Successfully reported this slideshow.
Sections 7-7 and 7-8
Rational Exponents
HOW DO YOU WORK WITH RATIONAL EXPONENTS?
3
                         16   4


WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY
                        GOING...
3
                         16   4


WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY
                        GOING...
3
                         16   4


WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY
                        GOING...
3
                         16   4


WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY
                        GOING...
3
                         16   4


WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY
                        GOING...
3
                         16   4


WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY
                        GOING...
3
                         16   4


WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY
                        GOING...
Rational Exponent
Theorem
Rational Exponent
  Theorem
For any nonnegative real number x and positive integers m and n,
Rational Exponent
  Theorem
For any nonnegative real number x and positive integers m and n,

          m       1
        ...
Rational Exponent
  Theorem
For any nonnegative real number x and positive integers m and n,

          m       1
        ...
Example 1
        SIMPLIFY.
Example 1
        SIMPLIFY.
                 3
            36   2
Example 1
        SIMPLIFY.
                 3
            36   2


                 1
                     3
        = (3...
Example 1
        SIMPLIFY.
                 3
            36   2


                 1
                         3
        ...
Example 1
        SIMPLIFY.
                 3
            36   2


                 1
                         3
        ...
Example 2
  APPROXIMATE TO THE NEAREST THOUSANDTH.

                       3
                  16   5
Example 2
  APPROXIMATE TO THE NEAREST THOUSANDTH.

                       3
                  16   5
Example 2
  APPROXIMATE TO THE NEAREST THOUSANDTH.

                       3
                  16   5




                ...
Exploration
        FIND 251 AND 252.
Exploration
        FIND 251 AND 252.

              1
           25 = 25
Exploration
        FIND 251 AND 252.

              1
           25 = 25
              2
           25 = 625
Exploration
                FIND 251 AND 252.

                      1
                   25 = 25
                      2
...
Exploration
                  FIND 251 AND 252.

                        1
                     25 = 25
                  ...
Exploration
                FIND 251 AND 252.

                        1
                   25 = 25
                      ...
Exploration
                FIND 251 AND 252.

                        1
                   25 = 25
                      ...
Exploration
                FIND 251 AND 252.

                        1
                   25 = 25
                      ...
Exploration
                FIND 251 AND 252.

                        1
                   25 = 25
                      ...
Exploration
                 FIND 251 AND 252.

                           1
                      25 = 25
               ...
Example 3
  LIST IN ORDER FROM SMALLEST TO LARGEST.
                  4       3
              2       1       3
          ...
Example 3
  LIST IN ORDER FROM SMALLEST TO LARGEST.
                  4       3
              2       1       3
          ...
Example 4
        SOLVE.
Example 4
        SOLVE.
        5
       x = 243
        4
Example 4
             SOLVE.
             5
        x = 243
             4


         5       4    4
       ( x ) = 243
 ...
Example 4
             SOLVE.
             5
        x = 243
             4


         5       4        4
       ( x ) = 2...
Example 4
             SOLVE.
             5
        x = 243
             4


         5       4        4
       ( x ) = 2...
Example 4
             SOLVE.
             5
        x = 243
             4


         5       4        4
       ( x ) = 2...
***CAUTION***
     BE CAREFUL WHEN WORKING WITH EVEN ROOTS OF NUMBERS.

                          2
                     (...
Example 5
        SIMPLIFY.
Example 5
        SIMPLIFY.
                 −3
            25   2
Example 5
        SIMPLIFY.
                 −3
            25   2

                 1
        = (25 )  2    −3
Example 5
        SIMPLIFY.
                 −3
            25   2

                 1
        = (25 )  2    −3


        ...
Example 5
        SIMPLIFY.
                 −3
            25   2

                 1
        = (25 )  2      −3


      ...
Example 5
        SIMPLIFY.
                 −3
            25   2

                 1
        = (25 )  2    −3


        ...
Rational Exponent
Theorem (Negatives)
Rational Exponent
Theorem (Negatives)
                                                        1
    −m          1         ...
Rational Exponent
Theorem (Negatives)
                                                        1
    −m          1         ...
Choose what you do
first!
Choose what you do
first!
                             −m       1
    NEGATIVE EXPONENT:   x    n
                         ...
Choose what you do
first!
                             −m       1
    NEGATIVE EXPONENT:   x    n
                         ...
Choose what you do
first!
                                  −m       1
         NEGATIVE EXPONENT:   x    n
               ...
Example 6
        SOLVE.
Example 6
        SOLVE.
            −2
        x =95
Example 6
              SOLVE.
              −2
         x =9 5

         −2    −5      −5
       ( x ) = (9)
         5  ...
Example 6
              SOLVE.
              −2
         x =9 5

         −2    −5          −5
       ( x ) = (9)
        ...
Example 6
              SOLVE.
              −2
         x =9 5

         −2    −5          −5
       ( x ) = (9)
        ...
Example 6
              SOLVE.
              −2
         x =9 5

         −2    −5            −5
       ( x ) = (9)
      ...
Example 6
              SOLVE.
              −2
         x =9 5

         −2    −5            −5
       ( x ) = (9)
      ...
Homework
Homework


                P. 461 #1-21 ODD, P. 466 #1-23 ODD




“IF WE ATTEND CONTINUALLY AND PROMPTLY TO THE LITTLE THA...
Upcoming SlideShare
Loading in …5
×

AA Section 7-7/7-8

756 views

Published on

Rational Exponents

Published in: Education, Technology, Business
  • Be the first to comment

  • Be the first to like this

AA Section 7-7/7-8

  1. 1. Sections 7-7 and 7-8 Rational Exponents
  2. 2. HOW DO YOU WORK WITH RATIONAL EXPONENTS?
  3. 3. 3 16 4 WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY GOING ON?
  4. 4. 3 16 4 WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY GOING ON? 3 16 4
  5. 5. 3 16 4 WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY GOING ON? 3 16 4 = 16 () (3) 1 4
  6. 6. 3 16 4 WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY GOING ON? 3 16 4 = 16 () (3) 1 4 1 3 = (16 ) 4
  7. 7. 3 16 4 WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY GOING ON? 3 3 16 4 16 4 = 16 () (3) 1 4 1 3 = (16 ) 4
  8. 8. 3 16 4 WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY GOING ON? 3 3 16 4 16 4 = 16 () (3) 1 4 = 16 ( )( 3 ) 1 4 1 3 = (16 ) 4
  9. 9. 3 16 4 WE CAN SOLVE THIS USING A CALCULATOR, BUT WHAT IS REALLY GOING ON? 3 3 16 4 16 4 = 16 () (3) 1 4 = 16 ( )( 3 ) 1 4 1 1 3 3 = (16 ) 4 = (16 ) 4
  10. 10. Rational Exponent Theorem
  11. 11. Rational Exponent Theorem For any nonnegative real number x and positive integers m and n,
  12. 12. Rational Exponent Theorem For any nonnegative real number x and positive integers m and n, m 1 m x = (x ) n n , the mth power of the nth root of x
  13. 13. Rational Exponent Theorem For any nonnegative real number x and positive integers m and n, m 1 m x = (x ) n n , the mth power of the nth root of x m 1 m x = (x ) n n , the nth root of the mth power of x
  14. 14. Example 1 SIMPLIFY.
  15. 15. Example 1 SIMPLIFY. 3 36 2
  16. 16. Example 1 SIMPLIFY. 3 36 2 1 3 = (36 ) 2
  17. 17. Example 1 SIMPLIFY. 3 36 2 1 3 = (36 ) 2 3 = (6)
  18. 18. Example 1 SIMPLIFY. 3 36 2 1 3 = (36 ) 2 3 = (6) = 216
  19. 19. Example 2 APPROXIMATE TO THE NEAREST THOUSANDTH. 3 16 5
  20. 20. Example 2 APPROXIMATE TO THE NEAREST THOUSANDTH. 3 16 5
  21. 21. Example 2 APPROXIMATE TO THE NEAREST THOUSANDTH. 3 16 5 ≈ 5.278
  22. 22. Exploration FIND 251 AND 252.
  23. 23. Exploration FIND 251 AND 252. 1 25 = 25
  24. 24. Exploration FIND 251 AND 252. 1 25 = 25 2 25 = 625
  25. 25. Exploration FIND 251 AND 252. 1 25 = 25 2 25 = 625 3 WHERE IS 25 IN RELATION TO THESE TWO VALUES? 2
  26. 26. Exploration FIND 251 AND 252. 1 25 = 25 2 25 = 625 3 WHERE IS 25 IN RELATION TO THESE TWO VALUES? 2 3 25 2
  27. 27. Exploration FIND 251 AND 252. 1 25 = 25 2 25 = 625 3 WHERE IS 25 IN RELATION TO THESE TWO VALUES? 2 3 1 3 25 = (25 ) 2 2
  28. 28. Exploration FIND 251 AND 252. 1 25 = 25 2 25 = 625 3 WHERE IS 25 IN RELATION TO THESE TWO VALUES? 2 3 1 3 3 25 = (25 ) = (5) 2 2
  29. 29. Exploration FIND 251 AND 252. 1 25 = 25 2 25 = 625 3 WHERE IS 25 IN RELATION TO THESE TWO VALUES? 2 3 1 25 = (25 ) = (5) = 125 2 2 3 3
  30. 30. Exploration FIND 251 AND 252. 1 25 = 25 2 25 = 625 3 WHERE IS 25 IN RELATION TO THESE TWO VALUES? 2 3 1 25 = (25 ) = (5) = 125 2 2 3 3 25 < 125 < 625
  31. 31. Exploration FIND 251 AND 252. 1 25 = 25 2 25 = 625 3 WHERE IS 25 IN RELATION TO THESE TWO VALUES? 2 3 1 25 = (25 ) = (5) = 125 2 2 3 3 25 < 125 < 625 3 1 2 25 < 25 < 25 2
  32. 32. Example 3 LIST IN ORDER FROM SMALLEST TO LARGEST. 4 3 2 1 3 6 ,6 ,6 ,6 ,6 3 2
  33. 33. Example 3 LIST IN ORDER FROM SMALLEST TO LARGEST. 4 3 2 1 3 6 ,6 ,6 ,6 ,6 3 2 4 3 1 2 3 6 ,6 ,6 ,6 ,6 3 2
  34. 34. Example 4 SOLVE.
  35. 35. Example 4 SOLVE. 5 x = 243 4
  36. 36. Example 4 SOLVE. 5 x = 243 4 5 4 4 ( x ) = 243 4 5 5
  37. 37. Example 4 SOLVE. 5 x = 243 4 5 4 4 ( x ) = 243 4 5 5 1 4 x = (243 ) 5
  38. 38. Example 4 SOLVE. 5 x = 243 4 5 4 4 ( x ) = 243 4 5 5 1 4 x = (243 ) 5 4 x =3
  39. 39. Example 4 SOLVE. 5 x = 243 4 5 4 4 ( x ) = 243 4 5 5 1 4 x = (243 ) 5 4 x =3 x = 81
  40. 40. ***CAUTION*** BE CAREFUL WHEN WORKING WITH EVEN ROOTS OF NUMBERS. 2 (−3) = 9 = 3 ≠ −3 NEVER SIMPLIFY THE FRACTIONAL EXPONENTS. WORK WITH THEM AS THEY ARE!!!
  41. 41. Example 5 SIMPLIFY.
  42. 42. Example 5 SIMPLIFY. −3 25 2
  43. 43. Example 5 SIMPLIFY. −3 25 2 1 = (25 ) 2 −3
  44. 44. Example 5 SIMPLIFY. −3 25 2 1 = (25 ) 2 −3 =5 −3
  45. 45. Example 5 SIMPLIFY. −3 25 2 1 = (25 ) 2 −3 =5 −3 1 = 3 5
  46. 46. Example 5 SIMPLIFY. −3 25 2 1 = (25 ) 2 −3 =5 −3 1 = 3 5 1 = 125
  47. 47. Rational Exponent Theorem (Negatives)
  48. 48. Rational Exponent Theorem (Negatives) 1 −m 1 1 m −1 m m −1 n x n = (( x ) ) = (( x ) ) = (( x ) ) n n −1 1 1 1 −1 m m −1 m n = (( x ) ) = (( x ) ) = (( x ) ) n −1 n
  49. 49. Rational Exponent Theorem (Negatives) 1 −m 1 1 m −1 m m −1 n x n = (( x ) ) = (( x ) ) = (( x ) ) n n −1 1 1 1 −1 m m −1 m n = (( x ) ) = (( x ) ) = (( x ) ) n −1 n IN OTHER WORDS, BREAK IT DOWN INTO THREE STEPS
  50. 50. Choose what you do first!
  51. 51. Choose what you do first! −m 1 NEGATIVE EXPONENT: x n = m x n
  52. 52. Choose what you do first! −m 1 NEGATIVE EXPONENT: x n = m x n ROOT: MAKES THE NUMBER SMALLER
  53. 53. Choose what you do first! −m 1 NEGATIVE EXPONENT: x n = m x n ROOT: MAKES THE NUMBER SMALLER EXPONENT >1 (NUMERATOR): MAKES NUMBER LARGER
  54. 54. Example 6 SOLVE.
  55. 55. Example 6 SOLVE. −2 x =95
  56. 56. Example 6 SOLVE. −2 x =9 5 −2 −5 −5 ( x ) = (9) 5 2 2
  57. 57. Example 6 SOLVE. −2 x =9 5 −2 −5 −5 ( x ) = (9) 5 2 2 1 5 −1 x = ((9 ) ) 2
  58. 58. Example 6 SOLVE. −2 x =9 5 −2 −5 −5 ( x ) = (9) 5 2 2 1 5 −1 x = ((9 ) ) 2 5 −1 x = (3 )
  59. 59. Example 6 SOLVE. −2 x =9 5 −2 −5 −5 ( x ) = (9) 5 2 2 1 5 −1 x = ((9 ) ) 2 5 −1 x = (3 ) x = 243 −1
  60. 60. Example 6 SOLVE. −2 x =9 5 −2 −5 −5 ( x ) = (9) 5 2 2 1 5 −1 x = ((9 ) ) 2 5 −1 x = (3 ) x = 243 −1 x = 243 1
  61. 61. Homework
  62. 62. Homework P. 461 #1-21 ODD, P. 466 #1-23 ODD “IF WE ATTEND CONTINUALLY AND PROMPTLY TO THE LITTLE THAT WE CAN DO, WE SHALL ERE LONG BE SURPRISED TO FIND HOW LITTLE REMAINS THAT WE CANNOT DO.” - SAMUEL BUTLER

×