Finding and Estimating Square Roots

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Finding and Estimating Square Roots Powered By Docstoc
					            Finding and Estimating
                Square Roots
ALGEBRA 1 LESSON 10-3

                                                        (For help, go to Lessons 1-2 and 8-5.)


          Simplify each expression.
          1. 112           2. (–12)2      3. –(12)2              4. 1.52


          5. 0.62          6. 1    2      7.    2   2            8. 4     2

                               2
                                               –3                     5




                                       10-3
            Finding and Estimating
                Square Roots
ALGEBRA 1 LESSON 10-3




Solutions
       1. 112 = 11 • 11 = 121                    2. (–12)2 = (–12)(–12) = 144

          3. (–12)2 = –(12)(12) = –144           4. 1.52 = (1.5)(1.5) = 2.25

                                                         2
          5. 0.62 = (0.6)(0.6) = 0.36            6. 1        = 1• 1 = 1
                                                     2         2   2   4
                        2                                2
          7. – 2            = –2    2
                                   –3   = 4      8. 4        = 4 • 4 = 16
               3               3          9          5         5   5       25




                                              10-3
   10.3: Finding and estimating
Vocabulary: square roots
Square root: A number a is a square root of b if a2 = b


Radicand: The number inside of the radical sign.


         The index gives the degree of the root. It is
Index:
         the small number in the “hook” of the radical
         sign.
Principal Root: When a number has two roots, the principle
               root is the positive root.
Property of Square Roots:


        a a
          2
            Finding and Estimating
                Square Roots
ALGEBRA 1 LESSON 10-3




                           Simplify each expression.

          a.     25 = 5                positive square root

                   9     3                                    3      3
          b. ±        = ±5             The square roots are     and – .
                   25                                         5      5


          c. –     64 = –8             negative square root


          d.     –49 is undefined      For real numbers, the square root of a negative
                                       number is undefined.

          e. ±      0= 0               There is only one square root of 0.



                                             10-3
Rational:                              Irrational:
A number that can be                   A number that canNOT be
written in RATIO form.                 written as a RATIO.

                 Tell whether each expression is rational or irrational.

   a. ±    144 = ± 12                rational


           1
   b. –    5
             = –0.44721359 . . .     irrational


   c. –    6.25 = –2.5               rational


          1
   d.       = 0.3                    rational
          9


   e.     7 = 2.64575131 . . .       irrational

                                     10-3
    Table of Perfect Squares: page 750


N   N2             N        N2
1   1             9         81
2   4             10        100
3   9             11        121
4   16            13        169
5   25            14        196
6   36            15        225
7   49            16        256
8   64            17        289
                                                                 N      N2
                                                                 1      1
                                                                 2      4
                                                                 3      9
                                                                 4      16
                                                                 5      25
                                                                 6      36
            Between what two consecutive integers is   28.34 ?   7      49
                                                                 8      64
                              28.34 is between the two consecutive
25 <     28.34 <   36
                              perfect squares 25 and 36.



                              The square roots of 25 and 36 are 5 and
5<     28.34 < 6
                              6, respectively.



28.34 is between 5 and 6.
                               10-3
        Find   28.34 to the nearest hundredth.


28.34   5.323532662        Use a calculator.


        5.32               Round to the nearest hundredth.




                          10-3
                Suppose a rectangular field has a length three times its width
       x. The formula d = x2 + (3x)2 gives the distance of the diagonal of a
       rectangle. Find the distance of the diagonal across the field if x = 8 ft.




d=    x2 + (3x)2

d=    82 + (3 • 8)2         Substitute 8 for x.

d=    64 + 576               Simplify.

d=    640

d    25.3                   Use a calculator. Round to the nearest tenth.


The diagonal is about 25.3 ft long.


                                  10-3
            Finding and Estimating
                Square Roots
ALGEBRA 1 LESSON 10-3




          1. Simplify each expression.
                                                 4      2
               a.       196 14        b. ±             ±5
                                                25


          2. Tell whether each expression is rational, irrational, or undefined.
               a. ±       3           b.     –25                   c. –   2.25
                          5
                    irrational             undefined                 rational
          3. Between what two consecutive integers is –      54?
                    –8 and –7
          4. The formula s = 13.5d estimates the speed s in miles per hour that a
             car was traveling, when it applied its brakes and left a skid mark d feet
             long on a wet road. Estimate the speed of a car that left a 120 foot
             long skid mark.
                    about 40.25 mph


                                             10-3
              Finding and Estimating
                  Square Roots
ALGEBRA 1 LESSON 10-3




pages 526–528 Exercises   12. ± 0.1         24. –12.25
    1. 13                 13. irrational    25. 0.93
    2. 20                 14. rational      26. ± 20
    3. 1                  15. irrational    27. 0
         3
    4. 30                 16. rational      28. ± 25
    5. 0.5                17. 5 and 6       29. ±3
                                                    7
    6. 6                  18. 5 and 6       30. ± 1.3
         7
    7. –1.1               19. –12 and –11   31. ±1
                                                    9
    8. 1.4                20. 13 and 14     32. ± 27
    9. 0.6                21. 3.46          33. ± 1.5
    10. –12               22. –14.25        34. ± 16
    11. 5                 23. 107.47        35. ± 0.1
          4


                             10-3
            Finding and Estimating
                Square Roots
ALGEBRA 1 LESSON 10-3




36. ± 8                 47. 6.40                    54. false;   1=1
         11
37. ± 202               48. 8.66                    55. true

38. 1            49. Answers may vary. Sample:     56. true
                     The first expression means    57. False; answers may vary.
39.   a. 8450 km     the neg. square root of 1 and
      b. 7684 km                                       Sample: 4 + 9 = 4 + 9.
                                                                          /
                     the second expression means
40.   21             the pos. square root of 1.    58. False; answers may vary.
                 50. Answers may vary.                 Sample: 12 and 3 are
41.   – 2
                                                       irrational but 36 is rational.
        5            Sample: 3 and 4
42.   1.41                                         59. a. 4 units2
                 51. 4
43.   1.26       52. a. 5 s                            b. 1 unit2
                                                           2
44.   –5.48          b. 10 s                           c. 2 units2
                     c. No; the object takes
45.   –33               twice as long to fall.         d. 2 units
46.   –0.8       53. False; zero has one square root.


                                       10-3
            Finding and Estimating
                Square Roots
ALGEBRA 1 LESSON 10-3




60. 1.3                 67.          70.
61. 6
62. 2
      9
63. 9
64. 11                  68.          71.
65. 128
66.



                        69.          72. d 2 – 81
                                     73. 9t 2 – 25
                                     74. 8075



                              10-3
               Finding and Estimating
                   Square Roots
ALGEBRA 1 LESSON 10-3




75. x2 + 26x + 169
76. 16y2 – 56y + 49
77. 10,201
78. 813,604
79. 36k2 – 49
80. 144b2 – 168b + 49
81. –2
82. 3
      16
83. – 3
           4
84. –972
85. 1
    12
86. 81
     4

                           10-3

				
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