1. 3535 18 0 9 3

2. Location of Exponent An exponent is the small number high and to the right of a regular or base number. 3 4 Base Exponent

3. What an Exponent Represents An exponent tells how many times a number is multiplied by itself. 3 4 = 3 x 3 x 3 x 3

4. How to read an Exponent This exponent is read three to the fourth power. 3 4 Base Exponent

5. How to read an Exponent This exponent is read three to the 2 nd power or three squared. 3 2 Base Exponent

6. How to read an Exponent This exponent is read three to the 3rd power or three cubed. 3 3 Base Exponent

7. Read These Exponents 3232 2323 6565 7474

8. What is the Exponent? 2 x 2 x 2 =2 3

9. What is the Exponent? 3 x 3 =3 2

10. What is the Exponent? 5 x 5 x 5 x 5 = 5 4

11. What is the Base and the Exponent? 8 x 8 x 8 x 8 =8 4

12. What is the Base and the Exponent? 7 x 7 x 7 x 7 x 7 = 7 5

13. What is the Base and the Exponent? 9 x 9 =9 2

14. How to Multiply Out an Exponent to Find the Standard Form = 3 x 3 x 3 x 33 9 27 81 4

15. How to Multiply Out an Exponent to Find the Standard Form = 3 x 3 x 3 x 33 9 4 9 81

16. What is the Standard Form? 4242 = 16

17. What is the Standard Form? 2323 = 8

18. 5353 = 125

19. Exponents Are Often Used in Area Problems to Show the Feet Are Squared Length x width = area 15ft. 30ft A pool is a rectangle. Length = 30 ft. Width = 15 ft. Area = 30 x 15 = 450 ft 2.

20. Exponents Are Often Used in Volume Problems to Show the Centimeters Are Cubed length x width x height = volume 10 20 A box is a rectangle. Length = 10 cm Width = 10 cm Height = 20 cm Volume = 10 x 10 x 20 = 2,000 cm 3

21. Here Are Some Areas Change Them to Exponents 40 feet squared = 40 ft 56 sq. inches = 56 in 38 m. squared = 38 m 56 sq. cm = 56 cm 2 2 2 2

22. Here Are Some Volumes Change Them to Exponents 30 feet cubed = 30 ft 26 cu. inches = 26 in 44 m. cubed = 44 m 56 cu. cm. = 56 cm 3 3 3 3

23. 2.3 x 10 5

24. 210,000,000,000,000,000,000,000 miles (22 zeros) This number is written in decimal notation. When numbers get this large, it is easier to write them in scientific notation. How wide is our universe?

25. Scientific Notation A number is expressed in scientific notation when it is in the form: a x 10 n where a is between 1 and 10 and n is an integer

26. Write the width of the universe in scientific notation. 210,000,000,000,000,000,000,000 miles Where is the decimal point now? Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1 After the last zero.

27. 2.10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 10 23

28. Express 0.0000000902 in scientific notation. Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative. 9.02 x 10 -8

29. Write 28750.9 in scientific notation. 1.2.87509 x 10 -5 2.2.87509 x 10 -4 3.2.87509 x 10 4 4.2.87509 x 10 5

30. Express 1.8 x 10 -4 in decimal notation. 0.00018 Express 4.58 x 10 6 in decimal notation. 4,580,000

31. Write in PROPER scientific notation. 1.Move the decimal point so that the number is between 1 and 10. 2. To do this the decimal point had to be moved 2 places to the left. 3 This means you would add 2 to the exponent 9 with a result of 11. 2.346 x 10 11 234.6 x 10 9 Notice the number is not between 1 and 10

32. Write 531.42 x 10 5 in scientific notation. 1..53142 x 10 2 2.5.3142 x 10 3 3.53.142 x 10 4 4.531.42 x 10 5 5.53.142 x 10 6 6.5.3142 x 10 7 7..53142 x 10 8

33. Write the following in scientific notation. 1. 100,2300,00 2. 48.9 x 10 6 3..000401 4. 701.4 x 10 -8

35. Basic Terminology Exponent – In exponential notation, the number of times the base is used as a factor. Base – In exponential notation, the number or variable that undergoes repeated MULTIPLICATION. BASE EXPONENT 2 2 = 2 X 2 X 2 X 2

36. IMPORTANT EXAMPLES -3 4 means –(3 x 3 x 3 x 3) = -81 (-3) 4 means (–3) x (–3) x (–3) x (–3) = 81 -3 3 means –(3 x 3 x 3) = -27 (-3) 3 means (–3) x (–3) x (–3) = -27

37. Try these: – 2 4 – 5 2 (– 5) 2

38. Variable Expressions x 5 = (x)(x)(x)(x)(x) y 3 = (y)(y)(y)

39. Substitution and Evaluating STEPS 1.Write out the original problem. 2.Show the substitution with parentheses. 3.Work out the problem. Solve x 3 if x = 4 4 3 = 64 x3x3

40. Evaluate the variable expression when x = 1, y = 2, and w = -3 Step 1 Step 2 Step 3

41. Evaluate the variable expression when x = 1, y = 2, and w = -3 Step 1 Step 2 Step 3

42. Evaluate the variable expression when x = 1, y = 2, and w = -3 Step 1 Step 2 Step 3

43. MULTIPLICATION PROPERTIES PRODUCT OF POWERS This property is used to combine 2 or more exponential expressions with the SAME base. 2 3 x 2 5 = (2 x 2 x 2) x (2 x 2 x 2 x 2 x 2) = 2 8 = 256 a 3 x a 4 = (a x a x a) x (a x a x a x a) = a 7 d 3 d 6 = (d x d x d) x (d x d x d x d x d x d) = a 3+6 = a 9

44. MULTIPLICATION PROPERTIES POWER TO A POWER This property is used to write and exponential expression as a single power of the base. (5 2 ) 3 =(5 2 )(5 2 )(5 2 ) =5 2+2+2 =5656 (b 2 ) 4 =(b 2 )(b 2 )(b 2 )(b 2 ) =b 2+2+2+2 =b (2x4) (4 3 ) 5 =4 3x5 = 4 15 =b8=b8

45. MULTIPLICATION PROPERTIES POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions. (-6 x 5) 2 = (-6) 2 x (5) 2 = 36 x 25 = 900 900

46. MULTIPLICATION PROPERTIES POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions. (5xy) 3 = (5 3 )(x 3 )(y 3 )= 125x 3 y 3

47. MULTIPLICATION PROPERTIES POWER OF PRODUCT (4a 2 ) 3 x a 5 = (4 3 )(a 2 ) 3 x a 5 = (64)(a 2x3) x a 5 = (64)(a 6 ) x a 5 = (64)(a 6+5 ) 64a 11

48. Simplify each expression. 1. (2xy) 3 x 2 2. (2x 3 y) 4 x 4 y 5 3. (3x 2 y 3 ) 2 (4x 2 ) 2 4. (2x) 2 (3x+y) 2

49. MULTIPLICATION PROPERTIES SUMMARY PRODUCT OF POWERS add the exponents y a x y b = y a+b POWER TO A POWER multiply the exponents (m a ) b = m ab POWER OF PRODUCT distribute the exponents (cd) a = c a d a

50. ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1. 2 4 = 2 x 2 x 2 x 2 = 16 2 3 = 2 x 2 x 2 = 8 2 2 = 2 x 2 = 4 2 1 = 2 2 0 = 1 16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 =2 2 ÷ 2 = 1 Dividing by the base number

51. ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1. 3 3 = 27 3 2 = 9 3 1 = 3 16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 =2 2 ÷ 2 = 1 Dividing by the base number 3 0 = 1 3 -1 = 3 -2 = 3 -3 =

52. ZERO AND NEGATIVE EXPONENTS

54. Simplify each expression.

55. DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base.

56. DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base.

57. DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base.

58. DIVISION PROPERTIES QUOTIENT OF POWERS

59. REVIEW ZERO, NEGATIVE, AND DIVISION PROPERTIES Zero power Quotient of powers Power of a quotient Negative Exponents