1) The document discusses rational power functions and provides the standard frequencies of musical notes starting from A above middle C. 2) It gives an expression for calculating the frequency of any note n notes above A as 440*(2^(1/12))^n. 3) Using this expression, it calculates the frequency of the G note that is two notes below A.

1. Section 6-2
Rational Power Functions

2. Warm-up
The frequencies of the notes beginning with A above
middle C are usually:
1 2 3
A = 440, A # = 440i2 ,B = 440i2 ,C = 440i2 ,
12 12 12
4 5
C # = 440i2 ,D = 440i2
12 12
1. Calculate the frequencies (to the nearest integer) of
A#, B, C, C#, and D.

3. Warm-up
The frequencies of the notes beginning with A above
middle C are usually:
1 2 3
A = 440, A # = 440i2 ,B = 440i2 ,C = 440i2 ,
12 12 12
4 5
C # = 440i2 ,D = 440i2
12 12
1. Calculate the frequencies (to the nearest integer) of
A#, B, C, C#, and D.
466, 494, 523, 554, 587

4. Warm-up
2. Write an expression for the frequency of the note n
notes above A.
3. Use the expression you have written to find the
frequency of the G that is two notes below A.

5. Warm-up
2. Write an expression for the frequency of the note n
notes above A.
n
440i2 12
3. Use the expression you have written to find the
frequency of the G that is two notes below A.

6. Warm-up
2. Write an expression for the frequency of the note n
notes above A.
n
440i2 12
3. Use the expression you have written to find the
frequency of the G that is two notes below A.
−2
440i2 12

7. Example 1
Rewrite as a radical:

8. Example 1
Rewrite as a radical:
4
x 7

9. Example 1
Rewrite as a radical:
4
x 7
4
= ⎛x ⎞1
7
⎝ ⎠

10. Example 1
Rewrite as a radical:
4
x 7
( x)
4 4
= ⎛x ⎞1
7
⎝ ⎠
7
=

11. Example 1
Rewrite as a radical:
4
x 7
( x)
4 4
= ⎛x ⎞1
7
⎝ ⎠
7
=
or

12. Example 1
Rewrite as a radical:
4
x 7
( x)
4 4
= ⎛x ⎞1
7
⎝ ⎠
7
=
or
( )
1
4
= x
7

13. Example 1
Rewrite as a radical:
4
x 7
( x)
4 4
= ⎛x ⎞1
7
⎝ ⎠
7
=
or
( )
1
4
= x 7
7
4
= x

14. Rational Exponent Theorem

15. Rational Exponent Theorem
( )
m m
m
x =
n ⎛x ⎞ =
1
n
n
x
⎝ ⎠

16. Rational Exponent Theorem
( )
m m
m
x =
n ⎛x ⎞ =
1
n
n
x
⎝ ⎠
the m power of the positive n root of x
th th

17. Rational Exponent Theorem
( )
m m
m
x =
n ⎛x ⎞ =
1
n
n
x
⎝ ⎠
the m power of the positive n root of x
th th
( )
1
m
m n m
x = x = x
n
n

18. Rational Exponent Theorem
( )
m m
m
x =
n ⎛x ⎞ =
1
n
n
x
⎝ ⎠
the m power of the positive n root of x
th th
( )
1
m
m n m
x = x = x
n
n
the positive nth root of the mth power of x

19. Postulates for Powers

20. Postulates for Powers
m n m+n
x ix = x

21. Postulates for Powers
m n m+n
x ix = x
( xy )
m
m m
=x y

22. Postulates for Powers
m n m+n
x ix = x
( xy )
m
m m
=x y
m
x m−n
n
=x ,x ≠ 0
x

23. Postulates for Powers
m n m+n
x ix = x
( xy )
m
m m
=x y
m
x m−n
n
=x ,x ≠ 0
x
()
m
m
x
y
= x
m
y

24. A couple others...

25. A couple others...
0
x =1

26. A couple others...
0
x =1
x −n
= 1
n
x

27. Example 2
Evaluate:
2
64 3

28. Example 2
Evaluate:
2
64 3
By hand:

29. Example 2
Evaluate:
2
64 3
By hand:
2
⎛ 64 ⎞
1
3
⎝ ⎠

30. Example 2
Evaluate:
2
64 3
By hand:
2
⎛ 64 ⎞
1
2
⎝
3
⎠ =4

31. Example 2
Evaluate:
2
64 3
By hand:
2
⎛ 64 ⎞
1
⎝
3
⎠ = 4 = 16 2

32. Example 2
Evaluate:
2
64 3
By hand:
2
⎛ 64 ⎞
1
⎝
3
⎠ = 4 = 16 2
By Calculator:

33. Example 2
Evaluate:
2
64 3
By hand:
2
⎛ 64 ⎞
1
⎝
3
⎠ = 4 = 16 2
By Calculator:

34. Example 3
Evaluate:
−3
32 5

35. Example 3
Evaluate:
−3
32 5
By hand:

36. Example 3
Evaluate:
−3
32 5
By hand:
−1
⎛ ⎛ 1 ⎞ 3⎞
⎜⎝ 325 ⎟
⎝ ⎠ ⎠

37. Example 3
Evaluate:
−3
32 5
By hand:
−1
( )
⎛ ⎛ 1 ⎞ 3⎞ 3
−1
⎜⎝ 325 ⎟
⎠ ⎠
= 2
⎝

38. Example 3
Evaluate:
−3
32 5
By hand:
−1
( )
⎛ ⎛ 1 ⎞ 3⎞ −1
⎜⎝ 325 ⎟
⎠ ⎠
= 2 3
=8
−1
⎝

39. Example 3
Evaluate:
−3
32 5
By hand:
−1
( )
⎛ ⎛ 1 ⎞ 3⎞ −1
⎜⎝ 325 ⎟
⎠ ⎠
= 2 3
=8
−1
= 1
8
⎝

40. Example 3
Evaluate:
−3
32 5
By hand:
−1
( )
⎛ ⎛ 1 ⎞ 3⎞ −1
⎜⎝ 325 ⎟
⎠ ⎠
= 2 3
=8
−1
= 1
8
⎝
By calculator:

41. Example 3
Evaluate:
−3
32 5
By hand:
−1
( )
⎛ ⎛ 1 ⎞ 3⎞ −1
⎜⎝ 325 ⎟
⎠ ⎠
= 2 3
=8
−1
= 1
8
⎝
By calculator:

42. Example 4
y = h( x) = x
3
2
a. Find an equation for the inverse of h.

43. Example 4
y = h( x) = x
3
2
a. Find an equation for the inverse of h.
2
y=x 3

44. Example 4
y = h( x) = x
3
2
a. Find an equation for the inverse of h.
2
y=x 3
b. Is the inverse a function?

45. Example 4
y = h( x) = x
3
2
a. Find an equation for the inverse of h.
2
y=x 3
b. Is the inverse a function?
Yes

46. Example 4
y = h( x) = x
3
2
a. Find an equation for the inverse of h.
2
y=x 3
b. Is the inverse a function?
Yes
...but only on D = {x: x ≥ 0}

47. Example 4
y = h( x) = x
3
2
a. Find an equation for the inverse of h.
2
y=x 3
b. Is the inverse a function?
Yes
...but only on D = {x: x ≥ 0}
c. Plot h and its inverse

48. Example 4
y = h( x) = x
3
2
a. Find an equation for the inverse of h.
2
y=x 3
b. Is the inverse a function?
Yes
...but only on D = {x: x ≥ 0}
c. Plot h and its inverse

49. Example 4
y = h( x) = x
3
2
a. Find an equation for the inverse of h.
2
y=x 3
b. Is the inverse a function?
Yes
...but only on D = {x: x ≥ 0}
c. Plot h and its inverse

50. Example 4
y = h( x) = x
3
2
a. Find an equation for the inverse of h.
2
y=x 3
b. Is the inverse a function?
Yes
...but only on D = {x: x ≥ 0}
c. Plot h and its inverse

51. Homework

52. Homework
p. 380 #1-22