Division Properties of Exponents

2. Warm Up
Simplify.
1. (x2)3 2.
3. 4.
5. 6.
Write in Scientific Notation.
7.
8.

3. 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.

5. Example 1: Finding Quotients of Powers
Simplify.
A. B.

6. Example 1: Finding Quotients of Powers
Simplify.
C. D.

7. Helpful Hint
Both and 729 are considered to be simplified.

8. Check It Out! Example 1
Simplify.
a. b.

9. Check It Out! Example 1
Simplify.
c. d.

10. 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.

11. Writing Math
You can “split up” a quotient of products into a
product of quotients:
Example:

12. 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.

13. Example 3: Application
The Department of Education spent about
pesos 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.

14. Example 3 Continued
The Department of Education spent about
pesos 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 Php5,800.

15. Check It Out! Example 3
In 1990, the Philippines public debt was about
pesos. 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.

16. Check It Out! Example 3 Continued
In 1990, the Philippines public debt was about
pesos. 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 Php12,800.

17. 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.

19. Example 4A: Finding Positive Powers of Quotient
Simplify.
Use the Power of a Quotient
Property.
Simplify.

20. 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:

21. 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:

22. Example 4C Continued
Simplify.
Use the Power of a Product
Property:

23. Check It Out! Example 4a
Simplify.
Use the Power of a Quotient
Property.
Simplify.

24. Check It Out! Example 4b
Simplify.

25. Check It Out! Example 4c
Simplify.

26. 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,

28. Example 5A: Finding Negative Powers of Quotients
Simplify.
Rewrite with a positive exponent.
Use the Powers of a Quotient
Property .
and

29. Example 5B: Finding Negative Powers of Quotients
Simplify.

30. 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

31. Example 5C: Finding Negative Powers of Quotients
Simplify.
Square and cube terms.
1
1 2
Divide out common
factors.
1
24
12
Simplify.

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

33. Check It Out! Example 5a
Simplify.
Rewrite with a positive
exponent.
Use the power of a Quotient
Property.
93=729 and 43 = 64.

34. 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.

35. 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.

36. Lesson Quiz: Part I
Simplify.
1. 2.
3. 4.
5.

37. 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