By Nicholas D. Kazarinoff

Mathematical research is essentially a scientific research and exploration of inequalities — yet for college kids the examine of inequalities frequently is still a overseas nation, tricky of entry. This e-book is a passport to that nation, providing a historical past on inequalities that would arrange undergraduates (and even highschool scholars) to deal with the suggestions of continuity, spinoff, and integral.

Beginning with causes of the algebra of inequalities and conditional inequalities, the textual content introduces a couple of historical theorems and their functions. Explorations of inequalities and calculus disguise the quantity *e,* examples from the calculus, and approximations by way of polynomials. the ultimate sections current smooth theorems, together with Bernstein's facts of the Weierstrass approximation theorem and the Cauchy, Bunyakovskii, Hölder, and Minkowski inequalities. a variety of figures, difficulties, and examples look in the course of the publication, supplying scholars an outstanding starting place for extra reports of calculus.

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**Example text**

5 Ί Ϫ243 ϭ Ϫ3 3 4 813 ϭ 27 10. Ί 12. 32 4 81͒ ϭ 27 9. ͑Ί 11. Rational Exponent Form 1͞5 Ϫ216 ϭ Ϫ6 7. 3 Ί ϭ2 5 64yϪ5 35. Ί 36. 5 4 16xϪ4 y 8 z 4 39. Ί 5 243x 5 yϪ5 z15 40. Ί In Exercises 41– 48, rewrite the expression by rationalizing the denominator. Simplify your answer. 41. 3 Ϫ27 15. ϪΊ 3 0 16. Ί 18. 4 81 Ί 3 4 5624 20. Ί 21. 161͞2 22. 271͞3 23. 363͞2 24. 253͞2 25. Ί2 и Ί3 26. Ί2 27. 28. 30. 43. 45. 47. 3 Ϫ125 19. ͑Ί ͒ 29. 5 96x5 34. Ί 38. Ί3x 2 yz 6 3 64 14. Ί Ϫ3͞4 ͑16 81 ͒ ͑Ϫ 641 ͒Ϫ1͞3 33.

Am͒n ϭ amn 6. aϪn ϭ 8. ab Ϫn ϭ Product of Powers x7 ϭ x7Ϫ4 ϭ x3 x4 Quotient of Powers ͑5x͒3 ϭ 53x3 ϭ 125x3 Power of a Product 2x Power of a Quotient 3 ϭ 23 8 ϭ 3 x3 x ͑ y3͒Ϫ4 ϭ y3͑Ϫ4͒ ϭ yϪ12 1 an 7. a0 ϭ 1, и 34 ϭ 32ϩ4 ϭ 36 1 y4 yϪ4 ϭ a Definition of negative exponent ͑x2 ϩ 1͒0 ϭ 1 0 ba , Power of a Power 32 n a 0, b 0 Խ Խ ԽԽ Ϫ3 ϭ 23 Definition of zero exponent 3 Խ22Խ ϭ Խ2Խ2 ϭ 22 9. a2 ϭ a 2 ϭ a2 Notice that these properties of exponents apply for all integers m and n, not just positive integers.

Add or subtract with like denominators: a c a±c ± ϭ b b b 5. Add or subtract with unlike denominators: 6. Multiply fractions: 7. Divide fractions: a b c 0 a c ad ± bc ± ϭ b d bd ac и d ϭ bd a c a Ϭ ϭ b d b d ad и c ϭ bc , c 0 In Property 1 (equivalent fractions) the phrase “if and only if” implies two statements. One statement is: If a͞b ϭ c͞d, then ad ϭ bc. The other statement is: If ad ϭ bc, where b 0 and d 0, then a͞b ϭ c͞d. 2 Example 5 a. x Ϫ b. ✓CHECKPOINT 5 Simplify the expression x 2x ϩ .