1. 13
Exponents and Polynomials

2. 13.5 Integer Exponents and the Quotient Rule
Objectives
1. Use 0 as an exponent. 2. Use negative numbers as exponents. 3. Use the quotient rule for exponents. 4. Use combinations of rules.

3. Use 0 as an Exponent
Zero Exponent
For any nonzero real number a,
a0 = 1.
Example: 170 = 1

4. Use 0 as an Exponent
Example 1 Evaluate.
(a) 380
= 1
(b) (–9)0
= 1
(c) –90
= –1(9)0
= –1(1)
= –1
(d) x0
= 1
(e) 5x0
= 5·1
= 5
(f) (5x)0
= 1

5. Use Negative Numbers as Exponents
Negative Exponents
For any nonzero real number a and any integer n,
Example:

6. Use Negative Numbers as Exponents
Example 2
Simplify by writing with positive exponents. Assume that all
variables represent nonzero real numbers.
(a) 9–3
Notice that we can change the base to its reciprocal if we also change the sign of the exponent.

7. Use Negative Numbers as Exponents
Example 2 (concluded)
Simplify by writing with positive exponents. Assume that all
variables represent nonzero real numbers. 7

8. Use Negative Numbers as Exponents
CAUTION
A negative exponent does not indicate a negative number. Negative exponents lead to reciprocals.
Expression Example
a–n
Not negative
–a–n
Negative

9. Use Negative Numbers as Exponents
Changing from Negative to Positive Exponents
For any nonzero numbers a and b and any integers m and n,
Examples:

10. Use Negative Numbers as Exponents
CAUTION
Be careful. We cannot use the rule to
change negative exponents to positive exponents if the
exponents occur in a sum or difference of terms. For
example,
would be written with positive exponents as

11. Use the Quotient Rule for Exponents
For any nonzero number a and any integers m and n,
(Keep the same base and subtract the exponents.)
Example:

12. Use the Quotient Rule for Exponents
CAUTION
A common error is to write This is incorrect.
By the quotient rule, the quotient must have the same base, 5, so
We can confirm this by using the definition of exponents to write out the factors:

13. Use the Quotient Rule for Exponents
Example 3
Simplify. Assume that all variables represent nonzero real numbers.

14. Use the Quotient Rule for Exponents
Example 4
Simplify. Assume that all variables represent nonzero real numbers. 14

15. Use the Quotient Rule for Exponents
Example 4 (concluded)
Simplify. Assume that all variables represent nonzero real numbers. 15

16. Use the Quotient Rule for Exponents
Definitions and Rules for Exponents
For any integers m and n:
Product rule am · an = am+n
Zero exponent a0 = 1 (a ≠ 0)
Negative exponent
Quotient rule

17. Use the Quotient Rule for Exponents
Definitions and Rules for Exponents (concluded)
For any integers m and n:
Power rules (a) (am)n = amn
(b) (ab)m = ambm
(c)
Negative-to-Positive
Rules 17

18. Use Combinations of Rules
Example 5
Simplify each expression. Assume all variables represent nonzero real numbers.

19. Use Combinations of Rules
Example 5 (continued)
Simplify each expression. Assume all variables represent nonzero real numbers. 19

20. Use Combinations of Rules
Example 5 (concluded)
Simplify each expression. Assume all variables represent nonzero real numbers. 20

21. Use Combinations of Rules
Note
Since the steps can be done in several different orders, there are many equally correct ways to simplify expressions like those in Example 5.