nA

1. Rational
Exponents
In other words,
exponents that are
fractions.

2. Definition of
For any real number b
and any integer n > 1,
except when b < 0 and n
is even
b
1
n
b
1
n
 b
n

3. Examples:
 36
1. 36
1
2  6
 64
3
2. 64
1
3  4

4. 
1
49






1
2
3. 49

1
2 
1
49
 8
3
4.
1
8







1
3
 2
Examples:

1
7
 8
 
1
3

5. Definition of Rational Exponents
For any nonzero number b and
any integers m and n with n > 1,
except when b < 0 and n is even
b
m
n
 bm
n
 b
n
 
m

6. NOTE: There are 3 different
ways to write a rational
exponent
27
4
3
 274
3
 27
3
 
4

7. 2. 27
4
3
Examples:
 27
3
 
4
1. 36
3
2
 3
 
4
 81
 36
 
3
 6
 
3
 216
3. 81
3
4  81
4
 
3
 3
 
3
 27

8. Simplifying Expressions
No negative exponents
No fractional exponents in the
denominator
No complex fractions (fraction
within a fraction)
The index of any remaining radical
is the least possible number

9. Examples: Simplify each expression
 4
2
6
 a
3
6
 b
5
6
4
1
3
 a
1
2
 b
5
6
 42
6
 a3
6
 b5
6
 16a3
b5
6
Get a common
denominator -
this is going to
be our index
Rewrite as
a radical

10. Examples: Simplify each expression
 x
1
2

3
4

1
5
 x
10
20

15
20

4
20
 x
29
20
x
1
2
 x
3
4
 x
1
5
 x
20
20
 x
9
20  x x9
20
Remember
we add
exponents

11. 
w
1
5
w
5
5

w
1
5
w
Examples: Simplify each expression
w
4
5 
1
w






4
5

1
4
5
w
4
5

1
w
4
5

w
1
5
w
1
5

w
1
5
w
4
5

1
5

w
5
w

12. 
xy
7
8
y
Examples: Simplify each expression
 x
1
y






1
8
 x
1
y
1
8

x
y
1
8

y
7
8
y
7
8
To rationalize
the denominator
we want an
integer exponent

x y7
8
y
xy
1
8

13.  2
1
2

1
4
 2
2
4

1
4
 2
1
4
Examples: Simplify each expression

32
1
10
4
1
8

25
 
1
10
22
 
1
8

2
5
10
2
2
8

2
1
2
2
1
4
32
10
4
8
 2
4

14. Examples: Simplify each expression

5

1
2
2 5
1
2
5
1
2
2 5

1
2

5

1
2
5
1
2

1
2
 5

1
2

1
2

1
2
 51

1
2

1
5

1
10

15. Examples: Simplify each expression

m
1
2
 1
m
1
2
 1











m 1
m 1
1
m
1
2
1

m
1
2
1
m1
Multiply by
conjugate
and use
FOIL

16. Examples: Simplify each expression
6
2
5
y
8
5
 6
2
5
y
5
5
y
3
5
 6
2
5
 y y
3
5
 y 6
2
5
 y
3
5
 y 6
2
y
3
5
 y 36y
3
5

17. Examples: Simplify each expression
c
2
3
 c
c
1
2
 c
2
3
 c
1
2
 c c
1
2
 c
7
6
 c
3
2
 c
1
2
c
2
3
 c







18. Examples: Simplify each expression
 c
6
6
c
1
6
 c
2
2
c
1
2
 c c
6
 c c
 c
7
6
 c
3
2

19. Examples: Simplify each expression

2x
3
2
x2










2
2x
2
x
3
2








2
 2x
3
2
2






2
 2x
1
2






2
 2
2
x
1
2
2

1
22
 x 
x
4