1. 6.4 Rational Exponents

2. Rational Exponents
• We can write a radical expression in an
equivalent form using a fractional (rational)
exponent instead of a radical sign:
1
n
x=x n

3. Rational Exponents
• Examples:
1 Notice that the
36 = 36 2
denominator
is the index of
1 the radical
3
64 = 64 3
1
4
16 = 16 4

4. Example: Simplify
1
216 3

5. Example: Simplify
1 1
7 ×7
2 2

6. Example: Simplify
1 1
5 × 125
4 4

7. Exponent Rules
1
• If n
x=x n
1 m
 
( x)
1
=( x )
m
m n
then
n
x m
=x  =
n n
 

8. Exponent Rules
• If the nth root of a is a real number, m is an
m
integer, and is in lowest terms, then
n
m
( a)
1 m
a =
n n
a and a = a =
n n m n

9. Example: Write in Radical Form
3
x 7

10. Example: Write in Radical Form
7
−
y 2

11. Example: Write in Exponential Form
5
a

12. Example: Write in Exponential Form
( b)
3
5

13. Properties of Rational Exponents
• All the properties of integer exponents apply
to rational exponents
• To write an expression in simplest form, write
every exponent as a positive number

14. Example: Write in Simplest Form
3 ( 3)
4

15. Example: Write in Simplest Form
3
x
3 2
x

16. Example: Write in Simplest Form
4
( −32 ) 5

17. Example: Write in Simplest Form
2
( −8x xy ) 3

18. Example: Write in Simplest Form
3
( 16 y )
−8 − 4

19. Homework
• P386 #12 – 87 multiples of 3