1. exponential Fun: An Interactive Tutorial on the Properties of Exponents Click here to begin

2. objective to learn how to multiply and divide expressions with exponents, including zero and negative exponents. Tennessee State Standards: 1.0 Number and Operations 1.1 demonstrate an understanding of the subsets, properties, and operations of the real number system 1.1d use exponents to simplify a monomial written in expanded form 1.2 demonstrate an understanding of the relative size of rational and irrational numbers 1.2f select ratios and proportions to represent real-world problems (e.g. scale drawings, sampling, etc.) 2.0 Algebra 2.2 use algebraic thinking to generalize a pattern by expressing the pattern in functional notation 2.2i Evaluate an algebraic expression given values for one or more variables using grouping symbols and/or exponents less than four

3. directions Read the tutorial and complete the quiz at the end to test your knowledge! takes you to the directions page takes you to the directions page takes you to the main menu takes you to the main menu takes you to the next page takes you to the next page takes you back to the previous slide takes you back to the previous slide stops the lesson stops the lesson

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5. expressions containing exponents an expression like 4 6 is called a power. The exponent 6 represents the number of times the base 4 is used as a factor: exponent base 4 6 = 4 ● 4 ● 4 ● 4 ● 4 ● 4 power 6 factors of 4

6. multiplying exponents: product of powers property Let a and b be numbers and let m and n be integers: to multiply two powers that have the same base add the exponents together: a 2 a 3 = a 2 ● a 3 = a ● a ● a ● a ● a 2 factors 3 factors = a 3 + 2 = a 5

7. multiplying exponents: power of a power property Let a and b be numbers and let m and n be integers: to find the power of a power, you multiply exponents: (a 2 ) 3 = a 2 ● a 2 ● a 2 3 factors = a ● a ● a ● a ● a ● a 6 factors = a 6 = (a 2 ) 3

8. multiplying exponents: power of a power property to find the power of a power, you multiply exponents Important Note: when you use the power of product property, it is the quantity within the parentheses that is raised to the power, not the individual terms. For example: (a + 1) 3 has 3 factors: (a + 1) 3 = (a + 1)(a + 1)(a + 1)

9. multiplying exponents: power of a product property Let a and b be numbers and let m and n be integers: to find the power of a product, find the power of each factor and multiply: (a ● b) 3 = a 3 ● b 3 (ab) 3 = a 3 b 3

10. zero and negative exponents Let a and b be numbers and let m and n be integers: Let a be a nonzero number and let n be a positive integer: A nonzero number to the zero power is 1: a 0 = 1, a = 0. a -n is the reciprocal of a n : a -n = 1/a n, a = 0.

11. dividing with exponents: quotient of powers property Let a and b be numbers and let m and n be integers: to divide powers having the same base, subtract exponents: a m /a n = a m – n, a = 0

12. dividing with exponents: power of a quotient property Let a and b be numbers and let m and n be integers: to find a power of a quotient, find the power of the numerator and the power of the denominator and divide: (a/b) m = a m /b m, b = 0

13. Time for a quiz! Now that you have learned how to multiply and divide expressions with exponents, including zero and negative exponents, check your understanding with these quiz questions!

14. quiz question one Simplify the following expression: (4x 2 y) 3 ● x 5 d. 64x 10 y 3 c. 64x 11 y 3 a. 4x 10 y 3 b. 4x 7 y 3

15. Great job! The correct answer is c: (4x 2 y) 3 ● x 5 = 4 3 ● (x 2 ) 3 ● y 3 ● x 5 power of a product = 64 ● x 6 ● y 3 ● x 5 power of a power = 64x 11 y 3 product of powers

16. Try again … Remember the rules: 1. 1. To multiply two powers that have the same base, add the exponents together. 2. 2. To find the power of a power, you multiply exponents. 3. 3. To find the product of a power, find the power of each factor and multiply.

17. quiz question two Simplify the following expression: (abc 2 ) 3 (a 2 b) 2 d. a 7 b 5 c 5 c. a 12 b 6 c 6 a. a 5 b 5 c 5 b. a 7 b 5 c 6

18. Great job! The correct answer is b: (abc 2 ) 3 (a 2 b) 2 = a 3 b 3 (c 2 ) 3 ● a 4 b 2 power of a product = a 3 b 3 c 6 ● a 4 b 2 power of a power = a 7 b 5 c 6 product of powers

19. Try again … Remember the rules: 1. 1. To multiply two powers that have the same base, add the exponents together. 2. 2. To find the power of a power, you multiply exponents. 3. 3. To find the product of a power, find the power of each factor and multiply.

20. quiz question three You are offered a job that pays 2x dollars or 2 x dollars for x hours of work. Assuming you must work at least 2 hours, which method of payment would you choose? a. 2x dollars b. 2 x dollars

21. Great job! The correct answer is b. If you work more than 2 hours, the pay is much better at the rate of 2 x dollars per hour. For example, at 2 x dollars per hour, you would earn $256 for 8 hours: 2 8 = 2 ● 2 ● 2 ● 2 ● 2 ● 2 ● 2 ● 2 = 256 while at 2x dollars per hour, you would earn $16: 2 ● 8 = 16

22. Try again … Remember the rules: an expression like 4 6 is called a power. The exponent 6 represents the number of times the base 4 is used as a factor: exponent base 4 6 = 4 ● 4 ● 4 ● 4 ● 4 ● 4 power 6 factors of 4

23. quiz question four Evaluate the exponential expression: 3 -2 ● 3 2 d. 1 c. 3 a. 0 b. 6

24. Great job! The correct answer is d: 3 -2 ● 3 2 = 3 -2 + 2 product of powers = 3 0 add exponents = 1 a 0 is 1

25. Try again … Remember the rules: 1. 1. To multiply two powers that have the same base, add the exponents together. 2. 2. A nonzero number to the zero power is 1: a 0 = 1, a = 0.

26. quiz question five Evaluate the exponential expression: (5a) -2 d. a 2 /5 c. 25a/2 a. 5a/2 b. 1/25a 2

27. Great job! The correct answer is b: (5a) -2 = 5 -2 ● a -2 power of a product = 1/5 2 ● 1/a 2 reciprocals of 5 2 and a 2 = 1/25a 2 multiply fractions

28. Try again … Remember the rules: 1. 1. To find the product of a power, find the power of each factor and multiply. 2. 2. a -n is the reciprocal of a n : a -n = 1/a n, a = 0

29. quiz question six Evaluate the exponential expression: 9 4 ● 9 2 /9 7 d. 9 6/7 c. 1/9 a. 9 b. 1

30. Great job! The correct answer is c: 9 4 ● 9 2 /9 7 = 9 4+2 /9 7 = 9 6 /9 7 product of powers = 9 -1 quotient of powers = 1/9

31. Try again … Remember the rules: 1. 1. To multiply two powers that have the same base, add the exponents together. 2. 2. To divide powers having the same base, subtract exponents.

32. quiz question seven Evaluate the exponential expression: (7/4) -3 d. 343/64 c. 64/343 a. -7/4 b. 4/7

33. Great job! The correct answer is c: (7/4) -3 = 7 -3 /4 -3 power of a quotient = 4 3 /7 3 definition of negative exponents = 64/343

34. Try again … Remember the rules: To find a power of a quotient, find the power of the numerator and the power of the denominator and divide.

35. quiz question eight: put it all together! Evaluate the exponential expression: 2x 2 y/3x ● 9xy 2 /y 4 d. 6x/y 2 c. x 2 /6y a. 6x 2 /y b. 18x 2 /y

36. Great job! The correct answer is a: 2x 2 y/3x ● 9xy 2 /y 4 = (2x 2 y)(9xy 2 )/(3x)(y 4 ) = 18x 3 y 3 /3xy 4 product of powers = 6x 2 y -1 quotient of powers = 6x 2 /y definition of negative exponents

37. Try again … Remember the rules: 1. 1. to multiply two powers that have the same base add the exponents together 2. 2. to divide powers having the same base, subtract exponents 3. 3. a -n is the reciprocal of a n : a -n = 1/a n, a = 0

38. Summary Now you know how to multiply and divide expressions with exponents, including zero and negative exponents.

39. resources Algebra I Teacher's Edition. Larson, R., Boswell, L., Kanold, T.D. and Stiff, L. 2004, Evanston, McDougal Littell. all images under license from Fotolia.com Fotolia.com

40. exit good job! You have completed the tutorial. Please click on the exit button to exit out of the program.