7.2 Properties of Rational
Exponents
Algebra 2
Mrs. Spitz
Spring 2009
Objectives/Assignment
Use properties of rational exponents to
evaluate and simplify expressions.
 Use properties of rational exponents to
solve real-life problems, such as finding
the surface area of a mammal.

Assignment: pp. 411-412 #5-81 every 3rd
Review of Properties of Exponents
from section 6.1
am * an = am+n
 (am)n = amn
 (ab)m = ambm
1
-m
a = a
a
 a = am-n
a
a
   =
b

m
m
n
m
 
m
bm
These all work
for fraction
exponents as
well as integer
exponents.
Ex: 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: Simplify.
3
25

5=
a.
3
3
Ex: 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: 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.
More Examples
x2  x
a.
b.
c.
d.
6
x 
x
11
y11 
y
4
6
r8 
4
r4 r4  4 r 4  4 r 4
 r r  r 2
Ex: 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: Write the expression in simplest
form. Assume all variables are positive.
4
4 9
a. 12d e f
14
 12  d  e  f
4
4
4
 12  d  e
4
 de 2 f
b.
5
34
9
4
24
e f
12ef
4
34
14
f
2
2
2
3
2 3
g2
g
h
g

h
5
5

10
h 7  h7  h3 
h
5
g 2 h3
h2
No tents in the basement!
3
2
3
c. 15d e f
 3d
4
5df
2
31 3
e f
1( 4 )
2
2 3
 3d e f 5
** Remember, solutions must be in the same form as the
original problem (radical form or rational exponent form)!!
8
d.
4
x y
6
z
11
4
8
11 2
x y z
6 2
z z
Can’t have a tent in the basement!!

8
4
11 2
x y z
8
z

2
x y
24
z
3 2
y z
2
Ex: Perform the indicated operation.
Assume all variables are positive.
a.
8 x 3 x  5 x
1
4
b. 3gh  6 gh
1
4
 3gh
c. 2 6 x  x 4 6 x
4
5
1
4
 2x 6x  x 6x
4
4
 3x 4 6 x
d.
6 s  2 s  s  6  2  1 s  5 s
e.
3 6y  2y
7
3
 3y
23
 5y
23
6y  2y
23
6y
6y
23
6y