Transcript Slide 1
7-4 Division Properties of Exponents Warm Up Simplify.
1. (x
3.
5.
2 ) 3
x
6
2 .
4.
6.
Write in Scientific Notation.
7.
8.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents
Objective
Use division properties of exponents to evaluate and simplify expressions.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents
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 .
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 1: Finding Quotients of Powers Simplify.
A. B. Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 1: Finding Quotients of Powers Continued Simplify.
C.
D.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Helpful Hint
Both and 729 are considered to be simplified.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Check It Out!
Example 1 Simplify.
a. b. Holt McDougal Algebra 1
7-4 Division Properties of Exponents Check It Out!
Example 1 Continued Simplify.
c.
d.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents 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.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Writing Math
You can “split up” a quotient of products into a product of quotients: Example:
Holt McDougal Algebra 1
7-4 Division Properties of Exponents 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.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 3: 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 total debt by the number of students.
Write as a product of quotients.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents 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.
To find the average spending per student, divide the total debt by the number of students.
= 0.58 ×10 9–5
Simplify each quotient.
= 0.58 ×10 4
Simplify the exponent.
= 5800
Write in standard form.
The average spending per student is $5800.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents 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.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Check It Out!
Example 3 Continued 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.
Simplify each quotient.
Simplify the exponent.
Write in standard form.
The average debt per person was $12,800.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents
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.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 4A: Finding Positive Powers of Quotient Simplify.
Use the Power of a Quotient Property.
Simplify.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 4B: Finding Positive Powers of Quotient Simplify.
Use the Power of a Quotient Property.
Use the Power of a Product Property: Simplify and use the Power of a Power Property:
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 4C: Finding Positive Powers of Quotient Simplify.
Use the Power of a Quotient Property.
Use the Power of a Product Property:
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 4C Continued Simplify.
Simplify and use the Power of a Power Property: .
Use the Power of a Power Property: (x 3 y 3 ) 2 = x 3
2 y 3
2 .
Simplify.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Check It Out!
Example 4a Simplify.
Use the Power of a Quotient Property and the Power of a Power Property.
Simplify.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Check It Out!
Example 4b Simplify.
Use the Power of a Product Property and the Power of a Power Property.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Check It Out!
Example 4c Simplify.
Use the Power of a Product Property.
Use the Power of a Power Property.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents
Therefore, .
Holt McDougal Algebra 1
Write the fraction as division.
Use the Power of a Quotient Property.
Multiply by the reciprocal.
Simplify.
Use the Power of a Quotient Property.
7-4 Division Properties of Exponents Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 5A: Finding Negative Powers of Quotients Simplify.
Rewrite with a positive exponent.
Use the Power of a Quotient Property.
and
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 5B: Finding Negative Powers of Quotients Simplify.
Rewrite with a positive exponent.
Use the Power of a Quotient Property.
Use the Power of a Power Property (y 3 Property (2x ) 2 2 ) 2 = y 3 = 2
2 2 x = y 2
2 6 . Use the Power of a Product Simplify.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 5C: Finding Negative Powers of Quotients Simplify.
Rewrite each fraction with a positive exponent.
Use the Power of a Quotient Property.
Use the Power of a Product Property: (2n) 3 = 2 3 n 3 and (6m) 3 = 6 3 m 3 .
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Example 5C: Finding Negative Powers of Quotients Simplify.
Holt McDougal Algebra 1 1 1 1 2 24 12
Divide out common factors.
Simplify.
7-4 Division Properties of Exponents Helpful Hint
Whenever all of the factors in the numerator or the denominator divide out, replace them with 1.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Check It Out!
Example 5a Simplify.
Rewrite with a positive exponent.
Use the Power of a Quotient Property.
9 3 = 729 and 4 3 = 64.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Check It Out!
Example 5b Simplify.
Rewrite with a positive exponent.
Use the Power of a Quotient Property.
Use the Power of a Product Property: (b 2 c 3 ) 4 = b 2 •4 c 3•4 = b 8 c 12 and (2a) 4 = 2 4 a 4 = 16a 4 .
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Check 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.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Lesson Quiz: Part I Simplify.
1.
2.
3.
4.
5.
Holt McDougal Algebra 1
7-4 Division Properties of Exponents Lesson Quiz: Part II Simplify.
6. Simplify (3 10
12
) ÷ (5 10
5
) and write the answer in scientific notation.
6 10
6
7. The Republic of Botswana has an area of 6 10
5
square kilometers. Its population is about 1.62 Botswana? Write your answer in standard form.
10
6
. What is the population density of 2.7 people/km
2 Holt McDougal Algebra 1