Notes 6-2
52
Section 6-2 Rational Power Functions
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Rational Power Functions
Transcript of Notes 6-2
Section 6-2Rational Power Functions
Warm-upThe frequencies of the notes beginning with A above
middle C are usually:
A = 440,A# = 440i21
12 ,B = 440i22
12 ,C = 440i23
12 ,
C # = 440i24
12 ,D = 440i25
12
1. Calculate the frequencies (to the nearest integer) of A#, B, C, C#, and D.
Warm-upThe frequencies of the notes beginning with A above
middle C are usually:
A = 440,A# = 440i21
12 ,B = 440i22
12 ,C = 440i23
12 ,
C # = 440i24
12 ,D = 440i25
12
1. Calculate the frequencies (to the nearest integer) of A#, B, C, C#, and D.
466, 494, 523, 554, 587
Warm-up2. 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.
Warm-up2. Write an expression for the frequency of the note n
notes above A.
440i2n
12
3. Use the expression you have written to find the frequency of the G that is two notes below A.
Warm-up2. Write an expression for the frequency of the note n
notes above A.
440i2n
12
3. Use the expression you have written to find the frequency of the G that is two notes below A.
440i2−212
Example 1Rewrite as a radical:
Example 1Rewrite as a radical:
x47
Example 1Rewrite as a radical:
x47
= x
17⎛
⎝⎞⎠
4
Example 1Rewrite as a radical:
x47
= x
17⎛
⎝⎞⎠
4
= x7( )4
Example 1Rewrite as a radical:
x47
= x
17⎛
⎝⎞⎠
4
= x7( )4
or
Example 1Rewrite as a radical:
x47
= x
17⎛
⎝⎞⎠
4
= x4( )
17
= x7( )4
or
Example 1Rewrite as a radical:
x47
= x
17⎛
⎝⎞⎠
4
= x4( )
17
= x7( )4
or
= x47
Rational Exponent Theorem
Rational Exponent Theorem
x
mn = x
1n⎛
⎝⎞⎠
m
= xn( )m
Rational Exponent Theorem
x
mn = x
1n⎛
⎝⎞⎠
m
= xn( )mthe mth power of the positive nth root of x
Rational Exponent Theorem
x
mn = x
1n⎛
⎝⎞⎠
m
= xn( )mthe mth power of the positive nth root of x
x
mn = xm( )
1n = xmn
Rational Exponent Theorem
x
mn = x
1n⎛
⎝⎞⎠
m
= xn( )mthe mth power of the positive nth root of x
x
mn = xm( )
1n = xmn
the positive nth root of the mth power of x
Postulates for Powers
Postulates for Powers
xmixn = xm+n
Postulates for Powers
xy( )m = xmym
xmixn = xm+n
Postulates for Powers
xy( )m = xmym
xm
xn= xm−n ,x ≠ 0
xmixn = xm+n
Postulates for Powers
xy( )m = xmym
xm
xn= xm−n ,x ≠ 0
xmixn = xm+n
xy( )m = xm
ym
A couple others...
A couple others...
x0 =1
A couple others...
x0 =1
x−n = 1
xn
Example 2Evaluate:
6423
Example 2Evaluate:
6423
By hand:
Example 2Evaluate:
6423
By hand:
64
13⎛
⎝⎞⎠
2
Example 2Evaluate:
6423
By hand:
64
13⎛
⎝⎞⎠
2
= 42
Example 2Evaluate:
6423
By hand:
64
13⎛
⎝⎞⎠
2
= 42 =16
Example 2Evaluate:
6423
By hand:
64
13⎛
⎝⎞⎠
2
By Calculator:
= 42 =16
Example 2Evaluate:
6423
By hand:
64
13⎛
⎝⎞⎠
2
By Calculator:
= 42 =16
Example 3Evaluate:
32−35
Example 3Evaluate:
32−35
By hand:
Example 3Evaluate:
32−35
By hand:
32
15⎛
⎝⎞⎠
3⎛
⎝⎜
⎞
⎠⎟
−1
Example 3Evaluate:
32−35
By hand:
32
15⎛
⎝⎞⎠
3⎛
⎝⎜
⎞
⎠⎟
−1
= 23( )−1
Example 3Evaluate:
32−35
By hand:
32
15⎛
⎝⎞⎠
3⎛
⎝⎜
⎞
⎠⎟
−1
= 23( )−1
= 8−1
Example 3Evaluate:
32−35
By hand:
32
15⎛
⎝⎞⎠
3⎛
⎝⎜
⎞
⎠⎟
−1
= 23( )−1
= 8−1
=18
Example 3Evaluate:
32−35
By hand:
32
15⎛
⎝⎞⎠
3⎛
⎝⎜
⎞
⎠⎟
−1
= 23( )−1
= 8−1
=18
By calculator:
Example 3Evaluate:
32−35
By hand:
32
15⎛
⎝⎞⎠
3⎛
⎝⎜
⎞
⎠⎟
−1
= 23( )−1
= 8−1
=18
By calculator:
Example 4
y = h x( ) = x
32
a. Find an equation for the inverse of h.
Example 4
y = h x( ) = x
32
a. Find an equation for the inverse of h.
y = x23
Example 4
y = h x( ) = x
32
a. Find an equation for the inverse of h.
y = x23
b. Is the inverse a function?
Example 4
y = h x( ) = x
32
a. Find an equation for the inverse of h.
y = x23
b. Is the inverse a function?Yes
Example 4
y = h x( ) = x
32
a. Find an equation for the inverse of h.
y = x23
b. Is the inverse a function?Yes
...but only on D = {x: x ≥ 0}
Example 4
y = h x( ) = x
32
a. Find an equation for the inverse of h.
y = x23
b. Is the inverse a function?Yes
...but only on D = {x: x ≥ 0}c. Plot h and its inverse
Example 4
y = h x( ) = x
32
a. Find an equation for the inverse of h.
y = x23
b. Is the inverse a function?Yes
...but only on D = {x: x ≥ 0}c. Plot h and its inverse
Example 4
y = h x( ) = x
32
a. Find an equation for the inverse of h.
y = x23
b. Is the inverse a function?Yes
...but only on D = {x: x ≥ 0}c. Plot h and its inverse
Example 4
y = h x( ) = x
32
a. Find an equation for the inverse of h.
y = x23
b. Is the inverse a function?Yes
...but only on D = {x: x ≥ 0}c. Plot h and its inverse
Homework
Homework
p. 380 #1-22
