Transcript 7-2 Properties of Rational Exponents (Day1)

7-2
Properties of Rational
Exponents
(Day 1)
Objective: Ca State Standard 7.0: Students add,
subtract, multiply, divide, reduce, and evaluate
rational expressions with monomial and
polynomial denominators and simplify
complicated rational expressions, including those
with negative exponents in the denominator.
Properties of Rational Exponents
Let a and b be real numbers and let m and n be
rational numbers.
Property
Product of Powers Property
a a  a
m
n
m n
1 3 

2
2 2
2
2
3 3  3
3 9
1
3
a 
m
n
a
2
mn
3 
 2 
2 
 
4   4
 
3
2
 ab 
m
9  4
1
2
 4  64
3
a b
m
1
m
 9 4
2
1
2
 3 2  6
Negative Exponents Property
a
m
25
1
1
 m ,a  0
a
2

1
25
1
2
1

5
Quotient of Powers Property
m
a
mn

a
,
a

0
n
a
6
6
5
1
2
2
5 1 
4

2
2 2
2
6
 6  6  36
Power of a Quotient Property
m
a
a
   m ,b  0
b
b
 8 
 
 27 
m
1
3

8
1
27
3
1
3
2

3
1
If m  for some interger n > 1 the third and
n
sixth properties can be written using radical notation
as follows:
n
a b  n a  n b
Product Property
n
n
a
a
 n
b
b
Quotient Property
Example 1
Use the properties of Rational Expressions to
simplify the expression.
1
1
5 2 5 4  5

1
8 5
2
1
 8
2
3
1 1
2 4
1 2
2
5
5
3
4
1 2
3
 8 5
1
2
3
 85
2
3
2
4
3
4

1
1

4
4
 2  3 


7
7
1
3

1
7
7
1
3
4

 
 6
4
1
1 1
7
3
4
6
7
1
4 4
2
3
1
6 
6
1
 12 3
 1
43

1
2
2
  12  3 
     
  4  

 
1
3 
1
3
2
1 2 
2

3
3
3
3
Example 2:
Using Properties of Radicals
3
4  16  4 16  64  4
4
162

4
2
3
3
3
162
4
4
 81  3
2
For a radical to be in simplest form you must not
only apply the properties of radicals but must
remove any perfect nth powers (other than 1) and
rationalize the denominator.
3
54  27  2
3
 27  2
3
3
3 2
3
Factor out perfect cubes
Product Property
Simplify
5
3 53 8


4
4 8
24
5
32
5
24
 5
32
Make the denominator a
perfect fifth power
Simplify
Quotient Property
5
24
2
Simplify
Two radicals expressions are like radicals if they
have the same index and the same radicand.
3
2 and 4 3 2 are like radicals. To add or subtract
like radicals use the distributive property.
Adding and Subtracting Roots and Radicals
   
7 6
1
5
2 6
1
5
   
 (7  2) 6
1
5
9 6
1
5
3
16  2  8  2  2
3
3
3
 8 2  2
3
3
2 2 2
3
3
3
  2  1
1 2
3
 2
3
 2
3
Homework: page 411 # 23-53(odd)