Page 1 of 5 10.1 Simplifying Square Roots Goal Simplify square roots. Key Words radial radiand Square roots are written with a radial symbol m. An epression written with a radial symbol is alled a radial epression, or radial. The number or epression inside the radial symbol is the radiand. radial symbol 2 5 radial radiand The radial symbol always indiates the nonnegative square root of a number. For eample, 2 5 5 beause 5 2 25. Find the square root of 52. Round your answer to the nearest tenth. Chek that your answer is reasonable. 1 Use a Calulator to Find Square Roots Calulator keystrokes Display Rounded value 52 or 52 7.21110 5 2 7.2 This is reasonable, beause 52 is between the perfet squares 49 and 64. So, 5 2 should be between 4 9 and 6 4, or 7 and 8. The answer 7.2 is between 7 and 8. STUDY TIP Reall that for any number a 0, ( a ) 2 a. Use the Pythagorean Theorem to find the length of the hypotenuse to the nearest tenth. 2 Find Side Lengths a 2 b 2 2 Write Pythagorean Theorem. ( 2 ) 2 ( 3 ) 2 2 Substitute 2 for a and 3 for b. 2 3 2 Simplify. 5 2 Add. 5 Take the square root of eah side. 2.2 Use a alulator. 2 3 10.1 Simplifying Square Roots 537
Page 2 of 5 SKILLS REVIEW To review the Produt Property of Radials, see p. 669. Multiplying Radials You an use the Produt Property of Radials to multiply radial epressions. a p b a b, where a 0 and b 0. 3 Multiply Radials Multiply the radials. Then simplify if possible. a. 3 p 7 b. 2 p 8 a. 3 p 7 3 p 7 b. 2 p 8 2 p 8 2 1 1 6 4 Simplifying Radials You an also use the Produt Property of Radials to simplify radial epressions. a b a p b, where a 0 and b 0. To fator the radiand, look for perfet square fators. 4 Simplify Radials Simplify the radial epression. STUDY TIP When you fator a radiand, write the perfet square fators first. a. 1 2 b. 4 5 a. 1 2 4 p 3 b. 4 5 9 p 5 4 p 3 9 p 5 2 3 3 5 Find the square root. Round your answer to the nearest tenth. Chek that your answer is reasonable. 1. 2 7 2. 4 6 3. 8 4. 9 7 Multiply the radials. Then simplify if possible. 5. 3 p 5 6. 1 1 p 6 7. 3 p 2 7 8. 5 3 p 3 Simplify the radial epression. 538 Chapter 10 Right Triangles and Trigonometry Evaluate, Multiply, and Simplify Radials 9. 2 0 10. 8 11. 7 5 12. 1 1 2
Page 3 of 5 10.1 Eerises Guided Pratie Voabulary Chek 1. What is the radiand in the epression 2 5? Math the radial epression with its simplified form. 2. 3 6 A. 6 3. 3 p 2 B. 3 2 4. 3 p 6 C. 4 2 Skill Chek 5. 3 2 D. 6 Use the figure shown at the right. 6. Use the Pythagorean Theorem to find the length of the hypotenuse in radial form. 7. Use a alulator to find the length of the hypotenuse to the nearest tenth. 2 4 Simplify the epression. 8. 4 9 9. 2 8 10. 7 2 11. 5 4 Pratie and Appliations Etra Pratie See p. 693. Finding Square Roots Find the square root. Round your answer to the nearest tenth. Chek that your answer is reasonable. 12. 1 3 13. 6 14. 9 1 15. 3 4 16. 1 0 6 17. 1 4 8 18. 6 2 19. 1 8 6 Pythagorean Theorem Find the length of the hypotenuse. Write your answer in radial form. 20. 21. 22. 2 13 5 6 19 10 Homework Help Eample 1: Es. 12 19 Eample 2: Es. 20 25 Eample 3: Es. 26 34 Eample 4: Es. 35 45 Pythagorean Theorem Find the missing side length of the right triangle. Round your answer to the nearest tenth. 23. 24. 25. 1 6 11 4 13 5 10.1 Simplifying Square Roots 539
Page 4 of 5 Multiplying Radials Multiply the radials. Then simplify if possible. 26. 7 p 2 27. 5 p 5 28. 3 p 1 1 29. 2 5 p 7 30. 1 0 p4 3 31. 1 1 p 2 2 Square a Radial Evaluate the epression. a. (3 7 ) 2 b. (2 1 1 ) 2 a. (3 7 ) 2 3 7 p3 7 b. (2 1 1 ) 2 2 1 1 p2 1 1 3p3p 7 p 7 2p2p 1 1 p 1 1 9p7 4p11 63 44 Squaring Radials Evaluate the epression. Use the eample above as a model. 32. (6 5 ) 2 33. (5 3 ) 2 34. (7 2 ) 2 Simplifying Radials Simplify the radial epression. 35. 1 8 36. 5 0 37. 4 8 38. 6 0 39. 5 6 40. 1 2 5 41. 2 0 0 42. 1 6 2 43. 4 4 You be the Judge Determine whether the epression an be simplified further. If so, eplain how you would do so. 44. 8 0 4 p 2 0 45. 8 p 1 2 8 p 1 2 4 p 2 0 2 2 0 4 p 2 p 4 p 3 4 6 Area Formula Use the area formula A lw to find the area of the retangle. Round your answer to the nearest tenth. I I C L A S S Z O N E. C O M HOMEWORK HELP Etra help with problem solving in Es. 46 51 is at lasszone.om 46. 47. 48. 2 5 10 8 3 14 49. 50. 51. 4 3 4 6 4 3 8 3 9 2 3 2 14 4 6 540 Chapter 10 Right Triangles and Trigonometry
Page 5 of 5 52. Area of an Equilateral Triangle The area of an equilateral triangle with side length s is given by the formula A 1 4 s 2 3. The flower bed shown is an equilateral triangle with a side length of 30 feet. Find its area. 30 ft 30 ft 30 ft 53. Challenge An equilateral triangle has an area of 1 square meter. What is the length of eah side? Round your answer to the nearest entimeter. Standardized Test Pratie 54. Multiple Choie Whih number is a perfet square? A 44 B 110 C 169 D 500 55. Multiple Choie 2 2 0 is between whih two integers? F 12 and 13 G 13 and 14 H 14 and 15 J 15 and 16 56. Multiple Choie Whih of the following epressions ould not be used to represent the length of the hypotenuse in the triangle shown at the right? 2 10 8 A 2 2 6 B 1 0 4 C about 10.2 D 6 1 0 Mied Review Finding Angle Measures Find the measure of a1. (Lesson 4.2) 57. 58. 59. 1 61 87 45 1 51 1 25 Isoseles Triangles Find the value of. (Lesson 4.3) 60. 61. 62. 9 4 2 3 64 4 Algebra Skills Distributive Property Use the distributive property to rewrite the epression without parentheses. (Skills Review, p. 671) 63. ( 5) 64. 4(2 1) 65. (3 4) 66. 5( 2) 67. 3(1 ) 68. 2 ( 6) 10.1 Simplifying Square Roots 541