1. 7-4 Division Properties of Exponents
7-4 Division Properties of Exponents
Warm Up
Lesson Presentation
Lesson Quiz
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2. 7-4 Division Properties of Exponents
Warm Up
Simplify.
1. (x2)3 2.
3. 4.
5. 6.
Write in Scientific Notation.
7.
8.
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3. 7-4 Division Properties of Exponents
Objective
Use division properties of exponents to
evaluate and simplify expressions.
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4. 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.
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5. 7-4 Division Properties of Exponents
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6. 7-4 Division Properties of Exponents
Example 1: Finding Quotients of Powers
Simplify.
A. B.
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7. 7-4 Division Properties of Exponents
Example 1: Finding Quotients of Powers
Simplify.
C. D.
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8. 7-4 Division Properties of Exponents
Helpful Hint
Both and 729 are considered to be simplified.
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9. 7-4 Division Properties of Exponents
Check It Out! Example 1
Simplify.
a. b.
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10. 7-4 Division Properties of Exponents
Check It Out! Example 1
Simplify.
c. d.
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11. 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.
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12. 7-4 Division Properties of Exponents
Writing Math
You can “split up” a quotient of products into a
product of quotients:
Example:
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13. 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.
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.
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14. 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.
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15. 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.
Simplify each quotient.
Simplify the exponent.
Write in standard form.
The average spending per student is $5,800.
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16. 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.
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17. 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 Algebra 1

18. 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.
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19. 7-4 Division Properties of Exponents
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20. 7-4 Division Properties of Exponents
Example 4A: Finding Positive Powers of Quotient
Simplify.
Use the Power of a Quotient
Property.
Simplify.
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21. 7-4 Division Properties of Exponents
Example 4B: Finding Positive Powers of Quotient
Simplify.
Use the Power of a Product
Property.
Use the Power of a Product
Property:
Simplify and use the Power
of a Power Property:
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22. 7-4 Division Properties of Exponents
Example 4C: Finding Positive Powers of Quotient
Simplify.
Use the Power of a Product
Property.
Use the Power of a Product
Property:
Use the Power of a Product
Property:
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23. 7-4 Division Properties of Exponents
Example 4C Continued
Simplify.
Use the Power of a Product
Property:
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24. 7-4 Division Properties of Exponents
Check It Out! Example 4a
Simplify.
Use the Power of a Quotient
Property.
Simplify.
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25. 7-4 Division Properties of Exponents
Check It Out! Example 4b
Simplify.
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26. 7-4 Division Properties of Exponents
Check It Out! Example 4c
Simplify.
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27. 7-4 Division Properties of Exponents
Remember that . What if x is a fraction?
Write the fraction as division.
Use the Power of a Quotient
Property.
Multiply by the reciprocal.
Simplify.
Use the Power of a Quotient
Property.
Therefore,
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28. 7-4 Division Properties of Exponents
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29. 7-4 Division Properties of Exponents
Example 5A: Finding Negative Powers of Quotients
Simplify.
Rewrite with a positive exponent.
Use the Powers of a Quotient
Property .
and
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30. 7-4 Division Properties of Exponents
Example 5B: Finding Negative Powers of Quotients
Simplify.
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31. 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:
(3)2 (2n)3 = 32  23n3
and (2)2  (6m)3 = 22  63m3
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32. 7-4 Division Properties of Exponents
Example 5C: Finding Negative Powers of Quotients
Simplify.
Square and cube terms.
1
1 2
Divide out common
factors.
1
24
12
Simplify.
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33. 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.
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34. 7-4 Division Properties of Exponents
Check It Out! Example 5a
Simplify.
Rewrite with a positive
exponent.
Use the power of a Quotient
Property.
93=729 and 43 = 64.
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35. 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 Power
Property: (b2c3)4= b2•4c3•4 =
b8c12 and (2a)4= 24a4= 16a4.
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36. 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.
Use the Power of a Product
Property: (3)2= 9.
Add exponents and divide
out common terms.
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37. 7-4 Division Properties of Exponents
Lesson Quiz: Part I
Simplify.
1. 2.
3. 4.
5.
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38. 7-4 Division Properties of Exponents
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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