256 CHAPTER 5 Number Theory and the Real Number System Example 8 Approximating Square Roots Use a scientific calculator to approximate the following square roots. Round your answer to the nearest hundredth. a) 7 b) 89 Solution a) 7 2.65 ≈ b) 89 9.43 ≈ Note that these approximations confirm the estimates that were made in Example 7. 7 Now try Exercise 71 Exercises Warm Up Exercises In Exercises 1–8, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. A real number that cannot be written as a ratio of two integers is known as a(n) _______ number. Irrational 2. The symbol is called the _______ sign. Radical 3. In the radical expression 3 5, the 5 is called the _______. Radicand 4. In the radical expression 3 5, the 3 is called the _______ of the radical expression. Coefficient 5. The principal (or positive) square root of a number n, written n, is the positive number that, when multiplied by _______, gives n. Itself 6. Any number that is the square of a natural number is called a(n) _______ square. Perfect 7. When the denominator of a fraction contains no radical expressions, it is said to be _______. Rationalized 8. Consider the irrational number the square root of two. a) We use the symbol 2 to represent the _______ value of the number. Exact b) The number 1.414213562 is a(n) _______ of the number 2. Approximation Practice the Skills In Exercises 9–18, determine whether the number is rational or irrational. 9. 25 Rational 10. 3 4 Rational 11. 10 Irrational 12. π Irrational 13. 3.575775777… Irrational 14. 0.212112111… Irrational 15. 22 7 Rational 16. 3.14159 Rational 17. 75 3 Rational 18. 5 5 Rational In Exercises 19–26, evaluate the expression. 19. 0 0 20. 1 1 21. 25 5 22. 36 6 23. 36 − 6− 24. 81 − 9− 25. 100 − 10 − 26. 225 − 15 − In Exercises 27–36, classify the number as a member of one or more of the following sets: the rational numbers, the integers, the natural numbers, the irrational numbers. 27. 0 Rational, integer 28. 5− Rational, integer 29. 11 Irrational 30. 4 13 Irrational 31. 0.040040004 Rational 32. 2.718 Rational 33. 0.123 Rational 34. 0.123123123 Rational 35. 123 Irrational 36. 0.123112311123… Irrational In Exercises 37– 44, simplify the radical. 37. 28 2 7 38. 72 6 2 39. 54 3 6 40. 90 3 10 41. 63 3 7 42. 75 5 3 43. 84 2 21 44. 90 3 10 In Exercises 45–52, perform the indicated operation. 45. 4 6 3 6 + 7 6 46. 8 3 10 3 − 2 3 − SECTION 5.4 LightField Studios/Shutterstock
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