Section 7.2 (Rational Exponents)
Example: Find (27)1/3 = (33)1/3 =

With a real number a , a
1
n
 n a (n is a positive integer)
Examples: Use radical notation to write the following and simplify if possible.
251/2=
641/3=
x1/5=
-251/2=
(-27y6)1/3=
7x1/5=
Example: Find 82/3 = (82)1/3 =

With a real number a ,
a
m
 (n a ) m  n a m (m and n are positive integers in lowest terms)
n
Example: Use radical notation to rewrite each expression and simplify if possible.
93/2=
1
 
4

3
-2563/4=
(-32)2/5=
(2x + 1)2/7=
7x1/5=
2
=
With a real number a , a
m
n
1

a
m
(am/n is a nonzero real number)
n
Example: Write each expression with a positive exponent, and then simplify.
27-2/3=
-256-3/4 =
Summary of Exponent Rules
Product rule for exponents:
Power rule for exponents:
Power rule for products & quotients:
Quotient rule for exponents:
Negative exponent:
am * an = am + n
(am)n = am * n
(ab)n = an bn
(a/c)n = an / cn , c  0
am / an = am – n , a  0
a-n = 1 / an , a  0
Examples: Use the properties of exponents to simplify.
x1/3 x1/4 =
9
9
2
5
12
=
5
2
(112/9)3 =
(3x 3 ) 3
=
x2
y-3/10 * y6/10 =
Examples: Multiply
x3/4(x1/4 – x3) =
(x1/4 + 1)(x1/4 – 7) =
Example: Factor x-1/3 from the expression 7x-1/3 – 5x5/3

Some radical expressions are easier to simplify when we first write them with rational exponents
Examples: Use rational exponents to simplify
10
y5 =
4
9=
9
a 6 n3 =
Examples: Use rational exponents to write as a single radical
y 3 y =
3
x
4
x
=
5 3 2 =