1. MATH 31 LESSONS PreCalculus 2. Powers

2. A. Power Laws Terminology: b x

3. bx bx base

4. b x exponent

5. bx bx the base and exponent together form a power

6. Power Laws 1.x a  x b = ?

7. x a  x b = x a + b If the bases are the same... Keep the base the same and add the exponents

8. x a  x b = x a + b Similarly, x a ÷ x b = x a  b

9. e.g. Simplify

12. (x a ) b = x a  b = x a b Multiply the exponents

13. (x a ) b = x a  b = x a b Similarly, (x y) n = x n y n

14. Note: This works only for multiplication and division. It does NOT work for addition or subtraction. (a + b 2 ) 3 ≠ a 3 + b 6 (x - y) 5 ≠ x 5 - y 5

15. e.g. Simplify

18. Power Laws 3.x 0 = ?x  1 = ?

19. x 0 = 1x  1 =

20. x 0 = 1x  1 = Similarly, If you move the power from the top to the bottom (or bottom to the top), it gets the opposite exponent

21. e.g. Simplify Express your answer with only positive exponents.

23. Power Laws 4. = ?

24. Remember, the root is always the one on the bottom of the fraction.

25. e.g. Evaluate

26. The root is on the bottom of the fraction

27. =5 3 =125

28. Ex. 1Simplify Answer with positive exponents. Try this example on your own first. Then, check out the solution.

32. Ex. 2Simplify Try this example on your own first. Then, check out the solution.

37. B. Factoring Power Expressions Method:  Convert all variables to exponential notation - bring all powers to the numerator  Convert all fractions to LCD  Factor out the smallest power - remove the factor by dividing (subtracting the exponents) - this should leave the exponents positive

38. Ex. 3Factorcompletely Try this example on your own first. Then, check out the solution.

39. Convert to exponential notation

40. Factor out the smallest power When you divide, you subtract exponents

41. Don’t stop here. What else can you do?

43. Ex. 4Factorcompletely Try this example on your own first. Then, check out the solution.

44. Convert to exponential notation

45. Factor out the smallest power

46. Don’t stop here. What else can you do?

48. Ex. 5Factorcompletely Try this example on your own first. Then, check out the solution.

49. Convert to exponential notation Bring all powers up to the top

50. Factor out the smallest power. Notice that to subtract  5 from the exponents, you add +5

51. Don’t stop here. What else can you do?

53. Ex. 6Factorcompletely Try this example on your own first. Then, check out the solution.

54. Convert to exponential notation Bring all powers up to the top

55. Factor out the smallest power Notice that to subtract  4 from the exponents, you add +4

56. Don’t stop here. What else can you do?

57. This is a difference of cubes

58. A 3 - B 3 = (A - B) (A 2 + AB + B 2 )

60. Ex. 7Factorcompletely Try this example on your own first. Then, check out the solution.

61. Convert to exponential notation

62. Convert the coefficients to the LCD

63. Factor out the common coefficients and the lowest power

64. Don’t stop here. What else can you do?

66. Ex. 8Factorcompletely Try this example on your own first. Then, check out the solution.

67. Convert to exponential notation Bring all powers to the top

68. Convert coefficients to LCD

69. Factor out the common coefficients and the lowest power

70. Don’t stop here. What else can you do?

72. Ex. 9Factorcompletely Try this example on your own first. Then, check out the solution.

73. Let A = x 2 + 5 Then, Use substitution to remove the common binomial from the expression. Makes it simpler.

74. Let A = x 2 + 5 Then,

75. Let A = x 2 + 5 Then, Factor out the lowest power

76. Let A = x 2 + 5 Then,

77. Since A = x 2 + 5 Then, Now, back substitute to return the expression to its original variable.

78. Since A = x 2 + 5 Then, Don’t forget to use brackets

79. Since A = x 2 + 5 Then, Don’t stop here. What else can you do?

80. Since A = x 2 + 5 Then,

81. Ex. 10Factorcompletely Try this example on your own first. Then, check out the solution.

82. Let A = x + 7 and B = 2x - 1 Then, Use substitution to remove the common binomials from the expression. Makes it simpler.

83. Let A = x + 7 and B = 2x - 1 Then,

84. Let A = x + 7 and B = 2x - 1 Then,

85. Since A = x + 7 and B = 2x - 1 Then, Now, back substitute to return the expression to its original variables.

86. Since A = x + 7 and B = 2x - 1 Then, Don’t forget to use brackets

87. Since A = x + 7 and B = 2x - 1 Then, Simplify inside the bracket

88. Since A = x + 7 and B = 2x - 1 Then,