1. 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.

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

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

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

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

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

8. 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
The second two terms have the same base, so add the exponents.
Simplify the exponent.

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

10. 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.

11. Example 3: Application
The Colorado Department of Education spent about dollars in fiscal year on public schools. There were about students enrolled in public school. What was the average spending per student? Write your answer in scientific notation.
To find the average spending per student, divide the total debt by the number of students.
Write as a product of quotients.

12. Example 3 Continued
The Colorado Department of Education spent about dollars in fiscal year on public schools. There were about students enrolled in public school. What was the average spending per student? Write your answer in scientific notation.
To find the average spending per student, divide the total debt by the number of students.
Simplify each quotient.
Simplify the exponent.

13. Check It Out! Example 4
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 scientific notation.
To find the average debt per person, divide the total debt by the number of people.
Write as a product of quotients.

14. Check It Out! Example 4 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 scientific notation.
To find the average debt per person, divide the total debt by the number of people.
Simplify each quotient.
Simplify the exponent.

15. 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.

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

18. 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:

19. 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:

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

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

22. Check It Out! Example 4b
Simplify.

23. Check It Out! Example 4c
Simplify.

24. . 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,

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

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

28. 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

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

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

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

32. 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.

33. 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.

34. 1. 5.
Lesson Quiz: Part I
Simplify. 2.
6. Simplify (3  1012) ÷ (5  105) and write the answer in scientific notation.
6  106