Warm UpEvaluate each expression for the given values of the variables.

1. x3y2 for x = –1 and y = 10

2. for x = 4 and y = (–7)

Write each number as a power of the given base.

–100

433. 64; base 4

(–3)34. –27; base (–3)

Evaluate expressions containing zero and integer exponents.

Simplify expressions containing zero and integer exponents.

Objectives

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.

3125 625 125 25 5

5

Power

Value

55 54 53 52 51 5–150 5–2

5 5 5

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

Base

x

Exponent

Remember!

4

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.

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

Reading Math

Example 1: Application

One cup is 2–4 gallons. Simplify this expression.

gal is equal to

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

Example 2: Zero and Negative Exponents

Simplify.

A. 4–3

B. 70

7º = 1Any nonzero number raised to the zero

power is 1.

C. (–5)–4

D. –5–4

In (–3)–4, the base is negative because the

negative sign is inside the parentheses. In –3–4

the base (3) is positive.

Caution

Check It Out! Example 2 Simplify.

a. 10–4

b. (–2)–4

c. (–2)–5

d. –2–5

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

Example 3B: Evaluating Expressions with Zero and Negative Exponents

Simplify 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.

Simplify the denominator.

Simplify.

Check It Out! Example 3a

Evaluate the expression for the given value of the variable.

p–3 for p = 4

Substitute 4 for p.

Simplify exponent.

Write the power in the denominator as a product.

Simplify the denominator.

Check It Out! Example 3b

Evaluate the expression for the given values of the variables.

for a = –2 and b = 6

2

Substitute –2 for a and 6 for b.

Simplify expressions with exponents.

Write the power in the denominator as a product.

Simplify the denominator.

Simplify.

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 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.

An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents.

Simplify.

Example 4: Simplifying Expressions with Zero and Negative Numbers

A. 7w–4 B.

Simplify.

Example 4: Simplifying Expressions with Zero and Negative Numbers

C.

and

Check It Out! Example 4

Simplify.

a. 2r0m–3

b.

c.

rº = 1 and .

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

5. –112

1

–121

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