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# Division Properties of Exponents

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Division Properties of Exponents

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### Division Properties of Exponents

1. 1. Warm UpSimplify.1. (x2)3 2.3. 4.5. 6.Write in Scientific Notation.7.8.
2. 2. A quotient of powers with the same basecan be found by writing the powers in afactored form and dividing out commonfactors.Notice the relationship between theexponents in the original quotient and theexponent in the final answer: 5 – 3 = 2.
3. 3. Example 1: Finding Quotients of PowersSimplify. A. B.
4. 4. Example 1: Finding Quotients of PowersSimplify.C. D.
5. 5. Helpful HintBoth and 729 are considered to be simplified.
6. 6. Check It Out! Example 1Simplify.a. b.
7. 7. Check It Out! Example 1Simplify. c. d.
8. 8. Example 2: Dividing Numbers in Scientific NotationSimplify and write theanswer 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.
9. 9. Writing MathYou can “split up” a quotient of products into aproduct of quotients:Example:
10. 10. Check It Out! Example 2Simplify and write theanswer 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.
11. 11. Example 3: ApplicationThe Department of Education spent about pesos in fiscal year 2004-05 onpublic schools. There were about students enrolled in public school. Whatwas the average spending per student? Writeyour 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.
12. 12. Example 3 ContinuedThe Department of Education spent about pesos in fiscal year 2004-05 onpublic schools. There were about students enrolled in public school. Whatwas the average spending per student? Writeyour 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.
13. 13. Check It Out! Example 3In 1990, the Philippines public debt was about pesos. The population of the UnitedStates was about people. What wasthe average debt per person? Write youranswer 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.
14. 14. Check It Out! Example 3 ContinuedIn 1990, the Philippines public debt was about pesos. The population of the UnitedStates was about people. What wasthe average debt per person? Write youranswer 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.
15. 15. A power of a quotient can be found by firstwriting the numerator and denominator aspowers.Notice that the exponents in the final answerare the same as the exponent in the originalexpression.
16. 16. Example 4A: Finding Positive Powers of QuotientSimplify. Use the Power of a Quotient Property. Simplify.
17. 17. Example 4B: Finding Positive Powers of QuotientSimplify. Use the Power of a Product Property. Use the Power of a Product Property: Simplify and use the Power of a Power Property:
18. 18. Example 4C: Finding Positive Powers of QuotientSimplify. Use the Power of a Product Property. Use the Power of a Product Property: Use the Power of a Product Property:
19. 19. Example 4C ContinuedSimplify. Use the Power of a Product Property:
20. 20. Check It Out! Example 4aSimplify. Use the Power of a Quotient Property. Simplify.
21. 21. Check It Out! Example 4bSimplify.
22. 22. Check It Out! Example 4cSimplify.
23. 23. 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,
24. 24. Example 5A: Finding Negative Powers of QuotientsSimplify. Rewrite with a positive exponent. Use the Powers of a Quotient Property . and
25. 25. Example 5B: Finding Negative Powers of QuotientsSimplify.
26. 26. Example 5C: Finding Negative Powers of QuotientsSimplify. 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
27. 27. Example 5C: Finding Negative Powers of QuotientsSimplify. Square and cube terms. 1 1 2 Divide out common factors. 1 24 12 Simplify.
28. 28. Helpful HintWhenever all of the factors in the numerator orthe denominator divide out, replace them with 1.
29. 29. Check It Out! Example 5aSimplify. Rewrite with a positive exponent. Use the power of a Quotient Property. 93=729 and 43 = 64.
30. 30. Check It Out! Example 5bSimplify. 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.
31. 31. Check It Out! Example 5cSimplify. 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.
32. 32. Lesson Quiz: Part ISimplify.1. 2.3. 4.5.
33. 33. Lesson Quiz: Part IISimplify. 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