Lesson 3
MA 152
Definition of a Rational Exponent (1):
1
If a is a real number and n is a natural number, then then a n is the real number b such that
1
b n  a , if b exists. Basically, a n is the same as n a .
Definition of a root: If n is a natural number greater than 1 and a  0 , then
nonnegative number whose nth power is a. ( n a ) n  a
a is the
1
1
If n is a natural number greater than 1 and a n exists, then
n
n
a  an .
Ex 1: Evaluate each.
1
a)
92 
1
3
b)
125 
c)
 81 
 
 16 
d)
 18 

1
4

1
e)
1
2
(4) 
Sometimes an absolute value symbol must be used, if the variable is not defined as a
positive value whenever there is an even power. However, we will assume that each
variable represents a positive.
Ex 2: Evaluate each.
1
2
a) (4 x ) 
4
1
b)
c)
(64 x 6 ) 6 
1
3 3
(1000 x ) 
Other Rational Exponents (2):
1
Definition: If m and n are positive integers, the fraction
m
n
m
is in lowest terms, and a n
n
m
n
exists, then a  a or ( a ) . This definition shows that a can be evaluated two
different ways; take the nth root first, then raise to the mth power or raise to the mth
power first, then take the nth root. It is usually to the student's advantage to take the
root first, then raise to the power.
m
n
n
m
Ex 1: Evaluate each.
a)
3
2
36 
4
3
b)
8 
c)
16 4 
3
All the rules of exponents previously studied now apply also with rational exponents.
Ex 2: Assume that all variables represent positive integers. Evaluate or simplify each
and write all answers without using negative exponents.
a)
4

3
2


2
5
b)
(32)
c)
 8 
 3 6
a b 
9
d)

3
2
(8 x 3 )
e)


6
1
3
(64 x y
2
3


5
12 6
3
2
3
) (x y ) 
6
Ex 3: Simplify each using the rules of exponents. Assume that all variables represent
positive integers. Write answers without using negative exponents.
c c 
1
6 4 2
a)

1
2
c c
3
2


 a 
b)  5 
 a2 


3
4

4
c)
4
3

2
( 16r s )
rs
3
2
1
2

In the radical expression n a , the symbol
is called the radical sign, a is the
radicand, and n is the index. (If the index is 2, a square root, usually the index is not
written. If the index is 3, the expression is called a cube root. Otherwise, it is an nth
root.)
If n is a natural number greater than 1 and a  0 , then
whose nth power is a. ( n a ) n  a
If a  0, then
n
a is the nonnegative number
 a  a.
a is the real number such that  a  .
a is the nonnegativ e real number such that
n is odd, then
If a < 0, and
n
n
n
n
n
n
n is even, the n a is not a real number.
Ex 4: Evaluate each.
a ) 4 81 
b)
 3 64 
c)
4
 16 
d)
3

1

8
Multiplication and Division Properties of Radicals:
If all expressions represent real numbers,
n
a
a
n
a n b  n ab and n  n
(b  0)
b
b
Note: These properties are for multiplication and division. Similar statements are not
true for addition or subtraction. ( n a  b  n a  n b for example)
Numbers that are squares of positive integers; such as 1, 4, 9, 16, ...; are called perfect
squares. Powers such as x 2 , x 4 , x 6 , ... are also called perfect squares.
Numbers that are cubes of positive integers; such as 1, 8, 27, 64,....; are called perfect
cubes. Powers such as x 3 , x 6 , x 9 ,... are also called perfect cubes.
To simplify a radical expression:
1. Factor the radicand into the greatest perfect factor possible and a 'leftover' factor.
2. Use the multiplication property of radicals to 'separate' into two radicals.
3. Simplify the 'perfect factor' radical.
Ex 5: Simplify each.
18 x 3 
a)
b)
3
 81a 7 b 5 
Racial expressions with the same index and the same radicand are called like radicals.
Like radicals can be combined similarly as like terms. *When trying to combine
radicals, always simplify radicals first!
Ex 6: Simplify and combine where possible.
a)
32  162 
b)
3a 3a  48a 3 
c)
43 16a 4  3a 3 54a 
Rationalizing Denominators: Rationalizing a denominator means to write the
expression without a radical in the denominator. To rationalize when there is a square
root in the denominator do the following steps.
1. First, simplify each radical.
2. Multiply numerator and denominator by the radical factor in the denominator.
3. Simplify numerator and denominator.
Ex 7: Rationalize the denominator.
2
a)

5
b)
c)
3

8
4
3m 3
