Download Rational Numbers as Exponents

Rational Numbers as
Exponents
Section 7-5
Objectives
• To calculate radical expressions in
two ways.
• To write expressions with rational
exponents as radical expressions and
vice versa.
• To simplify expressions containing
negative rational exponents.
• To use rational exponents to simplify
radical expressions.
Simplify and Compare
3
2
 8
and  4 
8 and
4
2
3
2
2
Theorem 7-6
For any nonnegative number a, any
natural number index k , and any integer m,
k
a 
m
 a
k
m
.
We can raise to a power and then
take a root or we can take a root
and then raise to a power.
3

27  729  9
2
3
27
3

2
  3  9
2
2  64  4
3
6
 2
3

3
6
 2 2 2 2 2 2  8 8 4
3

3
5x
 5x 
3
3
3
3
3
3
3
3
 5x 5x 5x  5x 5x
 (5x)
2
5x  5x 5x
3

8
2
6y

3
Definition
For any nonnegative number a and any
1
ak
natural number index k ,
means
k
a (the nonnegative kth root of a).
1
2
x  x
1
3
27  27  3
 abc 
7
3
1
5
 abc
5
7 xy   7 xy 
1
7
Definition
For any natural numbers m and k , and any
nonnegative number a,
 a
k
m
.
m
ak
k
m
means a or
 27 
 4
3
2
2
3



3
27
 4

2
3
or 27
3
or 4
3
2
Write with rational exponents
3

4
8  8
4
7xy

5

4
3
 7xy 
5
4
What is
Write as a radical
9
1
2
7
1
2
 x
3
Write with Rational Exponents
1
3
11
8
5
3x
1
3
Numerator  Exponent
Deno min ator  Index / Root
1
3
1
2
x  x
x  x
2
6
1
3
3
x  ( x)  x  ( x)
2
3
2
6
3
2
3
3
1
x  ( x ) x  ( x ) x  ( x)
3
2
3
3
http://www.youtube.com/watch?v=-T1punCdxas
Negative Rational Exponents
m
For any rational number
and any
k
positive real number a,
1
m.
ak
m
a k
means
1

2
4
1

4
5xy 
4

5

1
2
1
5xy 
4
5
Variables w/ Exponents
• Reminder: When Multiplying
Expressions with similar Bases the
Exponents ADD!
• When Dividing Expressions with
similar Bases the Exponents
SUBTRACT!
x *x  x
2
3
3
23
x
5
x
3 2
1
x x x
2
x
Variables w/ Exponents
• When Taking an Exponential of an
Expression with a similar Base the
Exponents MULTIPLY!
• Reminder: Negative Exponents
mean you TAKE THE RECIPROCAL!
(x )  x
2
x
2
3
2*3
x
6
1
1
 2 or 2  x 2
x
x
2
5
3
5
3 3  3
16
16
1
4
1
2
16
1 1

4 2
3
4
2 3

5 5
 16

3
5
5
1
4


 72   72  72


2
3
6
12
1
2
Using Rational Exponents to
Simplify Radical Expressions
• Convert radical expressions to
exponential expressions.
• Use the properties of exponents
to simplify.
• Convert back to radical notation
when appropriate.
Use Rational Exponents to Simplify
6
6
x 
3
4
3
6
x
1
6
4

1
2
x
1
2 6
 
 2
 x
2
6
2
1
3
2
 2
3
Use Rational Exponents to Simplify
4
6
2
4
1
2
a  a a  a
2
12
6
6
6
2 a  2 a  2 a  4a
12
6
1
2
2
1
4
2
4
10  3  10 3  10 3
4
1
4
 10 3  300
4
2
4
Use Rational Exponents to Simplify
4
 x  y
 x  y
3
 x  y
3
4
 x  y
1
2

  x  y
3 1

4 2
1
4
  x  y  4 x  y
Write as a Single Radical Expression
x  2  4 3y 
1
2
 x  2  3 y 
1
4
2
4
1
4
  x  2  3 y  
1
4
 x  2    3 y   


2
4
4
 x  2  3 y  
2
4
x
3x y  12 xy  12 y
2
2

 4x  4  3 y  
Write as a Single Radical Expression
1
2
1

2
5
6
a b c 
3
6
3

6
5
6
3 3 5
a b c  ab c
6
Write as a Single Radical Expression
2

3
1
2
5
6
4

6
3
6
5
6
x y z
4
x y z  x y z
6
3 5
Methods to Simplify Radical
Expressions
• Simplify by factoring
• Rationalize denominators
• Collecting like radical terms
• Using rational exponents
Now you can
• Calculate radical expressions in two
ways.
• Write expressions with rational
exponents as radical expressions and
vice versa.
• Simplify expressions containing
negative rational exponents.
• Use rational exponents to simplify
radical expressions.
1.
 6a 
3
 6a 
3

 6a 6a 6a  6a 6a
7.

3
2

3
4 3
d 
3
3 3
12c d
3
144 c
3
3
2

3
8 18 c
3
2c 18cd
2
12c d 
2
2

2
c
3
d
2

13.
19.
25.

1
2 2 5
ab


1
5 3 2
at
3


5

19  19
2 2
ab
5 3
at
1
3
31.
37.
7

3
7
2
7
2
7
3
6
3
6
x y z  x y z
3
6
2 2

3
12ab
1
2
1
2
1
2
3
6
 12 a b 
12 a b  12ab 
1
2
40. x
1

3
1

x
43.
46.
 5xy 
1
x
2

3
5

6
1
3
1

 x
 5xy 
2
3
5
6
2
3
1
2
2 1

3 2
49. 11 11 11
52.
8.3
8.3
3
4
2
5
 8.3
3 2

4 5
4 3

6 6
 11
15 8

20 20
 8.3
7
6
 11
 8.3
7
20
55.
58.
3
 5 7
 54 





3 8y 6 
5
5 3

47
1
3
6
3
5
15
28
8 y  2y
2
61.
4 16x12 y16 
4
4 12 4
16 x
12 16
2x 4 y 4
16
y 
 2x y
3 4
1
3
1
2
x x  2  x ( x  2) 
3
64.
3
6
2
6
x ( x  2)  x ( x  2) 
6
6
3
2
x ( x  4 x  4)  ( x  4 x  4 x )
3
2
6
5
4
3
3
( x  y)
2
4
( x  y)
3
67.
( x  y)
( x  y)
2
3
3
4
( x  y)
12

 ( x  y)
8 9

12 12
( x  y)
1
2 3

3 4

 ( x  y)
1

12

7
12
5
6
1
3
1

6
s t
70.
s t
3
12
s t
4
st
4
12
12

s
7 4

12 12
1
4
4
4
t
 s t 
5 1

6 6
