Algebra: An Elementary Text-Book for the Higher Classes of by George Chrystal

By George Chrystal

This Elibron Classics publication is a facsimile reprint of a 1904 version by way of Adam and Charles Black, London.

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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 88. 2 # (d) 1b 4 2 5 ϭ b 4 5 ϭ b 20 Exponents and Radicals 15 Law 3: (am)n ϭ amn (e) 13x2 3 ϭ 33x 3 ϭ 27x 3 Law 4: (ab)n ϭ anbn x 5 x5 x5 (f) a b ϭ 5 ϭ 2 32 2 Law 5: (a/b)n ϭ an/bn ■ Example 4 Simplifying Expressions with Exponents Simplify: x 3 y 2x 4 b (b) a b a 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 ϭ 54a 6b 14 x 3 y 2x 4 x 3 1y 2 2 4x 4 b ϭ 3 (b) a b a z y y z4 ϭ x 3 y 8x 4 y 3 z4 ϭ 1x 3x 4 2 a ϭ Law 4: (ab)n ϭ anbn Law 3: (am)n ϭ amn Group factors with the same base Law 1: aman ϭ amϩn Laws 5 and 4 Law 3 y8 1 b y 3 z4 x 7y 5 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.

If you work x hours then your pay is y dollars, where y is given by the algebraic formula y ϭ 8x So if you work 10 hours, your pay is y ϭ 8 # 10 ϭ 80 dollars. An equation is a sentence written in the language of algebra that expresses a fact about an unknown quantity x. For example, how many hours would you need to work to get paid 60 dollars? 5 Hours worked x We use the rules of algebra to find x. 5 hours. The coordinate plane allows us to sketch a graph of an equation in two variables. For example, by graphing the equation y ϭ 8x we can “see” how pay increases with hours worked.

A) Ϫ6 Ͻ Ϫ10 30. (a) 11–14 ■ Rewrite the expression using the given property of real numbers. Ϫ 1 4 (b) 1 ϩ 58 Ϫ 16 Ϫ 35 2 Ϫ 34 1 1 2 Ϫ 3 27. (a) 3 10. 71a ϩ b ϩ c 2 ϭ 71a ϩ b 2 ϩ 7c (b) 24. (a) A3 ϩ 14 B A1 Ϫ 45 B 5. 1x ϩ 2y 2 ϩ 3z ϭ x ϩ 12y ϩ 3z2 9. 2x13 ϩ y 2 ϭ 13 ϩ y2 2x ϩ 154 23. (a) 23 A6 Ϫ 32 B 26. (a) 4. 213 ϩ 5 2 ϭ 13 ϩ 5 2 2 Perform the indicated operations. 3 10 State the property of real numbers being used. 3. 7 ϩ 10 ϭ 10 ϩ 7 ■ Write each statement in terms of inequalities. 33. (a) x is positive (b) t is less than 4 (c) a is greater than or equal to p (d) x is less than 13 and is greater than Ϫ5 (e) The distance from p to 3 is at most 5 34.

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