LESSON Page 1 of 5 9.1 Square Roots BEFORE Vocabulary square root, p. 453 perfect square, p. 454 radical expression, p. 454 You found squares of numbers. Now WHY? You’ll find and approximate square roots of numbers. So you can find a person’s running speed, as in Ex. 59. Human Chess In September of every even-numbered year, people in Marostica, Italy, play an unusual chess game. Each chess piece is portrayed by a person. The people portraying the knights are even on horseback! The chessboard is a square with an area of 324 square meters. What is the length of each side of the board? To answer this question, you need to find the square root of 324. A square root of a number n is a number m such that m 2 n. Every positive number has two square roots. One square root is positive and the other is negative. The radical sign, , represents a nonnegative square root. The symbol , read “plus or minus,” refers to both square roots of a positive number. For example: 100 10 Positive square root of 100 100 10 Negative square root of 100 100 10 Positive or negative square root of 100 Zero has only one square root, itself. Example 1 Study Strategy In Example 1, it doesn’t make sense to find the negative square root of 324, because length cannot be negative. Finding a Square Root The chessboard described above is a square with an area of 324 square meters, so the length of each side of the chessboard is the positive square root of 324. 324 18 because 182 324. Answer The length of each side of the chessboard is 18 meters. Checkpoint Find the square roots of the number. 1. 16 2. 64 3. 144 Lesson 9.1 4. 256 Square Roots 453 Page 2 of 5 Approximating Square Roots A perfect square is a number that is the square of an integer. For example, 1, 4, and 9 are perfect squares. Note Worthy You may find it helpful to make a list of the first 20 perfect squares and their square roots in your notebook and memorize them. You can find a table of squares and square roots on p. 822. 1 12, 4 22, and 9 32 You can use perfect squares to approximate a square root of a number. Example 2 Approximating a Square Root Approximate 51 to the nearest integer. The perfect square closest to, but less than, 51 is 49. The perfect square closest to, but greater than, 51 is 64. So, 51 is between 49 and 64. This statement can be expressed by the compound inequality 49 < 51 < 64. 49 < 51 < 64 Identify perfect squares closest to 51. 49 < 51 < 64 7 < 51 <8 Take positive square root of each number. Evaluate square root of each perfect square. Answer Because 51 is closer to 49 than to 64, 51 is closer to 7 than ≈ 7. to 8. So, to the nearest integer, 51 Checkpoint 5. Approximate 125 to the nearest whole number. Example 3 Tech Help When you enter a square root on the calculator, you should also enter a right parenthesis to close the left parenthesis that the calculator enters. Although the calculator shows 8 digits for the decimal part of the square root in Example 3, the decimal actually continues without end. Using a Calculator Use a calculator to approximate 515 . Round to the nearest tenth. Keystrokes [] 515 (515) 22.69361144 Answer 515 ≈ 22.7 Radical Expressions A radical expression is an expression that involves a radical sign. The horizontal bar in a radical sign is a grouping symbol. When you evaluate a radical expression, evaluate the expression inside the radical symbol before finding the square root. Example 4 Evaluating a Radical Expression Evaluate 2a b 2 when a 11 and b 5. b 2 211 52 2a 454 Chapter 9 Substitute 11 for a and 5 for b. 236 Evaluate expression inside radical symbol. 2 p 6 12 Evaluate square root. Multiply. Real Numbers and Right Triangles Page 3 of 5 Checkpoint Evaluate the expression when a 12 and b 4. 6. a b 7. Example 5 b2 a 8. 3ab 1 Solving an Equation Using Square Roots Physics An amusement park ride includes a free fall drop of 272 feet. You can use the equation d 16t 2 to determine the time t in seconds that it takes a dropped object to fall a distance of d feet. How long does the free fall part of the ride take? Solution d 16t 2 272 16t 17 t 2 2 Write original equation. Substitute 272 for d. Divide each side by 16. 17 t Use definition of square root. 4.1 ≈ t Use a calculator to approximate square root. Answer Because only the positive solution makes sense in this situation, the free fall part of the ride takes about 4.1 seconds. 9.1 Exercises INTERNET More Practice, p. 811 CLASSZONE.COM eWorkbook Plus Guided Practice Vocabulary Check 1. Describe and give an example of a perfect square. 2. You know that one square root of a number x is 9. What is the other square root? What is the value of x? Skill Check Find the square roots of the number. 3. 4 4. 36 5. 121 6. 225 Approximate the square root to the nearest integer. 7. 10 8. 84 9. 151 10. 200 Solve the equation. 11. a 2 9 Guided Problem Solving 12. n 2 25 13. 361 x 2 14. 400 y 2 15. Eiffel Tower The base of the Eiffel Tower is a square with an area of 15,625 square feet. What is the length of a side of the base? 1 Write an equation that relates base area A and side length s. 2 Substitute 15,625 for A in the equation in Step 1 and solve for s. Lesson 9.1 Square Roots 455 Page 4 of 5 Practice and Problem Solving Homework Help Example 1 2 3 4 5 Exercises 16–24 25–32, 54–57 33–40 41–44 45–53, 58–59 Online Resources CLASSZONE.COM • More Examples • eTutorial Plus In the following exercises, you may find it helpful to use a calculator for approximating square roots. Find the square roots of the number. 16. 25 17. 169 18. 81 19. 289 20. 1024 23. 900 21. 484 22. 1600 24. Geometry The area of a square is 49 square feet. Find the side length. Approximate the square root to the nearest integer. 25. 38 26. 120 27. 148 28. 17 29. 78 30. 250 31. 15.3 32. 7.4 Use a calculator to approximate the square root. Round to the nearest tenth. 33. 3 34. 10 35. 86 36. 110 37. 33 38. 1325 39. 19.5 40. 6.92 Evaluate the expression when a 48 and b 12. 41. a b 42. a b 4 43. 3ab 44. b2 ( a 15) Solve the equation. Round to the nearest tenth if necessary. 45. x 2 49 46. y 2 676 47. 441 t 2 48. n2 576 49. 20 m 2 50. c 2 125 51. 5y 2 110 52. 200 16t 2 53. Critical Thinking Write an equation that has exactly two solutions, 1.5 and 1.5. In Exercises 54–57, match the number with a point on the number line. A 0 1 54. 15 B 2 3 55. 2 4 C 5 D 6 7 56. 95 8 9 10 11 57. 27 58. Photography You can use the following rule of thumb when photographing fireworks: The f-stop, a number that describes the size of the opening of the camera lens, should be the number closest to the square root of the film speed. You have a camera with f-stop numbers 2.8, 4, 5.6, 8, 11, 16, and 22. Which f-stop should you use to photograph fireworks if you are using a film speed of 64? of 100? 59. Running You can use the formula l 0.0625s2 to approximate the maximum running speed s (in meters per second) that a person with leg length l (in centimeters) can sustain. Find the maximum running speed for a person with a leg length of 64 centimeters. Solve the equation. Round to the nearest hundredth if necessary. Fireworks near the Space Needle in Seattle, Washington 456 Chapter 9 60. 15 2h 2 3 61. 162 0.5t 2 62. 1400 10z 2 2 63. 3x 2 5 30 64. 1.5n2 7 20 65. 2a 2 1 98 Real Numbers and Right Triangles Page 5 of 5 66. Consider the function y x . a. Make a table of ordered pairs (x, y) for x 0, 1, 4, 9, 16, and 25. b. Plot the ordered pairs from part (a) on a coordinate plane. c. Writing Is y x a linear function? Explain. 67. Extended Problem Solving A tsunami is an ocean wave that moves very fast in deep water, but slows as it reaches shallow water. As the wave slows, it rises to great heights, often causing enormous destruction on land. A tsunami’s speed s (in feet per second) and the depth d of the water (in feet) are related by the equation s 2 32d. Suppose an earthquake at sea produces a tsunami in water 15,000 feet deep. In the Real World Tsunamis An earthquake occurred off the coast of Chile in 1960, generating a tsunami. The tsunami reached Japan, about 10,000 miles away, 22 hours later. To the nearest mile per hour, how fast did the tsunami travel? a. Calculate Find the original speed of the wave to the nearest mile per hour. b. Apply The wave enters a harbor 45 feet deep. Find the change in the wave’s speed from the original speed. Give your answer to the nearest mile per hour. 68. The cube root of a number n is the number m such a 0 1 8 27 64 that m3 n. For example, because 23 8, the cube 3 2. The table root of 8 is 2. You write this as 8 shows some whole numbers and their cube roots. a. Use the table to approximate each cube root to 3 3 3 the nearest integer: 3 , 55 , 22 . 3 a 0 1 2 3 4 b. Critical Thinking Do negative numbers have cube roots? Explain. c. Solve the equation x 3 125. 69. Challenge Solve (x 2)2 1 37. Describe the steps you use. Mixed Review Write the prime factorization of the number. (Lesson 4.1) 70. 45 71. 98 72. 484 73. 700 Write the fraction in simplest form. (Lesson 4.3) 21 74. 48 13 75. 52 30 76. 125 30 77. 162 Use the percent equation to answer the question. (Lesson 7.4) Standardized Test Practice 78. What percent of 240 is 42? 79. What number is 80% of 60? 80. 7 is 3.5% of what number? 81. What percent of 20 is 1.3? 82. Multiple Choice What is 500 to the nearest integer? A. 20 B. 21 C. 22 D. 23 83. Multiple Choice What is the value of the expression mn 2 when m 4 and n 5? F. 10 G. 20 H. 80 Lesson 9.1 I. 100 Square Roots 457

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