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Unit II -- Geo1 Worksheet 2 (pre 9.1) To simplify square roots, you must first be very familiar with “perfect squares.” Perfect squares are numbers that have integer square roots. Let’s complete the perfect square chart: Perfect Square The square root of the perfect square 1 1 4 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 Worksheet 2 (pre 9.1) Unit II -- Geo1 Simplifying With Square Roots: To simplify square roots, first check if the number is a perfect square. If it is, then the number that you multiply by itself to get the perfect square, is your answer. For example, 100 = 10 because 100 is a perfect square and when you multiply 10 by itself you get 100. If the number is not a perfect square, you look for the highest perfect square that is a factor of that number and you rewrite the square root as a product of the perfect square and the other factor. For example, if you want to simplify 72 , the highest perfect square that is a factor of 72 is 36 so you rewrite the square root to look like this: 36 • 2 . Now, we know that the square root of 36 is 6 so we pull out the 6 out of the square root and write our answer as follows: 6 2 . Always make sure that the number inside the square root is completely reduced, meaning there is no perfect square (besides 1) that is a factor of that number! Simplify the following: 1. 49 5. 6 8 9. 13. 48 18 17. 3 44 2. 3 64 6. 27 10. 78 48 14. 20 18. !5 98 3. !4 625 4. 8 7. !10 27 8. 150 11. 90 12. 456 15. 180 16. 162 19. 48 20. !6 54 Adding and Subtracting With Square Roots: One can only combine square roots if the roots are the same. These are considered “like” terms. The root stays the same and the numbers in front of the roots are combined. Make sure to first simplify the roots and then combine like terms. Example: !2 72 ! 6 44 + 5 32 First simplify the roots as follows !2 36 • 2 ! 6 4 •11 + 5 16 • 2 = !12 2 !12 11 + 20 2 = 8 2 !12 11 Simplify the following: 21. 6 13 + 7 13 22. 2 11 ! 8 11 23. 2 12 + 5 3 24. 2 27 ! 4 12 25. 4 8 ! 3 5 26. 8 32 + 4 50 2 Worksheet 2 (pre 9.1) Unit II -- Geo1 27. 6 20 + 45 28. 6 13 + 7 9 30. 2 48 ! 27 31. 150 ! 2 96 160 ! 360 + 250 33. 29. 2 63 + 8 45 ! 6 28 32. 3 135 ! 2 450 34. 5 32 + 35. 3 1440 ! 2 75 ! 192 28 ! 3 128 36. !2 175 + 243 + 5 63 Multiplying With Square Roots: To multiply square roots multiply the outside numbers not in the radical and multiply the inside numbers that are in the radical. Make sure your answer is completely simplified. You may also choose to simplify the radicals first and then you might have to simplify again after multiplying. Example: 2 32 • 4 8 First simplify to 8 2 • 8 2 = 64 4 = 64 • 2 =128 . Notice that when you square a radical, you end up with the number itself without the radical. Example: 5 3 • 24 = 5 3 • 2 6 = 10 18 = 30 2 Simplify the following: 37. 3 2 • 5 7 2 •5 2 40. 43. ! 2 ( ( 2+ 3 ) 46. 3 11 ) 39. 41. ( 42. !3 !2 5 + 4 6 ) ) ( ( 52. 5 2 6 2 ! 3 6 ( )( ( 3!8 5 ) ) 55. 3 6 ! 2 3 6 + 2 ) 2 59. (3 48. 7 8 • 3 10 5+ 7 53. ( 5 !2 56. ( 7 + 2 10 14 ! 7 )( )( ) ( ) ) ) 51. ! 5 3 ! 2 13 5+4 54. ( 57. (3 2 14 + 2 7 3 ) 45. 6 2 • 3 5 47. 5 6 • 2 3 50. ! 3 2 ( 2 !3 3 ( 49. (!3 10)(! 18) 58. (!2 3 )(!9 5 ) 44. 3 2 5 ! 2 2 (2 5 ) 38. ) 60. x2 )( 7!4 ) 7 !1 )( ) 2 !6 3 2!6 61. x8 Unit II -- Geo1 62. x9 63. Worksheet 2 (pre 9.1) x 11 64. x 25 y 3 18 x100 y 27 65. Dividing With Square Roots: Simplify the following: 66. 70. 1 5 67. 20 500 12 600 2 24 3 2 68. 69. 2 6 8 18 71. 27 51 18 17 72. 3 74. x2 x = 21 18 75. x = 10 78. 5x + 4 = 7 1 1 !9 3 12 Equations: Solve the following for x: 73. 76. 4 x = x 12 3x = 6 77. 2x ! 4 = 8 79. (MCAS 2003) What is the simplified form of the expression 450 : a. 15 2 b. 45 2 c. 75 2 d. 225 2 80. (MCAS 2003) What is the solution to the equation x = 16 ? 81. (MCAS 2001) Which of the following statments is true: a. 95 = 10 b. 95 < 10 c. 4 95 > 10 d. 95 < 9