Transcript No Slide Title

Copyright © 2012 Pearson Education, Inc.
Slide 7- 1
7.2
Rational
Numbers as
Exponents
■
■
■
■
Rational Exponents
Negative Rational Exponents
Laws of Exponents
Simplifying Radical Expressions
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Rational Exponents
Consider a1/2a1/2. If we still want to add
exponents when multiplying, it must follow
that a1/2a1/2 = a1/2 + 1/2, or a1. This suggests
that a1/2 is a square root of a.
Copyright © 2012 Pearson Education, Inc.
Slide 7- 3
1/ n
a
n
= a
1/ n
a
means n a . When a is nonnegative, n can
be any natural number greater than 1. When a is
negative, n must be odd.
Note that the denominator of the exponent
becomes the index and the base becomes
the radicand.
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Slide 7- 4
Example
Write an equivalent expression using radical
notation.
a) m1/ 3
 
2 1/ 5
 xy z 
b) 9x
c)
8 1/ 2
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Slide 7- 5
Solution
1/ 3
a) m
3m
 
8 1/ 2
b) 9x
c)

The denominator of the exponent
becomes the index. The base becomes
the radicand.

9
1/
5
2
xy z
1/ 2 4
4
x  9 x  3x
4
 5 xy 2 z
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Slide 7- 6
Example Write an equivalent expression
using exponential notation.
a) 3 4 x
2
5
y
b) 5
x
Solution
1/ 3
a) 3 4 x   4 x 
2 1/ 5
 5y
5
y
b) 5


 x 
x


2
The index becomes the
denominator of the
exponent. The radicand
becomes the base.
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Slide 7- 7
Positive Rational Exponents
For any natural numbers m and n (n ≠1) and any
real number a for which n a exists,
a
m/n
 
means n a
m
n m
, or a .
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Slide 7- 8
Example
Write an equivalent expression using radical
notation and simplify.
a) 8
2/3
b) 93 / 2
Solution
a) 8
2/3
3/ 2
b) 9
 
2
3
 8  8  22  4;
3 2

 9
3
3
 3  27.
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Slide 7- 9
Example Write an equivalent expression
using exponential notation.
a)
b)
3 5
x

5 3 xy

2
Solution
a)
b)
3 5
x  x5 / 3

5 3xy

2
  3xy 
2/5
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Slide 7- 10
Negative Rational Exponents
For any rational number m/n and any nonzero
real number a for which a m / n exists,
1
m / n
a
means
.
m/n
a
Caution! A negative exponent does not indicate that the
expression in which it appears is negative.
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Slide 7- 11
Example
Write an equivalent expression with positive
exponents and simplify, if possible.
a) 82 / 3
b) 9
3 / 2 1/ 5
 3t 
c)  
 2r 
x
3 / 4
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Slide 7- 12
Solution
2 / 3
a) 8
b) 9
1
1


2/3 4
8
3/ 2 1/ 5
x

8-2/3 is the reciprocal of 82/3.
1
1/ 5
x
3/ 2
9
1 1/ 5 x1/ 5
 x 
27
27
 3t 
c)  
 2r 
3/ 4
3/ 4
 2r 
 
 3t 
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Slide 7- 13
Laws of Exponents
The same laws hold for rational exponents
as for integer exponents.
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Slide 7- 14
Laws of Exponents
For any real numbers a and b and any rational
exponents m and n for which am, an, and bm are defined:
1. am  an  amn
2.
am
an
 a mn
 
3. a
m n
a
mn
4.  ab m  ambm
In multiplying, add exponents if the bases
are the same.
In dividing, subtract exponents if the
bases are the same. (Assumea  0.)
To raise a power to a power, multiply the
exponents.
To raise a product to a power, raise each
factor to the power and multiply.
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Slide 7- 15
Example
Use the laws of exponents to simplify.
a) 7 2 / 5  71/ 5
1/ 2
b)
m
1/ 4
m
c)
x

1/ 2 1/ 3 3 / 4
y
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Slide 7- 16
Solution
a) 72 / 5  71/ 5  72 / 51 / 5  73 / 5
1/ 2
b)
m
1/ 4
 m1/ 21/ 4  m2 / 4 1/ 4  m1/ 4
m
c)
x

1/ 2 1/ 3 3 / 4
y
x
x
(1/ 2)(3 / 4) ( 1/ 3)(3 / 4)
y
3 / 8 1/ 4
y

x
3/8
1/ 4
y
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Slide 7- 17
Simplifying Radical Expressions
To Simplify Radical Expressions
1. Convert radical expressions to exponential
expressions.
2. Use arithmetic and the laws of exponents to
simplify.
3. Convert back to radical notation as needed.
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Slide 7- 18
Example
Use rational exponents to simplify. Do not
use exponents that are fractions in the final
answer.
a) 4 (3 x)2
9

2 
3
b)  xy z 


c) 4 y
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Slide 7- 19
Solution
a) 4 (3 x) 2  (3 x)2 / 4
1/ 2
 (3x)
Convert to exponential notation
Simplify the exponent and
return to radical notation
 3x
9

2 
2 9/3
3
b)  xy z   ( xy z )


 ( xy 2 z )3  x3 y 6 z 3
 
1/
4
1/ 2
c) 4 y  4 y1/ 2  y
 y1/ 8  8 y
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Slide 7- 20