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Simplifying Quotients of Powers

Learn how to simplify expressions involving quotients of powers and write them in scientific notation. Practice examples included.

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Simplifying Quotients of Powers

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Simplify. 1.(x2)3 3. 5. 2. x6 4. 6. Write in scientific notation. 7. 8.

  3. 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. California Standards

  4. m factors n factors A quotient of powers with the same base can be found by writing the powers in factored form and dividing out common factors.

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

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

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

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

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

  10. Simplify and write the answer in scientific notation Write 0.5 in scientific notation as 5 x 10 . Additional Example 2: Dividing Numbers in Scientific Notation Write as a product of quotients. Simplify each quotient. Simplify the exponent. The second two terms have the same base, so add the exponents. Simplify the exponent.

  11. Simplify and write the answer in scientific notation. Check It Out! Example 2 Write as a product of quotients. Simplify each quotient. = 1.1 × 10–2 Simplify the exponent.

  12. 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. Additional Example 3: Economics Application To find the average spending per student, divide the dollars spent by the number of students. Write as a product of quotients.

  13. 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. Additional Example 3 Continued = 0.58 ×109–5 Simplify each quotient. = 0.58 ×104 Simplify the exponent. Write in standard form. = 5800 The average spending per student is $5800.

  14. Check It Out! Example 3 In 1990, the United States public debt was about 3.2 × 1012 dollars. The population of the United States in 1990 was about 2.5 × 108 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.

  15. Check It Out! Example 3 Continued In 1990, the United States public debt was about 3.2 × 1012 dollars. The population of the United States in 1990 was about 2.5 × 108 people. What was the average debt per person? Write your answer in standard form. Simplify each quotient. Simplify the exponent. Write in standard form. The average debt per person was $12,800.

  16. n factors n factors n factors A power of a quotient can be found by first writing factors and then writing the numerator and denominator as powers.

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

  18. Use the Power of a Product Property: Simplify and use the Power of a Power Property: Additional Example 4B: Finding Positive Powers of Quotient Simplify. Use the Power of a Quotient Property.

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

  20. Simplify and use the Power of a Power Property: . Use the Power of a Product Property: (x3y3)2 = x32y32. Additional Example 4C Continued Simplify. Simplify.

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

  25. Additional Example 5B: Finding Negative Powers of Quotients Simplify. Use the Power of a Quotient Property. Use the Power of a Power Property (y3)2 = y32 = y6. Use the Power of a Product Property (2x2)2 = 22x22. Simplify.

  26. Additional 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 = 23n3 and (6m)3 = 63m3.

  27. Additional Example 5C: Finding Negative Powers of Quotients Simplify. 1 2 1 Divide out common factors. 1 24 12 Simplify.

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

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

  30. 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: (b2c3)4= b2•4c3•4 = b8c12 and (2a)4= 24a4 = 16a4.

  31. 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.

  32. 1. 3.4. 5. Lesson Quiz: Part I Simplify. 2.

  33. 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  105square 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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