This document discusses working with rational exponents. It introduces the rational exponent theorem, which states that for any nonnegative real number x and positive integers m and n, x^(m/n) = (x^(1/n))^m, the mth power of the nth root of x. Several examples are worked through to demonstrate simplifying expressions with rational exponents using this theorem. Special cases for negative rational exponents are also discussed. Homework problems are assigned from the textbook.

1. Sections 7-7 and 7-8
Rational Exponents

2. HOW DO YOU WORK WITH RATIONAL EXPONENTS?

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

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

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. 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. 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. 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. 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. Rational Exponent
Theorem

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

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. 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. Example 1
SIMPLIFY.

15. Example 1
SIMPLIFY.
3
36 2

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

17. Example 1
SIMPLIFY.
3
36 2
1
3
= (36 ) 2
3
= (6)

18. Example 1
SIMPLIFY.
3
36 2
1
3
= (36 ) 2
3
= (6)
= 216

19. Example 2
APPROXIMATE TO THE NEAREST THOUSANDTH.
3
16 5

20. Example 2
APPROXIMATE TO THE NEAREST THOUSANDTH.
3
16 5

21. Example 2
APPROXIMATE TO THE NEAREST THOUSANDTH.
3
16 5
≈ 5.278

22. Exploration
FIND 251 AND 252.

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

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

25. Exploration
FIND 251 AND 252.
1
25 = 25
2
25 = 625
3
WHERE IS 25 IN RELATION TO THESE TWO VALUES?
2

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. 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. 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. 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. 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. 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. Example 3
LIST IN ORDER FROM SMALLEST TO LARGEST.
4 3
2 1 3
6 ,6 ,6 ,6 ,6
3 2

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. Example 4
SOLVE.

35. Example 4
SOLVE.
5
x = 243
4

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

37. Example 4
SOLVE.
5
x = 243
4
5 4 4
( x ) = 243
4 5 5
1
4
x = (243 ) 5

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. 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. ***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. Example 5
SIMPLIFY.

42. Example 5
SIMPLIFY.
−3
25 2

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

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

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

46. Example 5
SIMPLIFY.
−3
25 2
1
= (25 ) 2 −3
=5 −3
1
= 3
5
1
=
125

47. Rational Exponent
Theorem (Negatives)

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. 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. Choose what you do
first!

51. Choose what you do
first!
−m 1
NEGATIVE EXPONENT: x n
= m
x n

52. Choose what you do
first!
−m 1
NEGATIVE EXPONENT: x n
= m
x n
ROOT: MAKES THE NUMBER SMALLER

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. Example 6
SOLVE.

55. Example 6
SOLVE.
−2
x =95

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

57. Example 6
SOLVE.
−2
x =9 5
−2 −5 −5
( x ) = (9)
5 2 2
1
5 −1
x = ((9 ) ) 2

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. 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. 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. Homework

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