M098
Carson Elementary and Intermediate Algebra 3e
Section 10.2
Objectives
1.
2.
3.
4.
Evaluate rational exponents.
Write radicals as expressions raised to rational exponents.
Simplify expressions with rational number exponents using the rules of exponents.
Use rational exponents to simplify radical expressions.
Vocabulary
Rational Exponent
An exponent that is a fraction.
Prior Knowledge
Exponent Rules:
0
a = 1, where a ≠ 0
am  a n  a m  n
0
0 is indeterminate
a n 
am
am  an 
 am  n
n
a
n
 am   amn


1
an
1
 an

n
a
a
 
b
n
b
 
a
abn  anbn
n
a
 
b
n

an
bn
Fraction Operations: Add, subtract, multiply
New Concepts
1. Evaluate rational exponents.
Up to this point we have used our rules for exponents with integer exponents. Now we extend them to
rational (fraction) exponents as well.
What would a rational exponent mean? Let’s look at a couple of examples:
1/2
2 1/2
2 × 1/2
1
(16) = (4 ) = 4
=4 =4
1/2
2 1/2
2 × 1/2
1
(25) = (5 ) = 5
=5 =5
What other operation on 16 will give 4 for a result?
What other operation on 25 will give 5 for a result?
These are the same results we would get if we were to take the square root of the number.
(8)
1/3
3 1/3
= (2 )
3 × 1/3
=2
1
=2 =2
This result is the same as taking the cube root (third root) of 8.
(16)
1/4
4 1/4
= (2 )
=2
4 × 1/4
1
=2 =2
This result is the same as taking the fourth root of 16.
The denominator of the fractional exponent is the same as the index of the radical. This is simply
new notation for a familiar concept.
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M098
Carson Elementary and Intermediate Algebra 3e
Section 10.2
Example 1: Rewrite in radical notation and evaluate if possible.
a. 271 / 3  3 27  3
c.
b. 321 / 5  5 32  2
 81 / 3
 3  8  2
1/ 4
  81  3
e.  81
d.
4
 91 / 2
f. w
1/ 8


8
9
w
g. 22a1 / 2  22 a
So far all the rational exponents have had a one in the numerator. What would a fraction mean that has a
number other than 1? Again, let’s look at some examples, remembering that the rules we already know
must be consistent.
3/2
4
1/2 3
3
= (4 ) = 2 = 8
The numerator is the power to which we raise the root. Notice it does not matter if we do the power or the
root first.
3/2
4
3 1/2
= (4 )
1/2
= 64
=8
Generally it is better to do the root first because the numbers are smaller.
In general, for a
root.
x/y
, x (the numerator) represents the power and y (the denominator) represents the
Just keep reminding yourself: power over root. (The root of a tree grows on the bottom.)
Example 2: Rewrite in radical notation and evaluate if possible.
 
2
a. 272 / 3  3 27  32  9
c.
 815 / 4

1
e. 8  4 / 3 
8
g.

5
 4  81
 32 2 / 5
4/3


b. 253 / 2 
 5
3 8 

1
16
1

  32 
4
 32
2/5
f. 6a 5 / 6  6
1
5
4
d. m5 / 4  4 m  m5
Either answer is correct but the last one is
better form.
Not a real
number
1
 25 3  53  125
2

1
4
6
a5
h. 5r  25 / 7  7 5r  25
2. Write radical expressions in exponential form.
If the expression is given in radical form, it can be rewritten in exponential form by reversing the process.
Example 3: Rewrite in exponential form.
a. 8 33  331 / 8
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b.
7 5
r
 r5 / 7
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M098
Carson Elementary and Intermediate Algebra 3e
c.
8

7 3
n
8
n3 / 7
 8n  3 / 7
Section 10.2
 
8
d. 3 m  m8 / 3
3. Use rules of exponents to simplify expressions with rational exponents.
Math is consistent so all of the exponent rules that we already know must still apply to rational exponents.
We combine our fraction arithmetic skills and exponent skills to simplify these problems.
Example 4: n2 / 3  n 1 / 2
2 1

2
To multiply powers, use the product rule: Keep the base
and add the exponents.
n3
4 3

6
6
n
To add fractions, find a common denominator.
n1 / 6


Example 5: 8c 4 / 5  4c 3 / 2


4 3

5
32c 2
Multiply the coefficients and use the product rule: Keep
the base and add the exponents.
To add fractions, find a common denominator.

8 15

10
10
32c
 32c 23 / 10
x5 / 8
Example 6:
x1 / 2
5 1

x8 2
To divide powers, use the quotient rule: Keep the base
and subtract the exponents.
5 4

x8 8
To subtract fractions, find a common denominator.
x1 / 8

Example 7: m3 / 2
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
2/5
3 2

m2 5
To raise a power to a power, use the power rule: Keep the
base and multiply the exponents.
m3 / 5
To multiply fractions, reduce common factors and multiply
the numerators together and the denominators together.
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M098
Carson Elementary and Intermediate Algebra 3e

Example 8: 3a1 / 4b 3 / 2
Section 10.2

4
34 a 4
To raise a power to a power, use the power rule: Keep the
base and multiply the exponents. Remember to raise the
coefficient to the power also.
81ab6
To multiply fractions, reduce common factors and multiply
the numerators together and the denominators together.
1 4 3 4


1b 2 1
4. Use rational exponents to simplify radical expressions.
Rational exponents allow us to simplify some radical expressions and multiply and divide radicals that have
different indexes.
Example 9: 4 36
361 / 4
Rewrite in exponential form.
6 
Rewrite the base as a power.
2 1/ 4
6 4
Use the power rule: Keep the base and multiply the
exponents.
61 / 2
Reduce common factors and multiply.
2
1
Rewrite in radical form.
6
Example 10: 6 27
271 / 6
Rewrite in exponential form.
3 
Rewrite the base as a power.
31 / 2
Use the power rule: Keep the base and multiply the
exponents.
3 1/ 6
Rewrite in radical form.
3
Example 11:
10
a 2b 8
a b 
Rewrite in exponential form.
a 2 / 10 b 8 / 10
Use the power rule: Keep the base and multiply the
exponents.
a1 / 5b 4 / 5
Reduce the fractions.
ab 
Use the power rule in reverse.
5
Rewrite in radical form.
2 8 1 / 10
4 1/ 5
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M098
Carson Elementary and Intermediate Algebra 3e
Example 12:
5
x4 
3
x2
x4 / 5  x2 / 3
Rewrite in exponential form.
x12 / 15  x10 / 15
Change the fractions to a common denominator.
x 22 / 15
Use the product rule.
15
Rewrite in radical form.
x 22
6
Example 13:
Section 10.2
z4
z
z4 / 6
Rewrite in exponential form.
z1 / 2
z4 / 6
Change the fractions to a common denominator.
z3 / 6
z1 / 6
Use the quotient rule.
6
Rewrite in radical form.
z
Example 14: 4 5 z
5 z 1 / 4
Rewrite the outside radical in exponential form.
z 
Rewrite the other radical in exponential form.
z1 / 20
Use the power rule.
20
Rewrite in radical form.
1/ 5 1/ 4
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z
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