Chapter 7 Radicals, Radical Functions, and Rational Exponents
Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical...
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Transcript of Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical...
![Page 1: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/1.jpg)
Rational ExponentsRational Exponents
Fraction ExponentsFraction Exponents
![Page 2: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/2.jpg)
Radical expression and Exponents
By definition of Radical Expression.
The index of the Radical is 3.
51251255 33 so
![Page 3: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/3.jpg)
How would we simplify this expression?
What does the fraction exponent do to the number?
The number can be written as a Radical expression, with an index of the denominator.
2
1
9
2 9
![Page 4: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/4.jpg)
The Rule for Rational Exponents
46464 33
1
1
nn bb
![Page 5: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/5.jpg)
Write in Radical form
2
1
6
1
m
a
![Page 6: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/6.jpg)
Write in Radical form
mm
aa
2
1
66
1
![Page 7: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/7.jpg)
Write each Radical using Rational Exponents
w
b5
![Page 8: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/8.jpg)
Write each Radical using Rational Exponents
2
1
5
15
ww
bb
![Page 9: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/9.jpg)
What about Negative exponents
Negative exponents make inverses.
7
1
49
149
2
12
1
![Page 10: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/10.jpg)
What if the numerator is not 1Always remember:
Evaluate
5 25
2
3232
power
root
![Page 11: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/11.jpg)
What if the numerator is not 1
Evaluate
322
5 = 3225
25( )
25 = 2105
![Page 12: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/12.jpg)
What if the numerator is not 1
Evaluate
322
5 3225
25( )
25 2105 2
105
22 4
power
root
![Page 13: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/13.jpg)
For any nonzero real number b, and integer m and n
Make sure the Radical express is real, no b<0 when n is even.
bm
n = bmn or bn( )
m
![Page 14: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/14.jpg)
Simplify each expression
No fraction in the denominators.
3
2
7
4
7
1
x
yy
![Page 15: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/15.jpg)
Simplify each expression
No fraction in the denominators.
y1
7 ⋅y47 y
57 y57
x−23
1
x23
1
x23
![Page 16: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/16.jpg)
Simplify
64x63
![Page 17: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/17.jpg)
Simplify
64x63
(64x6 )1
3
641
3 (x6 )1
3 =4x63 =4x2
![Page 18: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/18.jpg)
Simplify
54x93
![Page 19: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/19.jpg)
Simplify in simplest radical form
54x93
27 • 2 • x93
27 • 2 • x9( )
13 =27
132
13 x
93
![Page 20: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/20.jpg)
Radical form:
54x93
27 • 2 • x93
27 • 2 • x9( )
13 =27
132
13 x
93 =3x3 23
![Page 21: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/21.jpg)
Simplifying exponential expressions
• If in radical form, transform to exponential form
• Simplify by applying the exponent to each base.
• Express with positive exponents or radical form, depending on the directions
![Page 22: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/22.jpg)
Express in simplest form
• Make exponential first!
25x−4
36y8
![Page 23: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/23.jpg)
Express in simplest form
• Make exponential first!
• Now simplify
25x−4
36y8=
25x−4
36y8
⎛
⎝⎜
⎞
⎠⎟
12=
2512 x
−42
3612 y
82
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
![Page 24: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/24.jpg)
Express in simplest form
• Make exponential first!
• Now simplify
25x−4
36y8=
25x−4
36y8
⎛
⎝⎜
⎞
⎠⎟
12=
2512 x
−42
3612 y
82
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
5x−2
6y4=
56x2y4
![Page 25: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/25.jpg)
Next example
• Use positive exponents only!
x4y12
z−26
![Page 26: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/26.jpg)
Next example
• Use positive exponents only!
• Or in radical form:
x4y12
z−26 =
x4y12
z−2
⎛
⎝⎜
⎞
⎠⎟
16=
x46 y
126
z−26
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=
x23y2
z−13
=x23y2z
13
y2 x2z3
![Page 27: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/27.jpg)
Simplify
1
1
2
1
2
1
y
y
![Page 28: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/28.jpg)
Simplify
1
1
1
1
2
1
2
1
2
1
2
1
y
y
y
y
![Page 29: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/29.jpg)
Simplify
1
1
1
1
1
11
2
1
2
11
2
1
2
1
2
1
2
1
y
yyy
y
y
y
y
![Page 30: Rational Exponents Fraction Exponents. Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649ee55503460f94bf4162/html5/thumbnails/30.jpg)
Simplify
1
12
1
1
1
1
1
1
2
1
1
2
1
2
11
2
1
2
1
2
1
2
1
y
yy
y
yyy
y
y
y
y