This document discusses integral exponents and how to evaluate expressions with zero and negative exponents. It provides examples of simplifying expressions with zero and negative exponents by using the definition that a negative exponent means to take the reciprocal of the base and raise it to the positive value of the exponent. It also explains that any nonzero number raised to the zero power is equal to 1, and expressions should be rewritten with only positive exponents.

1. Integral Exponents

2. Warm Up
Evaluate each expression for the given values of
the variables.
1. x3y2 for x = –1 and y = 10 –100
2. for x = 4 and y = (–7)
Write each number as a power of the given base.
3. 64; base 4 43
4. –27; base (–3) (–3)3

3. 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 55 54 53 52 51 50 5–1 5–2
3125 625 125 25 5
Value
5 5 5 5

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

5. Remember!
Base
x4
Exponent

7. 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.

8. Reading Math
2–4 is read “2 to the negative fourth power.”

9. Example 1: Application
One cup is 2–4 gallons. Simplify this expression.
cup is equal to

10. Check It Out! Example 1
A sand fly may have a wingspan up to 5–3 m. Simplify this
expression.
5-3 m is equal to

11. Example 2: Zero and Negative Exponents
Simplify.
A. 4–3
B. 70 Any nonzero number raised to the zero power is 1.
7º = 1
C. (–5)–4
D. –5–4

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

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

14. Example 3A: Evaluating Expressions with Zero and Negative Exponents
Evaluate the expression for the given value of the variables.
x–2 for x = 4
Substitute 4 for x.
Use the definition

15. Example 3B: Evaluating Expressions with Zero and Negative Exponents
Evaluate the expression for the given values of the variables.
–2a0b-4 for a = 5 and b = –3
Substitute 5 for a and –3 for b.
Evaluate expressions with exponents.
Write the power in the denominator as a
product.
Evaluate the powers in the
product.
Simplify.

16. Check It Out! Example 3a
Evaluate the expression for the given value of the variable.
p–3 for p = 4
Substitute 4 for p.
Evaluate exponent.
Write the power in the denominator as a
product.
Evaluate the powers in the
product.

17. Check It Out! Example 3b
Evaluate the expression for the given values of the variables.
for a = –2 and b = 6
Substitute –2 for a and 6 for b.
Evaluate expressions with exponents.
Write the power in the denominator as a
product.
Evaluate the powers in the
product.
Simplify.
2

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

19. Example 4: Simplifying Expressions with Zero and Negative Numbers
Simplify.
A. 7w–4 B.

20. Example 4: Simplifying Expressions with Zero and Negative Numbers
Simplify.
C.
and

21. Check It Out! Example 4
Simplify.
a. 2r0m–3
rº = 1 and .
b. c.

22. Lesson Quiz: Part I
1. A square foot is 3–2 square yards. Simplify this expression.
Simplify.
2. 2–6
3. (–7)–3
4. 60 1
5. –112 –121

23. Lesson Quiz: Part II
Evaluate each expression for the given value(s) of the variables(s).
6. x–4 for x =10
7. for a = 6 and b = 3