1. Lesson 8.3 Define and Use Zero and Negative Exponents After today’s lesson, you should be able to use zero and negative to simplify expressions. (CA 2.0)

3. PowerValue 1 1 1 1 1 1 1 1

4. Zero Exponent

6. Definition: Zero exponent

7. Examples: Simplify.

10. PowerValue

11. Negative Exponents Using Patterns to Discover the Meaning Behind Negative Exponents

12. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,000 10,000 1000 100 10 1

13. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙10 10,000 1000 100 10 1

14. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,000 1000 100 10 1

15. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙10 1000 100 10 1

16. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 1000 100 10 1

17. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙10 100 10 1

18. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 100 10 1

19. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙10 10 1

20. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 1

21. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 1

22. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 1

23. Complete the following table. Use only a base of 10. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0

24. Now let’s continue the pattern. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 1 100 1 1000 1 10,000

25. Now let’s continue the pattern. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 1 100 1 1000 1 10,000

26. Now let’s continue the pattern. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 1000 1 10,000

27. Now let’s continue the pattern. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 10∙10 1 1000 1 10,000

28. Now let’s continue the pattern. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 10∙10 10 -2 1 1000 1 10,000

29. Now let’s continue the pattern. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 10∙10 10 -2 1 1000 1 10∙10∙10 1 10,000

30. Now let’s continue the pattern. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 10∙10 10 -2 1 1000 1 10∙10∙10 10 -3 1 10,000

31. Now let’s continue the pattern. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 10∙10 10 -2 1 1000 1 10∙10∙10 10 -3 1 10,000 1 10∙10∙10∙10

32. Now let’s continue the pattern. Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 10∙10 10 -2 1 1000 1 10∙10∙10 10 -3 1 10,000 1 10∙10∙10∙10 10 -4

33. What should you learn from this? Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 10∙10 10 -2 1 1000 1 10∙10∙10 10 -3 1 10,000 1 10∙10∙10∙10 10 -4

34. What should you learn from this? Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 10∙10 10 -2 1 1000 1 10∙10∙10 10 -3 1 10,000 1 10∙10∙10∙10 10 -4 1) Any number raised to the zero power is always 1.

35. What should you learn from this? Stand. FormExpanded FormExpon. Form 100,00010∙10∙10∙10∙1010 5 10,00010∙10∙10∙1010 4 100010∙10∙1010 3 10010∙1010 2 10 10 1 110 0 1 10 10 -1 1 100 1 10∙10 10 -2 1 1000 1 10∙10∙10 10 -3 1 10,000 1 10∙10∙10∙10 10 -4 1) Any number raised to the zero power is always 1. 2) A negative exponent will create a fraction.

36. A negative exponent will result in the reciprocal of the base with a positive exponent.

37. 5 -2 =

38. A negative exponent will result in the reciprocal of the base with a positive exponent. 5 -2 = 1515 ( ) 2

39. A negative exponent will result in the reciprocal of the base with a positive exponent. 5 -2 = 1515 ( ) 2 = 1515 ∙ 1515 = 1 25

40. A negative exponent will result in the reciprocal of the base with a positive exponent. 4 -3 =

41. A negative exponent will result in the reciprocal of the base with a positive exponent. 4 -3 = 1414 ( ) 3

42. A negative exponent will result in the reciprocal of the base with a positive exponent. 4 -3 = 1414 ( ) 3 = 1414 ∙ 1414 = 1 64 ∙ 1414

43. A negative exponent will result in the reciprocal of the base with a positive exponent. 3 -4 =

44. A negative exponent will result in the reciprocal of the base with a positive exponent. 3 -4 = 1313 ( ) 4

45. A negative exponent will result in the reciprocal of the base with a positive exponent. 3 -4 = 1313 ( ) 4 = 1313 ∙ 1313 = 1 81 ∙ 1313 ∙ 1313

46. A negative exponent will result in the reciprocal of the base with a positive exponent. (-6) -3 =

47. ( ) A negative exponent will result in the reciprocal of the base with a positive exponent. = 1 -6 3 (-6) -3

48. ( ) A negative exponent will result in the reciprocal of the base with a positive exponent. = 1 -6 3 = ∙ = 1 216 ∙ 1 -6 - (-6) -3

49. Practice Time

50. Evaluate. 1)2 -3 =2)4 -4 = 3)(-2) -4 =4)(-5) -3 = 5)8 0 = 6)(-4) 0 =

51. Evaluate. 1)2 -3 =2)4 -4 = 3)(-2) -4 =4)(-5) -3 = 5)8 0 = 6)(-4) 0 = 1818

52. Evaluate. 1)2 -3 =2)4 -4 = 3)(-2) -4 =4)(-5) -3 = 5)8 0 = 6)(-4) 0 = 1818 1 256

53. Evaluate. 1)2 -3 =2)4 -4 = 3)(-2) -4 =4)(-5) -3 = 5)8 0 = 6)(-4) 0 = 1818 1 256 1 16

54. Evaluate. 1)2 -3 =2)4 -4 = 3)(-2) -4 =4)(-5) -3 = 5)8 0 = 6)(-4) 0 = 1818 1 256 1 16 1 125 -

55. Evaluate. 1)2 -3 =2)4 -4 = 3)(-2) -4 =4)(-5) -3 = 5)8 0 = 6)(-4) 0 = 1818 1 256 1 16 1 125 - 1

56. Evaluate. 1)2 -3 =2)4 -4 = 3)(-2) -4 =4)(-5) -3 = 5)8 0 = 6)(-4) 0 = 1818 1 256 1 16 1 125 - 11

57. Evaluate. Remember the order of operations. 1) 4 -2 + 2 -3 = 2) 5 0 – (6) -2 = 3)4(3 + 5) 0 + 4 2 = 4) (3 + 5) -2 =

58. Evaluate. Remember the order of operations. 1) 4 -2 + 2 -3 = 2) 5 0 – (6) -2 = 3)4(3 + 5) 0 + 4 2 = 4) (3 + 5) -2 = 3 16

59. Evaluate. Remember the order of operations. 1) 4 -2 + 2 -3 = 2) 5 0 – (6) -2 = 3)4(3 + 5) 0 + 4 2 = 4) (3 + 5) -2 = 3 16 35 36

60. Evaluate. Remember the order of operations. 1) 4 -2 + 2 -3 = 2) 5 0 – (6) -2 = 3)4(3 + 5) 0 + 4 2 = 20 4) (3 + 5) -2 = 3 16 35 36

61. Evaluate. Remember the order of operations. 1) 4 -2 + 2 -3 = 2) 5 0 – (6) -2 = 3)4(3 + 5) 0 + 4 2 = 20 4) (3 + 5) -2 = 3 16 1 64 35 36

62. Evaluate. Remember the order of operations. 5) 2 -1 + (-2) -2 = 6) 5 2 – 3 0 = 7)2(1 + 3) -2 = 8) (6 + 3) 2 + (6 + 3) -2 =

63. Evaluate. Remember the order of operations. 5) 2 -1 + (-2) -2 = 6) 5 2 – 3 0 = 7)2(1 + 3) -2 = 8) (6 + 3) 2 + (6 + 3) -2 = 3434

64. Evaluate. Remember the order of operations. 5) 2 -1 + (-2) -2 = 6) 5 2 – 3 0 = 7)2(1 + 3) -2 = 8) (6 + 3) 2 + (6 + 3) -2 = 3434 24

65. Evaluate. Remember the order of operations. 5) 2 -1 + (-2) -2 = 6) 5 2 – 3 0 = 7)2(1 + 3) -2 = 8) (6 + 3) 2 + (6 + 3) -2 = 3434 24 1818

66. Evaluate. Remember the order of operations. 5) 2 -1 + (-2) -2 = 6) 5 2 – 3 0 = 7)2(1 + 3) -2 = 8) (6 + 3) 2 + (6 + 3) -2 = 81 3434 24 1818 1 81

67. Definition: Negative exponent

68. Examples: Evaluate the expression.

76. Examples: Simplify the expression. Write your answer using only positive exponents.