7-1 Integer Exponents

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Lesson Presentation

California Standards

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7-1 Integer 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)

7-1 Integer Exponents

2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

California Standards

7-1 Integer 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 make a conjecture.

3125 625 125 25 5

5

Power

Value

55 54 53 52 51 5–150 5–2

5 5 5

7-1 Integer Exponents

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

7-1 Integer Exponents

x

Remember!

4

Exponent

Base

7-1 Integer Exponents

7-1 Integer Exponents

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 in

the table with a base of 0 instead of 5, you would

get 00 = . Also, 0–6 would be = . Since

division by 0 is undefined, neither value exists.

7-1 Integer Exponents

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

Reading Math

7-1 Integer ExponentsAdditional Example 1: Measurement Application

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

gal is equal to

7-1 Integer ExponentsCheck 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

7-1 Integer ExponentsAdditional 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

7-1 Integer Exponents

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

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

the base (3) is positive.

Caution

7-1 Integer ExponentsCheck It Out! Example 2

Simplify.

a. 10–4

b. (–2)–4

c. (–2)–5

d. –2–5

7-1 Integer ExponentsAdditional Example 3A: Evaluating Expressions with

Zero and Negative Exponents

Evaluate the expression for the given value of the variable.

x–2 for x = 4

Substitute 4 for x.

Use the definition

7-1 Integer ExponentsAdditional Example 3B: Evaluating Expressions with Zero

and Negative ExponentsEvaluate 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.

7-1 Integer ExponentsCheck 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.

7-1 Integer ExponentsCheck 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.

7-1 Integer Exponents

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.

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.

7-1 Integer Exponents

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

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.

7-1 Integer Exponents

Simplify.

Additional Example 4: Simplifying Expressions with Zero and Negative Numbers

A. 7w–4 B.

7-1 Integer Exponents

Simplify.

Additional Example 4: Simplifying Expressions with Zero and Negative Numbers

C.

and

7-1 Integer ExponentsCheck It Out! Example 4

Simplify.

a. 2r0m–3

b.

c.

r0 = 1 and .

7-1 Integer ExponentsLesson 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

7-1 Integer ExponentsLesson 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

Simplify.

8. 4y–5

10.

9. –3y6