 12 Law 4: (ab)n ϭ a nb n Law 3: (a m)n ϭ a mn Group factors with the same base Apago PDF ϭ 54a Enhancer b Law 1: a a 6 14 x 3 y 2x 4 x 3 1y 2 2 4x 4 (b) a b a b ϭ 3 z y y z4 ϭ x 3 y 8x 4 y 3 z4 ϭ 1x 3x 4 2 a ϭ x 7y 5 z4 m n ϭ a mϩn Laws 5 and 4 Law 3 y8 1 b y 3 z4 Group factors with the same base Laws 1 and 2 ■ When simplifying an expression, you will find that many different methods will lead to the same result; you should feel free to use any of the rules of exponents to arrive at your own method.
Can you locate 15 by a similar method? What about 16? List some other irrational numbers that can be located this way. (a) Will the post office accept a package that is 6 in. wide, 8 in. deep, and 5 ft long? What about a package that measures 2 ft by 2 ft by 4 ft? (b) What is the greatest acceptable length for a package that has a square base measuring 9 in. by 9 in? œ∑2 _1 Apago PDF Enhancer L 21 0 1 1 2 5 ft=60 in. x 6 in. y 8 in. Discovery • Discussion 77. Signs of Numbers Let a, b, and c be real numbers such that a Ͼ 0, b Ͻ 0, and c Ͻ 0.
4 Rational Exponents and Radicals 31 (b) If b Ͼ 0, then 225b Ϫ 2b 3 ϭ 2252b Ϫ 2b 2 2b ϭ 52b Ϫ b2b Property 1: 1ab ϭ 1a1b Property 5, b Ͼ 0 ϭ 15 Ϫ b2 2b Distributive Property ■ Rational Exponents To define what is meant by a rational exponent or, equivalently, a fractional exponent such as a 1/3, we need to use radicals. In order to give meaning to the symbol a 1/n in a way that is consistent with the Laws of Exponents, we would have to have 1a 1/n 2 n ϭ a 11/n2n ϭ a 1 ϭ a So, by the definition of nth root, n a 1/n ϭ 1 a In general, we define rational exponents as follows.