Integral Exponents

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)

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.

cup 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 = 1

Any 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

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.

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.

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.

Evaluate expressions with exponents.

Write the power in the denominator as a product.

Evaluate the powers in the product.

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