7-4 Division Properties of Exponents

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Lesson Presentation

California Standards

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7-4 Division Properties of Exponents

Warm UpSimplify.

1. (x2)3

3.

5.

2.

4.

6.

7.

Write in scientific notation.

8.

x6

7-4 Division Properties of 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

7-4 Division Properties of Exponents

A quotient of powers with the same base can be found by writing the powers in factored form and dividing out common factors.

m factors

n factors

7-4 Division Properties of Exponents

7-4 Division Properties of Exponents

Simplify.

Additional Example 1: Finding Quotients of Powers

A. B.

7-4 Division Properties of Exponents

C.

Simplify.

Additional Example 1: Finding Quotients of Powers

D.

7-4 Division Properties of ExponentsCheck It Out! Example 1

a.

Simplify.

b.

7-4 Division Properties of ExponentsCheck It Out! Example 1

Simplify.

c. d.

7-4 Division Properties of Exponents

You can “split up” a quotient of products into a product of quotients:

Writing Math

Example:

7-4 Division Properties of ExponentsAdditional Example 2: Dividing Numbers in Scientific

NotationSimplify and write the

answer in scientific notation

Write as a product of quotients.

Simplify each quotient.Simplify the exponent.Write 0.5 in scientific notation

as 5 x 10 .The second two terms have the same

base, so add the exponents.

Simplify the exponent.

7-4 Division Properties of ExponentsCheck It Out! Example 2

Simplify and write the answer in scientific notation.

Write as a product of quotients.

Simplify each quotient.

Simplify the exponent.= 1.1 × 10–2

7-4 Division Properties of ExponentsAdditional Example 3: Economics Application

The Colorado Department of Education spent about dollars in fiscal year 2004-05 on public schools. There were about students enrolled in public school. What was the average spending per student? Write your answer in standard form.

To find the average spending per student, divide the dollars spent by the number of students.

Write as a product of quotients.

7-4 Division Properties of ExponentsAdditional Example 3 Continued

The Colorado Department of Education spent about dollars in fiscal year 2004-05 on public schools. There were about students enrolled in public school. What was the average spending per student? Write your answer in standard form.

The average spending per student is $5800.

Simplify each quotient.

Simplify the exponent.

Write in standard form.

= 0.58 ×109–5

= 0.58 ×104

= 5800

7-4 Division Properties of ExponentsCheck It Out! Example 3

In 1990, the United States public debt was about 3.2 × 1012 dollars. The population of the United States in 1990 was about 2.5 × 108 people. What was the average debt per person? Write your answer in standard form.

To find the average debt per person, divide the total debt by the number of people.

Write as a product of quotients.

7-4 Division Properties of ExponentsCheck It Out! Example 3 Continued

Simplify each quotient.

Simplify the exponent.

Write in standard form.

The average debt per person was $12,800.

In 1990, the United States public debt was about 3.2 × 1012 dollars. The population of the United States in 1990 was about 2.5 × 108 people. What was the average debt per person? Write your answer in standard form.

7-4 Division Properties of ExponentsA power of a quotient can be found by first writing factors and then writing the numerator and denominator as powers.

n factors

n factors

n factors

7-4 Division Properties of Exponents

7-4 Division Properties of Exponents

Simplify.

Additional Example 4A: Finding Positive Powers of Quotient

Use the Power of a Quotient Property.

Simplify.

7-4 Division Properties of Exponents

Simplify.

Additional Example 4B: Finding Positive Powers of Quotient

Use the Power of a Quotient Property.

Use the Power of a Product Property:

Simplify and use the Power of a Power Property:

7-4 Division Properties of Exponents

Simplify.

Additional Example 4C: Finding Positive Powers of Quotient

Use the Power of a Quotient Property.

Use the Power of a Product Property:

7-4 Division Properties of Exponents

Simplify.

Additional Example 4C Continued

Simplify.

Simplify and use the Power of a Power Property: .Use the Power of a Product Property: (x3y3)2 = x32y32.

7-4 Division Properties of ExponentsCheck It Out! Example 4a

Simplify.

Use the Power of a Quotient Property.

Simplify.

7-4 Division Properties of ExponentsCheck It Out! Example 4b

Simplify.

7-4 Division Properties of ExponentsCheck It Out! Example 4c

Simplify.

7-4 Division Properties of Exponents

7-4 Division Properties of Exponents

Simplify.

Additional Example 5A: Finding Negative Powers of Quotients

Rewrite with a positive exponent.

and

Use the Power of a Quotient Property .

7-4 Division Properties of Exponents

Simplify.

Additional Example 5B: Finding Negative Powers of Quotients

Use the Power of a Quotient Property.

Use the Power of a Power Property (y3)2 = y32 = y6. Use the Power of a Product Property (2x2)2 = 22x22.

Simplify.

7-4 Division Properties of Exponents

Simplify.

Additional Example 5C: Finding Negative Powers of Quotients

Rewrite each fraction with a positive exponent.

Use the Power of a Quotient Property.

Use the Power of a Product Property: (2n)3 = 23n3 and (6m)3 = 63m3.

7-4 Division Properties of Exponents

1

241

21

12

Divide out common factors.

Simplify.

Additional Example 5C: Finding Negative Powers of Quotients

Simplify.

7-4 Division Properties of Exponents

Whenever all of the factors in the numerator or the denominator divide out, replace them with 1.

Helpful Hint

7-4 Division Properties of Exponents

Check It Out! Example 5a

Simplify.

93 = 729 and 43 = 64.

Use the Power of a Quotient Property.

Rewrite with a positive exponent.

7-4 Division Properties of ExponentsCheck It Out! Example 5b

Simplify.

Rewrite with a positive exponent.

Use the Power of a Product Property: (b2c3)4= b2•4c3•4 = b8c12 and (2a)4= 24a4 = 16a4.

Use the Power of a Quotient Property.

7-4 Division Properties of ExponentsCheck It Out! Example 5c

Simplify.

Rewrite each fraction with a positive exponent.

Use the Power of a Quotient Property.

Simplify.

Add exponents and divide out common terms.

7-4 Division Properties of Exponents

Lesson Quiz: Part I

1.

3. 4.

5.

2.

Simplify.

7-4 Division Properties of ExponentsLesson Quiz: Part II

Simplify.

6. Simplify (3 1012) ÷ (5 105) and write the answer in scientific notation. 6 106

7. The Republic of Botswana has an area of 6 105 square kilometers. Its population is about 1.62 106. What is the population density of Botswana? Write your answer in standard form.2.7 people/km2