UNIT 7.4 DIVISION PROPERTIES OF EXPONENTSUNIT 7.4 DIVISION PROPERTIES OF EXPONENTS

Warm UpSimplify.

1. (x2)3

3.

5.

2.

4.

6.

7.

Write in Scientific Notation.

8.

Use division properties of exponents to evaluate and simplify expressions.

Objective

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

Notice the relationship between the exponents in the original quotient and the exponent in the final answer: 5 – 3 = 2.

Simplify.

Example 1: Finding Quotients of Powers

A. B.

C.

Simplify.

Example 1: Finding Quotients of Powers

D.

Both and 729 are considered to be simplified.

Helpful Hint

Check It Out! Example 1

a.

Simplify.

b.

Check It Out! Example 1

Simplify.

c. d.

Example 2: Dividing Numbers in Scientific Notation

Simplify 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.

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

Example:

Writing Math

Check It Out! Example 2

Simplify and write the answer in scientific notation.

Write as a product of quotients.Simplify each quotient.

Simplify the exponent.Write 1.1 in scientific notation

as 11 x 10 .The second two terms have

the same base, so add the exponents.

Simplify the exponent.

Example 3: ApplicationThe 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 total debt by the number of students.

Write as a product of quotients.

Example 3 ContinuedThe 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 total debt by the number of students.

The average spending per student is $5,800.

Simplify each quotient.

Simplify the exponent.

Write in standard form.

Check It Out! Example 3

In 1990, the United States public debt was about dollars. The population of the United States was about 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.

In 1990, the United States public debt was about dollars. The population of the United States was about 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.

Check It Out! Example 3 Continued

Simplify each quotient.

Simplify the exponent.

Write in standard form.The average debt per person was $12,800.

A power of a quotient can be found by first writing the numerator and denominator as powers.

Notice that the exponents in the final answer are the same as the exponent in the original expression.

Simplify.

Example 4A: Finding Positive Powers of Quotient

Use the Power of a Quotient Property.

Simplify.

Simplify.

Example 4B: Finding Positive Powers of Quotient

Use the Power of a Product Property.

Use the Power of a Product Property:

Simplify and use the Power of a Power Property:

Simplify.Example 4C: Finding Positive Powers of Quotient

Use the Power of a Product Property.

Use the Power of a Product Property:

Use the Power of a Product Property:

Simplify.Example 4C Continued

Use the Power of a Product Property:

Check It Out! Example 4a

Simplify.

Use the Power of a Quotient Property.

Simplify.

Check It Out! Example 4b Simplify.

Check It Out! Example 4c Simplify.

Therefore,

Write the fraction as division.

Use the Power of a Quotient Property.

Multiply by the reciprocal.

Simplify.

Use the Power of a Quotient Property.

Remember that What if x is a fraction?.

Simplify.

Example 5A: Finding Negative Powers of Quotients

Rewrite with a positive exponent.

and

Use the Powers of a Quotient Property .

Simplify.

Example 5B: Finding Negative Powers of Quotients

Simplify.

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:

(3)2• (2n)3 = 32 • 23n3

and (2)2 • (6m)3 = 22 • 63m3

1

241

21

12

Divide out common factors.

Simplify.

Example 5C: Finding Negative Powers of Quotients

Simplify.

Square and cube terms.

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

Helpful Hint

Check It Out! Example 5a

Simplify.

93=729 and 43 = 64.

Use the power of a Quotient Property.

Rewrite with a positive exponent.

Check It Out! Example 5b

Simplify.

Rewrite with a positive exponent.

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

Use the Power of a Quotient Property.

Check It Out! Example 5c Simplify.

Rewrite each fraction with a positive exponent.

Use the Power of a Quotient Property.

Use the Power of a Product Property: (3)2= 9.

Add exponents and divide out common terms.

Lesson Quiz: Part I

1.

3. 4.

5.

2.

Simplify.

Lesson 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

 

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