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# AA Section 7-7/7-8

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Rational Exponents

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### 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