Download Section 10.2

Rational Exponents - Section 10.2
Radical expressions lead to the need for rational (fractional) exponents. Rational exponents are defined so that
they agree with the rules of exponents we have derived so far.
For example: If y = 9 1/2 , then what is y ? Note that by the property of exponents, we must have
2
y 2 = 9 1/2  = 9 1/22 = 9 1 = 9
So what is y?
y 2 = 9 so that y =
Thus, we must define 9 1/2 =
9 =3
9
If n a represents a real number and n ≥ 2 is an integer, then
a 1/n =
n
a
If n is even, then a must be nonnegative. If n is odd, then a can be any real number.
Rewrite and simplify, if possible:
−8 1/3
25 1/2
5xy 2 
1/4
Rewrite with rational exponents:
7
4
5xy
5
a3b
2
If n a represents a real number, m
n is a positive rational number reduced to lowest terms, and n ≥ 2 is an integer,
then
a m/n =
n
am
and
a m/n =  n a 
m
1
Use radical notation to rewrite each expression and simplify:
8 4/3
− 81 3/4
25 3/2
Rewrite with rational exponents:
5
x4
3xy
7
If a m/n is a nonzero number, then
a −m/n =
1
a m/n
Rewrite each expression with positive exponents and simplify, if possible:
100 −1/2
8 −1/3
32 −3/5
2
The same properties apply to rational exponents as to integer exponents.
If m and n are rational exponents and a and b are real numbers for which the following expressions are defined,
then
b m = b m−n
bn
b m ⋅ b n = b m+n
ab n = a n b n
a
b
n
n
= an
b
b m  n = b mn
b −n = 1n
b
b0 = 1
Recall that exponential expressions are simplified when:
1. No parentheses appear
2. No powers are raised to powers
3. Each base occurs only once
4. No negative or zero exponents appear
Simplify:
7 1/2 ⋅ 7 1/3
a 2/5 
3/4
50x 1/3
10x 4/3
x −3/5 y 1/4 
1/3
3
Use rational exponents to simplify as much as possible.
6
x3
8
x4y2
3
3
8a 12
x
3 x
x
4