6.2A Apply Properties of
Rational Exponents
Algebra II
Review of Properties of Exponents
from section 5.1
am * an = am+n
 (am)n = amn
 (ab)m = ambm
These all work
 a-m = 1
a
a
for rational
 a = am-n
(fraction)
a
a


   =
exponents as
b
b
well as integer
exponents.

m
m
n
m
m
m
Ex. 1 : Simplify. (no decimal answers)
61/2 * 61/3
= 61/2 + 1/3
= 63/6 + 2/6
= 65/6
b. (271/3 * 61/4)2
= (271/3)2 * (61/4)2
= (3)2 * 62/4
= 9 * 61/2
a.
(43 * 23)-1/3
= (43)-1/3 * (23)-1/3
= 4-1 * 2-1
c.
=¼
*½
= 1/ 8
1 3

d. 18 4 
 1 
 94 


=
18
9
3
4
3
4
=
 18 
 
9
3
4
=
2
3
4
** All of these examples were in rational exponent form to begin with, so the
answers should be in the same form!
Ex 2: Simplify.
3
25

5=
a.
3
3
Ex 3: Write the expression in
simplest form.
25 5
3
a.
= 125 = 5
4
64
4
=
16  4
=
4
16  4 4
= 24 4
3
32
3
4
b.
=
3
=
3
32
4
b.
4
7
8
8 = 2
4
=
** If the problem is
in radical form to
begin with, the
answer should be in
radical form as well.
=
4
4
7
8
7 42
4
8
2
4
=
4
14

2
Can’t have a tent in
the basement!
4
=
14
4
16
Ex 4: Perform the indicated operation
a.
5(43/4) – 3(43/4)
= 2(43/4)
b. 3 81  3 3
= 3 27  3  3 3
= 33 3  3 3
= 23 3
c. 3 625  3 5
= 3 125  5  3 5
3
3
= 5 5 5
3
=6 5
If the original problem is in radical form,
the answer should be in radical form as well.
If the problem is in rational exponent form, the
answer should be in rational exponent form.
6.2B Simplifying Expressions with
Variables
Ex. 1
x2  x
a.
b.
c.
d.
6
x 
11
y13 
4
7
r8 
4
x
6
y
11
x
y
2
r4 r4  4 r 4  4 r 4
 r r  r 2
Ex 2: Simplify the Expression.
Assume all variables are positive.
a. 27z  27  z  3z
3
9
3
3
(16g4h2)1/2
= 161/2g4/2h2/2
= 4g2h
b.
c.
5
x5
y10


x
y2
5
5
x5
y10
9
3
d.
18rs
2
3
1
4 3
6r t
 3r
3
4
1
2
3 3
3r s t
1
4
2
3
s t3
Ex. 3 Perform the indicated operation.
Assume all variables are positive.
a.)18 u  11 u
3
3
4 2/3
b.)15a b
 8a b
4 2/3
c.)10 5s  3s 80 s
4
7
4
3
Assignment
Ex. 4 Write the expression in
simplest form.
3
a.) 104
4
10
b.) 4
27