1. You have seen positive exponents
You have seen positive exponents. Recall that to simplify 32, use 3 as a factor 2 times: 32 = 3  3 = 9.
But what does it mean for an exponent to be negative or 0? You can use a table and look for a pattern to figure it out.
Power
Value 55 54 53 52 51 50
5–1
5–2
3125
625
125 25 5
 5
 5
 5
 5

2. When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5.

3. Base x
Exponent
Remember! 4

4. The words go in your foldable!!

5. Notice the phrase “nonzero number” in the previous table
Notice the phrase “nonzero number” in the previous table. This is because 00 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above the table with a base of 0 instead of 5, you would get 0º = . Also 0–6 would be = . Since division by 0 is undefined, neither value exists.

6. In (–3)–4, the base is negative because the negative sign is inside the parentheses. In –3–4 the base (3) is positive.
Caution

7. Check It Out! Example 2
Simplify.
a. 10–4
b. (–2)–4
c. (–2)–5
d. –2–5

8. What if you have an expression with a negative exponent in a denominator, such as ?
Definition of a negative exponent.
Substitute –8 for n.
Simplify the exponent on the right side.
An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents.
So if a base with a negative exponent is in a denominator, it is equivalent to the same base with the opposite (positive) exponent in the numerator.

9. Example 4: Simplifying Expressions with Zero and Negative Numbers
A. 7w–4

10. Example 1: Finding Products of Powers
Simplify. A.
Group powers with the same base together.
Add the exponents of powers with the same base. B.
Group powers with the same base together.
Add the exponents of powers with the same base.

11. Example 1: Finding Products of Powers
Simplify. C.
Group the positive exponents and add since they have the same base
Add the like bases. 1 D.
Group the first two and second two terms.
Divide the first group and add the second group. =
Multiply.

12. A number or variable written without an exponent actually has an exponent of 1.
Remember!
10 = 101
y = y1

13. To find a power of a power, you can use the meaning of exponents.
Notice the relationship between the exponents in the original power and the exponent in the final power:

15. Example 3: Finding Powers of Powers
Simplify.
Use the Power of a Power Property.
Simplify.
Use the Power of a Power Property.
Zero multiplied by any number is zero 1
Any number raised to the zero power is 1.

16. Example 3: Finding Powers of Powers
Simplify. C.
Use the Power of a Power Property.
Simplify the exponent of the first term.
Since the powers have the same base, add the exponents.
Write with a positive exponent.

17. The words go in your foldable!!
Powers of products can be found by using the meaning of an exponent.
The words go in your foldable!!

18. Example 4: Finding Powers of Products
Simplify. A.
Use the Power of a Product Property.
Simplify. B.
Use the Power of a Product Property.
Simplify.

19. Example 4: Finding Powers of Products
Simplify. C.
Use the Power of a Product Property.
Use the Power of a Product Property.
Simplify.

20. A quotient of powers with the same base can be found by writing the powers in a factored form and dividing out common factors.
Notice the relationship between the exponents in the original quotient and the exponent in the final answer: 5 – 3 = 2.

22. Example 1: Finding Quotients of Powers
Simplify. A. B.

23. Example 1: Finding Quotients of Powers
Simplify. C. D.

24. A power of a quotient can be found by first writing the numerator and denominator as powers.
Notice that the exponents in the final answer are the same as the exponent in the original expression.

25. The words go in your foldable!!

26. Example 4A: Finding Positive Powers of Quotient
Simplify.
Use the Power of a Quotient Property.
Simplify.

27. Example 4B: Finding Positive Powers of Quotient
Simplify.
Use the Power of a Product Property.
Use the Power of a Product Property:
Simplify and use the Power of a Power Property:

28. . Remember that What if x is a fraction?
Write the fraction as division.
Use the Power of a Quotient Property.
Multiply by the reciprocal.
Simplify.
Use the Power of a Quotient Property.
Therefore,

30. Example 5A: Finding Negative Powers of Quotients
Simplify.
Rewrite with a positive exponent.
Use the Powers of a Quotient Property .
and

31. Example 5B: Finding Negative Powers of Quotients
Simplify.