Download R.4 Review of Negative and Rational Exponents

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Slide R-1
Chapter R: Reference: Basic Algebraic
Concepts
R.1
Review of Exponents and Polynomials
R.2
Review of Factoring
R.3
Review of Rational Expressions
R.4
Review of Negative and Rational Exponents
R.5
Review of Radicals
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Slide R-2
R.4 Review of Negative and Rational
Exponents
Negative Exponent
If a is a nonzero real number and n is any integer, then
a
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n
1
 n .
a
Slide R-3
R.4 Using the Definition of a Negative
Exponent
Example Evaluate.
(a) 4
2
(b)
 2
 
5
3
(c) 42
Solution
3
2
1
1
125

(b)   


3
8
8
5
 2
 
125
5

1
1
2
(c) 4   2  
4
16
1
1
(a) 4  2 
4 16
2
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Slide R-4
R.4 Review of Negative and Rational
Exponents
Example Write without a negative exponent.
(a) x 5
(b)
xy 3
Solution
1
(a) x  5
x
5
( x  0)
1
x
(b) xy  x  3 = 3
y
y
3
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( y  0)
Slide R-5
R.4 Review of Negative and Rational
Exponents
Quotient Rule
For all integers m and n and all nonzero real numbers
a,
am
mn

a
.
n
a
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Slide R-6
R.4
Using the Quotient Rule
Example Simplify each expression. Assume variables
represent nonzero numbers.
125
(a)
12 2
(b)
a5
8
a
Solution
125
5 2
3
(a)
 12  12
2
12
a5
5 ( 8)
13

a

a
(b) 8
a
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Slide R-7
R.4 Rational Exponents
a1/n
n even If n is an even positive integer, and if a > 0,
then a1/n is the positive real number whose nth
power is a. That is, 1/ n n
a 
a.
n odd If n is an odd positive integer and if a is any
real number, then a1/n is the positive or
negative real number whose nth power is a.
That is, 1/ n n
a 
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a.
Slide R-8
R.4 Using the Definition of a1/n
Example Evaluate each expression.
(a) 361/ 2
(b) 6251/ 4
(c) (27)1/ 3
Solution
(a)
361/ 2  6
(b) 6251/ 4  5
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because 62  36.
(c) (27)1/ 3  3
Slide R-9
R.4 Rational Exponents
Rational Exponent
For all integers m, all positive integers n, and all real
numbers a for which a1/n is a real number,
a
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m/n
 a

1/ n m
.
Slide R-10
R.4 Using the Definition of am/n
Example Evaluate each expression.
(a) 1252 / 3
(b) 32 7 / 5
(c) (27) 2 / 3 (d) (4)5/ 2
Solution
(a) 1252/ 3  (1251/ 3 )2  52  25
(b) 327 / 5  (321/ 5 )7  27  128
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Slide R-11
R.4 Using the Definition of am/n
Solution
(c) (27)2/ 3  [(27)1/ 3 ]2  (3)2  9
1/ 2
5/ 2
(

4)
(

4)
(d)
is not a real number because
is not
a real number.
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Slide R-12
R.4 Review of Negative and Rational
Exponents
Definitions and Rules for Exponents
Let r and s be rational numbers. The following results
are valid for all positive numbers a and b.
a a  a
r
s
r s
r
a
r s
a
s
a
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(ab)  a b
r
r
r
a a
   r
b b
r
r
(a )  a
r s
rs
1
a  r
a
r
Slide R-13
R.4 Using the Definitions and Rules for
Exponents
Example Evaluate each expression.
27  27
(a)
273
1/ 3
5/ 3
(b) 6 y 2/ 3  2 y1/ 2
Solution
1/ 3
5/ 3
1/ 35/ 3
2
27

27
27
27
1
2 3
1
(a)

 3  27  27 
3
3
27
27
27
27
(b) 6 y 2/ 3  2 y1/ 2  12 y 2/ 31/ 2  12 y 7 / 6
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Slide R-14