1. Rational
Exponents

2. Definition of Rational
Exponents
For any nonzero number b and
any integers m and n with n > 1,
( )
m
m
n m n
b = b =
n
b
except when b < 0 and n is even

3. NOTE: There are 3 different
ways to write a rational
exponent
( )
4
4
3 4 3
27 = 27 =
3
27

4. Examples:
3
( )
3
36 = (6) = 216
3
2=
1. 36
4
2. 27 = ( 27) = (3) = 81
4 4
3
3
3
= ( 81 = (3) = 27
)
3 3
4
3. 814

5. Evaluating and
Simplifying
Expressions with
Rational
Exponents

6. Evaluate vs. Simplify
Evaluate – finding the numerical
value.
Simplify – writing the expression in
simplest form.

7. 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

8. In this case, we use
the laws of
exponents to simplify
expressions with
rational exponents.

9. Properties of Rational
Exponents
n Definition 1
−n 1 1 −
1
1 3 1 1
a =  = n of negative
8 3
3
=  = =
a a exponent
8 8 2
1 1
1
=a n Definition = 36 = 36 = 6
2
of negative 1
−
−n
a exponent 36 2

10. Properties of Rational
Exponents
(a )
1
Power-to-
m n  1
=a mn 2 1
1 1 
a-power  2 3  = ( 2)  3⋅ 2  = 2 6
 
Law  
 
( ab ) m Product-t0 1 1
=a b
m m
-a-power ( xy )
1
2 =x
 
2
y
 
2
Law

11. Properties of Rational
Exponents
Quotient-
1
m m to-a- 1
a  a  4
2 4 2
4 2
  = m power   = 1 =
 25 
=
25 5
b  b Law 25 2
−m m Quotient- 1 1
a b
−
to-a-  4   25 
2 25 5
2
  =  negative-
  =  = =
b a  25   4  4 2
power Law

12. Exampl
e:
Write as a Radical and
Evaluate
1
49 = 49 =
2 2
49
= 49 = 7

13. Example: Simplify each expression
1 1 5 2 3 5
4 ⋅a ⋅b = 4 ⋅a ⋅b
3 2 6 6 6 6
6 2 6 3 6 5 Rewrite
= 4 ⋅ a ⋅ b as a
Get a
6 3 5 radical
common
= 16a b denominator -
this is going

14. Example: Simplify each expression
1 3 1 1 3 1
+ +
x ⋅x ⋅x = x
2 4 5 2 4 5
10 15 4 29Remember
+ +
=x
we add
20 20 20
=x 20
exponents
20 9
20 9
= x ⋅x20 20 =x x

15. Example: Simplify each expression
4 4 1
−4 1 1 1 w 5 5 5
5 =  = 4 = 4 ⋅ 1
w  w 5 w5
w 5 w
1 1 1
5
w 5
w
5
w w
5 rationalize
To
= 4 1
= = = the
+
5
w
w denominator
w5 5 w
5
we want an

16. Example: Simplify each expression
1 7
−1
 1 8 1 x y 8
8 = x  = x⋅ 1 = 1 ⋅ 7
xy  y
y 8 y y
8 8
7
7
xy 8x y 8 To rationalize
= = the
y y denominator
we want an

17. Example: Simplify each expression
1
1
(2 ) = 2
5
10 5 10
32 32 10 10
= 1
= 1 2
(2 ) 2
8
4 2 8
1
4 8 8
1 1 2 1 1
2
2 − − 4
= 1
=2 2 4
=2 4 4
=2 = 2
4
2
4

18. Example: Simplify each expression
−1 1 1
− −
5 2 5 1 5
2 2
= 1
= ⋅ 1
2 5 2 2
2⋅ 5 2
5
1 1
1 − −
1 −1 1 1 1
= ⋅5 2 2
= ⋅5 = ⋅ =
2 2 2 5 10

19. Example: Simplify each expression
 1
 Multiply by
1  m + 1 conjugate and
2
1
⋅ 1  use FOIL
 2 
m − 1  m + 1
2
1
m +1
2 m +1
= =
m− 1 m− 1

20. Example: Simplify each expression
−2 −2
 −2   3
 −2
2x   3
−2 
 2x  = 2x 2
2

−3 =
 2   x2   
 x     
 
−2
 −1
 −1
⋅−2 1 x
=  2x  = 2 x = 2 ⋅ x =
2 −2 2
  2 4