1. Exponents 07/24/12lntaylor ©

2. Table of Contents Learning Objectives Bases Exponents Adding Bases with exponents Subtracting Bases with exponents Multiplying Bases with exponents Dividing Bases with exponents Exponent of an exponent Negative exponents Fractional exponents Explanation of a 0 exponent 07/24/12lntaylor ©

3. LO1: LO2: Define and locate bases and exponents Recognize bases and exponents which can be combined LO3:Add and subtract bases and exponents LO4:Multiply and divide bases and exponents 07/24/12lntaylor © TOC Learning Objectives LO5:Evaluate expressions with bases and exponents

4. Def1: Def2: Base – the number or variable whose exponents are expressed (4x 3 where x is the base; 4 is the coefficient and 3 is the exponent) Exponent – the number or variable to the upper right of the base which designates how many times the base is multiplied by itself (x 3 = x*x*x; 3x 4 = 3*x*x*x*x) Def3:Coefficient – the number or letter in front of the base (3x 4 where 3 is the coefficient) Def4:Any base with an exponent of 0 always = 1! (x 0 = 1; 12 0 = 1) 07/24/12lntaylor © TOC Definitions

5. Bases 07/24/12lntaylor © TOC

6. 3 Step 1 Note: Look for the base It can be a number Note:It can be a variable (letter) Note:It can be a combination of variables (letters) 07/24/12lntaylor © TOC 2 x 2 x 23 y 3

7. Now you try 10 2 – 6x + mn 07/24/12lntaylor © TOC

8. 10 Step 1 Note: x Look for the base It can be a number Note:It can be a variable (letter) Note:It can be a combination of variables (letters) 07/24/12lntaylor © TOC 2 + mn– 6

9. Now you try x 2 – 7xy + 18 07/24/12lntaylor © TOC

10. x Step 1 Note: xy Look for the base It can be a number Note:It can be a variable (letter) Note:It can be a combination of variables (letters) 07/24/12lntaylor © TOC 2 + 18– 7

11. Exponents 07/24/12lntaylor © TOC

12. 3 Step 1 Note: Look for the exponent It can be a number Note:It can be a variable (letter) Note:There can be more than one exponent 07/24/12lntaylor © TOC 2 x y 3 x 2m y Note:Everything gets an exponent!!!!! 1

13. Now you try x 2 + 10x n + 18y 07/24/12lntaylor © TOC

14. x Step 1 Note: Look for the exponent It can be a number Note:It can be a variable (letter) Note:There can be more than one exponent 07/24/12lntaylor © TOC 2 + 10 x n + 18 y Note:Everything gets an exponent!!!!! 111

15. Now you try 12x 2 + 14y m – 10 07/24/12lntaylor © TOC

16. 12 x Step 1 Note: Look for the exponent It can be a number Note:It can be a variable (letter) Note:There can be more than one exponent 07/24/12lntaylor © TOC 2 + 14 y- 10 Note:Everything gets an exponent!!!!! 11 1 m

17. Adding Bases with exponents 07/24/12lntaylor © TOC

18. x Step 1 Step 2 Look at each variable and its exponent Combine the variables and exponents that are exactly the same Step 3Rewrite the expression 07/24/12lntaylor © TOC 2 x- 10x 2 12+ 2 14x 2

19. Now you try 10x 2 + 12x 2 – x 07/24/12lntaylor © TOC

20. x Look at each variable and its exponent Combine the variables and exponents that are exactly the same Rewrite the expression 07/24/12lntaylor © TOC 2 x- x 2 10+ 12 22x 2 Step 1 Step 2 Step 3

21. Now you try 10xy 2 + 12xy 2 – x 2 07/24/12lntaylor © TOC

22. xy Step 1 Step 2 Look at each variable and its exponent Combine the variables and exponents that are exactly the same Step 3Rewrite the expression 07/24/12lntaylor © TOC 2 xy- x 2 2 10+ 12 22xy 2

23. Subtracting Bases with exponents 07/24/12lntaylor © TOC

24. x Step 1 Step 2 Look at each variable and its exponent Combine the variables and exponents that are exactly the same Step 3Rewrite the expression 07/24/12lntaylor © TOC 2 x- 10x 2 12- 2 10x 2

25. Now you try 10x 2 - 12x 2 – x 07/24/12lntaylor © TOC

26. x Step 1 Step 2 Look at each variable and its exponent Combine the variables and exponents that are exactly the same Step 3Rewrite the expression 07/24/12lntaylor © TOC 2 x- x 2 10- 12 - 2x 2

27. Now you try 12xy 2 – 12xy 2 – x 2 07/24/12lntaylor © TOC

28. xy Step 1 Step 2 Look at each variable and its exponent Combine the variables and exponents that are exactly the same Step 3Rewrite the expression 07/24/12lntaylor © TOC 2 xy- x 2 2 12- 12 0

29. Multiplying Bases with exponents 07/24/12lntaylor © TOC

30. x Step 1: Step 2: Multiply the coefficients (here all the coefficients are = 1) If the bases are identical, write it down If the bases are different, write them down Step 3:Add the exponents of similar bases 07/24/12lntaylor © TOC 2 (x ) 3 = x x 2 (xy ) 3 = x y 53 3 1

31. Now you try 2x 2 (6x 7 ) 07/24/12lntaylor © TOC

32. x Step 1: Step 2: Multiply the coefficients If the bases are identical, write it down If the bases are different, write them down Step 3:Add the exponents of similar bases 07/24/12lntaylor © TOC 2 ( x ) 7 x 9 2 6 12 =

33. Now you try 12xy 2 (6x 7 y 2 z) 07/24/12lntaylor © TOC

34. xy Step 1: Step 2: Multiply the coefficients If the base is identical, write it down If the bases are different, write them down Step 3:Add the exponents of similar bases 07/24/12lntaylor © TOC 2 ( x ) 7 x y z 8 12 6 72= y z 241

35. Now you try ¾xy 2 (4x 5 y 4 z) 07/24/12lntaylor © TOC

36. xy Step 1: Step 2: Multiply the coefficients If the bases are identical, write it down If the bases are different, write them down Step 3:Add the exponents of similar bases 07/24/12lntaylor © TOC 2 ( x ) 5 x y z 6 ¾ 4 3= y z 461

37. Dividing Bases with exponents 07/24/12lntaylor © TOC

38. x Step 1: Step 2: Divide the coefficients Determine where the bases go (numerator or denominator) by looking for the largest exponents Step 3:Subtract the exponents of similar bases 07/24/12lntaylor © TOC 7 x 2 x 5 12 6 2 = Note:Exponent answers are always positive Determine if the exponent is in the numerator or denominator

39. Now you try xy 2 4x 5 y 4 z 07/24/12lntaylor © TOC

40. xy Step 1: Step 2: Divide the coefficients Determine where the bases go (numerator or denominator) by looking for the largest exponents Step 3:Subtract the exponents of similar bases 07/24/12lntaylor © TOC 2 x 5 x y z 4 4 1 = y z 42 1 Note:Final exponent answers are always positive Double check your work 4

41. Now you try 6x 9 y 2 z 6 12x 7 y 12 z 6 07/24/12lntaylor © TOC

42. x y Step 1: Step 2: Divide the coefficients Determine where the bases go (numerator or denominator) by looking for the largest exponents Note here that z 6 / z 6 = 1 so there is no need to write z Step 3:Subtract the exponents of similar bases 07/24/12lntaylor © TOC 2 x 7 xyxy 2 12 1 = y z 1210 9 Note:Final exponent answers are always positive Double check your work and rewrite if necessary 2 6z 6 6 = x 2 2y 10

43. Now you try 9x 7 y 2 z 6 12x 7 y 2 z 6 07/24/12lntaylor © TOC

44. x y Step 1: Step 2: Divide the coefficients Determine where the bases go (numerator or denominator) by looking for the largest exponents Step 3:Subtract the exponents of similar bases 07/24/12lntaylor © TOC 2 x 7 12 3 = y z 2 7 Note:Final exponent answers are always positive Double check your work and rewrite if necessary 4 9z 6 6

45. Exponent of an exponent 07/24/12lntaylor © TOC

46. Step 1: Step 2: Look for the exponent next to a ( ); (2x) 5 Everything gets an exponent (no need here) Step 3:Rewrite all bases and multiply the exponents 07/24/12lntaylor © TOC (x ) 7 x 14 = Step 4:Simplify if necessary 2

47. Now you try ( 2x 7 y 2 z ) 3 07/24/12lntaylor © TOC

48. Step 1: Step 2: Look for the exponent next to a ( ) Everything gets an exponent Step 3:Rewrite all bases and multiply the exponents Be patient and let the computer work!!! 07/24/12lntaylor © TOC ( ) 7 2 3(1) = Step 4:Simplify 2 2xy z 3 3(7) 3(2)3(1) 1 1 xyz = 2 3 x 21 y 6 z 3 = 8x 21 y 6 z 3

49. Now you try ( 2x 4 y 5 z ) 7 07/24/12lntaylor © TOC

50. Step 1: Step 2: Look for the exponent next to a ( ) Everything gets an exponent Step 3:Rewrite all bases and multiply the exponents Be patient and let the computer work!!! 07/24/12lntaylor © TOC ( ) 4 2 7(1) = Step 4:Simplify 5 2xy z 7 7(4) 7(5)7(1) 1 1 xyz = 2 7 x 28 y 35 z 7 = 128x 28 y 35 z 7

51. Negative exponents 07/24/12lntaylor © TOC

52. Step 1: Step 2: Look for negative exponents Everything gets an exponent (no need here) Step 3:Remember that everything is a fraction Flip over only the base with the negative exponent Remove the negative ( - ) sign 07/24/12lntaylor © TOC x - 7 1 7 = Step 4:Simplify if necessary 1 x

53. Now you try 2x 4 y 5 z -7 07/24/12lntaylor © TOC

54. Step 1: Step 2: Look for negative exponents Everything gets an exponent Step 3:Remember that everything is a fraction Flip over only the base with the negative exponent Make sure you removed the negative ( - ) sign 07/24/12lntaylor © TOC 2 x 4 y 5 z - 7 2 1 x 4 y 5 7 = Step 4:Simplify if necessary 1z 1

55. Now you try 2x 4 y 5 z -7 x -8 yz 07/24/12lntaylor © TOC

56. Step 1: Step 2: Look for negative exponents Everything gets an exponent Step 3:Since you are dividing you can subtract the negative exponent (x 4 / x -8 is x 4- - 8 or x 12 ) Make sure you removed the negative ( - ) sign 07/24/12lntaylor © TOC 2 x 4 y 5 z - 7 2 1 x 12 y 4 8 = Step 4:Simplify if necessary x -8 y z z 1 1 1 2x 12 y 4 8 = z

57. Fractional Exponents 07/24/12lntaylor © TOC

58. 4 Step 1 Note: Look for the exponent It can be sometimes be a fraction (rational number) Step 2:The base and numerator go under the radical (square root symbol) Step 3:The denominator goes outside the radical 07/24/12lntaylor © TOC 1 2 √ 4 1 2 Note:An exponent of ½ means square root; 1/3 means cube root = 2

59. Now you try 8 1/3 07/24/12lntaylor © TOC

60. 8 Step 1 Note: Look for the exponent It can be sometimes be a fraction (rational number) Step 2:The base and numerator go under the radical (square root symbol) Step 3:The denominator goes outside the radical 07/24/12lntaylor © TOC 1/3 √ 8 1 3 Note:An exponent of ½ means square root; 1/3 means cube root = 2

61. Now you try 8 2/3 07/24/12lntaylor © TOC

62. 8 Step 1 Note: Look for the exponent It can be sometimes be a fraction (rational number) Step 2:The base and numerator go under the radical (square root symbol) Step 3:The denominator goes outside the radical 07/24/12lntaylor © TOC 2/3 √ 8 2 3 Note:An exponent of ½ means square root; 1/3 means cube root = √64 3 = 4

63. Exponent = 0 07/24/12lntaylor © TOC

64. 3 = 1 Step 1 Note: Proof of 3 0 = 1 Use 3 as the base and any exponent you wish – we will use 8 Divide identical bases and exponents Note:Anything divided by itself is 1 (the exception is of course base = 0) Note:Dividing identical bases and exponents requires subtraction 07/24/12lntaylor © TOC 0 38383838 = 3 8-8 = 1= 3 0 Note:This leaves an exponent of 0 The only way you get an exponent of 0 is to divide something by itself Therefore proving an exponent of 0 = 1 = 1

65. Now you try x 0 = 1 07/24/12lntaylor © TOC

66. x = 1 Step 1 Note: Proof of x 0 = 1 Use x as the base and any exponent you wish – we will use 10 Divide identical bases and exponents Note:Anything divided by itself is 1 (the exception is of course base = 0) Note:Dividing identical bases and exponents requires subtraction 07/24/12lntaylor © TOC 0 x 10 = x 10-10 = 1= x 0 Note:This leaves an exponent of 0 The only way you get an exponent of 0 is to divide something by itself Therefore proving an exponent of 0 = 1 = 1

67. Now you try 7x 0 y 0 = 7 07/24/12lntaylor © TOC

68. Step 1 Note: Proof of 7x 0 y 0 = 7 07/24/12lntaylor © TOC x 00 y 7= 7 7 (1)(1) = 7 Note:7 has an exponent of 1 1 Anything with a 0 exponent = 1 (except base 0 of course) Note:This expression = 7