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9.1 Roots and Radicals 9.1 OBJECTIVES 1. Use the radical notation to represent roots 2. Distinguish between rational and irrational numbers In Chapter 3, we discussed the properties of exponents. Over the next four sections, we will work with a new notation that “reverses” the process of raising to a power. From our work in Chapter 0, we know that when we have a statement such as x2 9 NOTE The symbol 1 first appeared in print in 1525. In Latin, “radix” means root, and this was contracted to a small r. The present symbol may have evolved from the manuscript form of that small r. it is read as “x squared equals 9.” Here we are concerned with the relationship between the variable x and the number 9. We call that relationship the square root and say, equivalently, that “x is the square root of 9.” We know from experience that x must be 3 (because 32 9) or 3 [because (3)2 9]. We see that 9 has two square roots, 3 and 3. In fact, every positive number will have two square roots. In general, if x2 a, we call x a square root of a. We are now ready for our new notation. The symbol 1 is called a radical sign. We saw above that 3 was the positive square root of 9. We also call 3 the principal square root of 9 and can write 19 3 to indicate that 3 is the principal square root of 9. Definitions: Square Root 1a is the positive (or principal) square root of a. It is the positive number whose square is a. Example 1 Finding Principal Square Roots Find the following square roots. (a) 149 7 (b) 4 2 A9 3 Because 7 is the positive number we must square to get 49. Because 2 4 is the positive number we must square to get . 3 9 © 2001 McGraw-Hill Companies CHECK YOURSELF 1 Find the following square roots. (a) 164 NOTE When you use the radical sign, you are referring to the positive square root: 125 5 (b) 1144 (c) 16 A 25 Each positive number has two square roots. For instance, 25 has square roots of 5 and 5 because 52 25 and (5)2 25 695 696 CHAPTER 9 EXPONENTS AND RADICALS If you want to indicate the negative square root, you must use a minus sign in front of the radical. 125 5 Example 2 Finding Square Roots Find the following square roots. (a) 1100 10 The principal root (b) 1100 10 The negative square root 9 3 (c) 4 A 16 CHECK YOURSELF 2 Find the following square roots. (a) 116 Be Careful! Do not confuse 19 with 19 The expression 19 is 3, whereas 19 is not a real number. Every number that we have encountered in this text is a real number. The square roots of negative numbers are not real numbers. For instance, 19 is not a real number because there is no real number x such that x2 9 Example 3 summarizes our discussion thus far. Example 3 Finding Square Roots Evaluate each of the following square roots. (a) 136 6 (b) 1121 11 (c) 164 8 (d) 164 is not a real number. (e) 10 0 (Because 0 0 0) CHECK YOURSELF 3 Evaluate, if possible. (a) 181 (b) 149 (c) 149 (d) 149 © 2001 McGraw-Hill Companies C A U TI ON 16 (c) A 25 (b) 116 ROOTS AND RADICALS SECTION 9.1 697 All calculators have square root keys, but the only integers for which the calculator gives the exact value of the square root are perfect square integers. For all other positive integers, a calculator gives only an approximation of the correct answer. In Example 4 you will use your calculator to approximate square roots. Example 4 Approximating Square Roots Use your calculator to approximate each square root to the nearest hundredth. NOTE The sign means “is approximately equal to.” (a) 145 6.708203932 6.71 (b) 18 2.83 (c) 120 4.47 (d) 1273 16.52 CHECK YOURSELF 4 Use your calculator to approximate each square root to the nearest hundredth. (a) 13 (b) 114 (c) 191 (d) 1756 As we mentioned earlier, finding the square root of a number is the reverse of squaring a number. We can extend that idea to work with other roots of numbers. For instance, the cube root of a number is the number we must cube (or raise to the third power) to get that number. For example, the cube root of 8 is 2 because 23 8, and we write 3 NOTE 18 is read “the cube 3 18 2 root of 8.” The parts of a radical expression are summarized as follows. Definitions: Parts of a Radical Expression 3 NOTE The index for 1a is 3. Every radical expression contains three parts as shown below. The principal nth root of a is written as Index NOTE The index of 2 for © 2001 McGraw-Hill Companies square roots is generally not written. We understand that 1a is the principal square root of a. n 1a Radical sign Radicand To illustrate, the cube root of 64 is written Index of 3 3 164 4 because 43 = 64. And Index of 4 4 181 3 is the fourth root of 81 because 34 81. 698 CHAPTER 9 EXPONENTS AND RADICALS We can find roots of negative numbers as long as the index is odd (3, 5, etc.). For example, 3 164 4 because (4)3 64. If the index is even (2, 4, etc.), roots of negative numbers are not real numbers. For example, 4 116 NOTE The even power of a real number is always positive or zero. is not a real number because there is no real number x such that x4 16. The following table shows the most common roots. Square Roots NOTE It would be helpful for your work here and in future mathematics classes to memorize these roots. 11 14 19 116 125 1 2 3 4 5 136 6 Cube Roots 149 164 181 1100 1121 7 8 9 10 11 3 11 3 18 3 127 3 164 3 1125 1 2 3 4 5 Fourth Roots 4 11 116 4 181 4 1256 4 1625 4 1 2 3 4 5 1144 12 You can use the table in Example 5, which summarizes the discussion so far. Example 5 Evaluating Cube Roots and Fourth Roots Evaluate each of the following. NOTE The cube root of a negative number will be negative. NOTE The fourth root of a negative number is not a real number. 5 (a) 132 2 because 25 32. 3 (b) 1125 5 because (5)3 125. 4 (c) 181 is not a real number. CHECK YOURSELF 5 Evaluate, if possible. 4 (b) 116 4 (c) 1256 3 (d) 18 The radical notation helps us to distinguish between two important types of numbers: rational numbers and irrational numbers. A rational number can be represented by a fraction whose numerator and denominator are integers and whose denominator is nonzero. The form of a rational number is a b NOTE Notice that each radicand is a perfect-square integer (that is, an integer that is the square of another integer). a and b are integers, b 0 Certain square roots are rational numbers also. For example, 14 125 and 164 represent the rational numbers 2, 5, and 8, respectively. © 2001 McGraw-Hill Companies 3 (a) 164 ROOTS AND RADICALS NOTE The fact that the square root of 2 is irrational will be proved in later mathematics courses and was known to Greek mathematicians over 2000 years ago. SECTION 9.1 699 An irrational number is a number that cannot be written as the ratio of two integers. For example, the square root of any positive number that is not itself a perfect square is an irrational number. Because the radicands are not perfect squares, the expressions 12, 13, and 15 represent irrational numbers. Example 6 Identifying Rational Numbers Which of the following numbers are rational and which are irrational? 2 A3 4 A9 17 116 125 2 are irrational numbers. And 116 and 125 are rational numbers because A3 4 2 4 . 16 and 25 are perfect squares. Also is rational because A9 3 A9 Here 17 and CHECK YOURSELF 6 Which of the following numbers are rational and which are irrational? (a) 126 NOTE The decimal representation of a rational number always terminates or repeats. For instance, 3 0.375 8 5 0.454545. . . 11 NOTE 1.414 is an approximation to the number whose square is 2. © 2001 McGraw-Hill Companies (c) 6 A7 (d) 1105 (e) 16 A9 An important fact about the irrational numbers is that their decimal representations are always nonterminating and nonrepeating. We can therefore only approximate irrational numbers with a decimal that has been rounded off. A calculator can be used to find roots. However, note that the values found for the irrational roots are only approximations. For instance, 12 is approximately 1.414 (to three decimal places), and we can write 12 1.414 With a calculator we find that (1.414)2 1.999396 The set of all rational numbers and the set of all irrational numbers together form the set of real numbers. The real numbers will represent every point that can be pictured on the number line. Some examples are shown below. 0 NOTE For this reason we refer to the number line as the real number line. (b) 149 3 34 2 15 8 10 4 The following diagram summarizes the relationships among the various numeric sets. Real numbers Rational numbers Fractions Irrational numbers Integers Negative integers Zero Natural numbers 700 CHAPTER 9 EXPONENTS AND RADICALS We conclude our work in this section by developing a general result that we will need later. Let’s start by looking at two numerical examples. 222 14 2 (1) 2(2) 14 2 because (2) 4 2 2 (2) Consider the value of 2x2 when x is positive or negative. NOTE This is because the In (1) when x 2: In (2) when x 2: principal square root of a number is always positive or zero. 222 2 2(2)2 2 2(2)2 (2) 2 Comparing the results of (1) and (2), we see that 2x2 is x if x is positive (or 0) and 2x2 is x if x is negative. We can write 2x2 x x when x 0 when x 0 From your earlier work with absolute values you will remember that x x x when x 0 when x 0 and we can summarize the discussion by writing 2x2 x for any real number x Example 7 Evaluating Radical Expressions could write 2( 4)2 216 4 Evaluate each of the following. (b) 2(4)2 4 4 (a) 252 5 CHECK YOURSELF 7 Evaluate. (b) 2(6)2 (a) 262 CHECK YOURSELF ANSWERS 4 4 2. (a) 4; (b) 4; (c) 3. (a) 9; (b) 7; (c) 7; 5 5 (d) not a real number 4. (a) 1.73; (b) 3.74; (c) 9.54; (d) 27.50 5. (a) 4; (b) 2; (c) not a real number; (d) 2 6. (a) Irrational; (b) rational (because 149 7); (c) irrational; (d) irrational 16 4 (e) because 7. (a) 6; (b) 6 A9 3 1. (a) 8; (b) 12; (c) © 2001 McGraw-Hill Companies NOTE Alternatively in (b), we Name Exercises 9.1 Section Date Evaluate, if possible. ANSWERS 1. 116 2. 1121 1. 2. 3. 1400 4. 164 3. 4. 5. 1100 6. 1100 5. 6. 7. 181 8. 181 7. 8. 9. 16 A9 1 A 25 10. 9. 10. 11. 4 A 5 12. 4 A 25 11. 12. 3 13. 127 4 14. 181 13. 14. 3 15. 127 4 16. 116 15. 16. 4 17. 181 3 18. 164 17. 18. 19. © 2001 McGraw-Hill Companies 3 19. 127 3 20. 18 20. 21. 4 21. 1625 3 22. 11000 22. 23. 1 23. A 27 3 24. 8 24. A 27 3 701 ANSWERS 25. Which of the following roots are rational numbers and which are irrational numbers? 26. 25. 119 26. 136 27. 1100 28. 17 3 30. 18 27. 28. 29. 30. 29. 19 31. 3 32. 33. 4 31. 116 32. 4 A9 34. 35. 33. 36. 4 A7 3 34. 15 37. 38. 3 35. 127 4 36. 181 39. 40. Use your calculator to approximate the square root to the nearest hundredth. 41. 42. 37. 111 38. 114 39. 17 40. 123 41. 146 42. 178 43. 44. 45. 702 43. 2 A5 44. 3 A4 45. 8 A9 46. 7 A 15 © 2001 McGraw-Hill Companies 46. ANSWERS 47. 118 47. 48. 131 48. 49. 127 49. 50. 165 50. 51. For exercises 51 to 56, find the two expressions that are equivalent. 52. 51. 116, 116, 4 52. 125, 5, 125 53. 54. 3 3 53. 1125, 1125, 5 5 5 54. 132, 132, 2 55. 56. 4 3 55. 110,000, 100, 11000 3 2 56. 10 , 110,000, 1100,000 57. 58. In exercises 57 to 62, label the statement as true or false. 59. 60. 16 4 57. 216x 4x 2 58. 2(x 4) x 4 61. 62. 4 4 59. 216x y is a real number 2 2 60. 2x y x y 63. 64. 65. 2x2 25 1x 5 61. x5 62. 12 16 18 66. 63. Dimensions of a square. The area of a square is 32 square feet (ft2). Find the length © 2001 McGraw-Hill Companies of a side to the nearest hundredth. 64. Dimensions of a square. The area of a square is 83 ft2. Find the length of the side to the nearest hundredth. 65. Radius of a circle. The area of a circle is 147 ft2. Find the radius to the nearest hundredth. 66. Radius of a circle. If the area of a circle is 72 square centimeters (cm2), find the radius to the nearest hundredth. 703 ANSWERS 67. 67. Freely falling objects. The time in seconds (s) that it takes for an object to fall from 1 1s, in which s is the distance fallen. Find the time required for 4 an object to fall to the ground from a building that is 800 ft high. 68. rest is given by t 69. 70. 68. Freely falling objects. Find the time required for an object to fall to the ground from 71. a building that is 1400 ft high. (Use the formula in exercise 67.) 72. In exercises 69 to 71, the area is given in square feet. Find the length of a side of the square. Round your answer to the nearest hundredth of a foot. 73. 74. 75. 69. 70. 71. 2 2 10 ft 13 ft 2 17 ft 72. Is there any prime number whose square root is an integer? Explain your answer. 73. Explain the difference between the conjugate, in which the middle sign is changed, of a binomial and the opposite of a binomial. To illustrate, use 4 17. 74. Determine two consecutive integers whose square roots are also consecutive 75. Determine the missing binomial in the following: (13 2)( 704 ) 1. © 2001 McGraw-Hill Companies integers. ANSWERS 76. Try the following using your calculator. 76. (a) Choose a number greater than 1 and find its square root. Then find the square root of the result and continue in this manner, observing the successive square roots. Do these numbers seem to be approaching a certain value? If so, what? (b) Choose a number greater than 0 but less than 1 and find its square root. Then find the square root of the result, and continue in this manner, observing successive square roots. Do these numbers seem to be approaching a certain value? If so, what? 77. 78. 77. (a) Can a number be equal to its own square root? 79. (a) (b) (c) (d) (e) a. (b) Other than the number(s) found in part a, is a number always greater than its square root? Investigate. b. 78. Let a and b be positive numbers. If a is greater than b, is it always true that the square c. root of a is greater than the square root of b? Investigate. d. e. 79. Suppose that a weight is attached to a string of length L, and the other end of the string is held fixed. If we pull the weight and then release it, allowing the weight to swing back and forth, we can observe the behavior of a simple pendulum. The period, T, is the time required for the weight to complete a full cycle, swinging forward and then back. The following formula may be used to describe the relationship between T and L. f. g. h. L T 2p Ag If L is expressed in centimeters, then g 980 cm/s2. For each of the following string lengths, calculate the corresponding period. Round to the nearest tenth of a second. © 2001 McGraw-Hill Companies (a) 30 cm (b) 50 cm (c) 70 cm (d) 90 cm (e) 110 cm Getting Ready for Section 9.2 [Section 1.7] Find each of the following products. (a) (4x2)(2x) (e) (27p6)(3p) (b) (9a4)(5a) (f) (81s4)(s3) (c) (16m2)(3m) (g) (100y4)(2y) (d) (8b3)(2b) (h) (49m6)(2m) 705 Answers 1. 4 3. 20 5. 10 15. 3 1 19. 3 21. 5 23. 25. Irrational 3 31. Rational 33. Irrational 35. Rational 41. 6.78 43. 0.63 45. 0.94 47. 4.24 11. Not a real number 51. 57. 67. 73. 13. 3 3 4 3 17. Not a real number 7. Not a real number 3 9. 27. Rational 4 37. 3.32 49. 5.20 29. Irrational 39. 2.65 3 53. 1125, 1125 55. 110,000, 11000 116, 4 False 59. True 61. False 63. 5.66 ft 65. 6.84 ft 7.07 s 69. 3.16 ft 71. 4.12 ft Conjugate: 4 17; opposite: 4 17 75. 13 2 77. a. 8x3 g. 200y5 b. 45a5 h. 98m7 © 2001 McGraw-Hill Companies 79. (a) 1.1 s; (b) 1.4 s; (c) 1.7 s; (d) 1.9 s; (e) 2.1 s c. 48m3 d. 16b4 e. 81p7 f. 81s7 706