# What is a value for ? using the product rule of exponents.

```Lesson 36
Section 7.2
1
2
What is a value for 9 ?
Consider This.
1
2
1
2
1 1

2 2
9 9  9
 91  9 using the product rule of exponents.
Now, Think!
What other number times itself equals 9? 3  3  9
1
2
Since both products equal 9, we can conclude that 9  3 .
Conclusion: A rational exponent is the same as a root!
1
Definition of a rational (fractional) exponent: a n  n a
Evaluate or write an equivalent root for each.
1
2
1. 16 
1
3
2. x 
1
5
3. (32) 
1
4. (4c) 4 
Write an equivalent expression using exponential notation.
5.
4
5
6.
3
ab 2 
7.
n
Expanded Definition of a rational (fractional) exponent: a
m
n
 n a m or
 a
n
m
Notice: The exponent n can be inside the radical (in the radicand) or outside of the
radical. The order does not matter. You can raise to the m power first and then take the
nth root. Or you can take the nth root and then raise to the m power.
1
The index of the root comes from the denominator. The exponent is the numerator.
Evaluate or write an equivalent root for each.
2
8. 27 3 
9. * 16

3
2

3
10. (mn) 4 
Write an equivalent expression using exponential notation.
11.
12.
83 
4

(ab
3

5

The *rules of exponents we had back at the beginning of the semester also apply with
these rational exponents. Here is a summary of those rules.
Product Rule: a m  a n  a m  n
am
Quotient Rule: n  a m  n
a
Negative Exponent Rule: a
 
 a mn
ab n
 a nb n
Power Rules: a m
n
n
1
1
a
 n or  n  a n or  
a
a
b
n
b
 
a
n
n
an
a

 
bn
b
Use the rules of exponents to simplify.
2
5
13. 8 3  8 3 
14.
2
2
5
8

1
8

2
  1  4 
15.  x 4  x 5  

 
 2
16.  4 3 
 
 
9
2
1
 2 3 2
17.  4 3 m 4  


  13 
18.  x 


4
3
```