College Algebra

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  1. Real numbers
    Collection of all rational and irrational numbers, both positive and negative, including zero.
  2. Collection of all rational and irrational numbers, both positive and negative, including zero.
    Real numbers
  3. Integers
    Counting numbers 1,2,3....together with 0 and the negative numbers -1,-2,-3....
  4. Counting numbers 1,2,3....together with zero and the negative numbers -1,-2,-3....
    Integers
  5. Natural numbers
    • All positive integers.
    • Also called counting numbers.
  6. All positive integers.
    Natural numbers
  7. Counting numbers
    Natural numbers
  8. Two consecutive negative signs
    Double negative
  9. Double negative
    Two consecutive negative signs
  10. Distributive law of multiplication
    a(b + c) = ab + ac
  11. a(b + c) = ab + ac is the _.
    distributive law of multiplication.
  12. Rational numbers
    Can be expressed as a ratio of integers or as a fraction.
  13. Can be expressed as a ratio of integers or as a fraction
    Rational numbers
  14. Multiplying fractions
    Image Upload 1
  15. A fraction Image Upload 2 is reduced if _.
    the numbers a and b have no common factors.
  16. A fraction Image Upload 3  is _ the numbers a and b have no common factors.
    reduced
  17. Another way to say the fraction Image Upload 4 is reduced is to say _.
    that it is in lowest terms.
  18. Image Upload 5
    Image Upload 6
  19. Image Upload 7
    Image Upload 8
  20. Image Upload 9
    Image Upload 10
  21. Image Upload 11
    Image Upload 12
  22. Image Upload 13
    Image Upload 14
  23. Image Upload 15
    Image Upload 16
  24. General rule for adding fractions:
    Image Upload 17
  25. Image Upload 18
    Image Upload 19
  26. Image Upload 20
    Image Upload 21 OR

    Image Upload 22
  27. General form for subtracting fractions
    Image Upload 23
  28. Subtract Image Upload 24 from Image Upload 25
    Image Upload 26
  29. Reciprocal of a number x when multiplied by x yeilds _.
    1
  30. Reciprocal of any integer n (except 0) is _.
    Image Upload 27
  31. Reciprocal of Image Upload 28 is _.
    n
  32. Find the reciprocal of Image Upload 29
    Image Upload 30
  33. Image Upload 31 
    Image Upload 32
  34. General rule for division
    Image Upload 33
  35. Irrational numbers
    • A number that is not rational.
    • Cannot be written as a fraction of integers.
  36. Cannot be written as a fraction of integers.
    Irrational
  37. Important class of irrational numbers
    Square roots of positive integers that are not perfect squares.
  38. Square roots of positive integers that are not perfect squares.
    Important class of irrational numbers.
  39. Integers from 2 and above are divided into two categories:
    • Prime
    • Composite
  40. Prime number
    • Not the product of two integers bigger than 1.
    • Divisible by itself and 1 only.
  41. Not the product of two integers bigger than 1.
    Prime number
  42. Divisible by itself and 1 only.
    Prime number
  43. Composite number
    Integer that is equal to or greater than 2 and is not prime.
  44. Integer that is equal to or greater than 2 and is not prime:
    Composite number
  45. Set of integers is closed under addition because _.
    whenever two integers are added, the result is another integer.
  46. Whenever two integers are added, the result is another integer, causing the set of integers to _.
    be closed.
  47. Set of integers is not closed under division, because _.
    it is not always true that an integer divided by another integer yields an integer.
  48. Set of negative integers is (close/not closed) under multiplication.
    not closed
  49. Absolute value of a number x is denoted _.
    |x|
  50. |x| denotes _.
    absolute value of a number
  51. Absolute value
    Distance from the number to 0.
  52. Distance from a number to 0.
    Absolute value
  53. |Image Upload 34|
    Image Upload 35
  54. Is every integer a rational number? Why?
    Yes. Each integer can be written as a fraction with 1 in the denominator, like 4 = Image Upload 36.
  55. Is Image Upload 37 a rational number? Is  Image Upload 38 rational? How about Image Upload 39?
    Image Upload 40 is irrational. Image Upload 41 is rational because it is the integer of 3. Image Upload 42 is not even real, so it is certainly not rational.
  56. List the prime numbers between 50 and 60.
    53 and 59 are the only primes between 50 and 60.
  57. Is Image Upload 43 rational?
    No. Image Upload 44 is close to both Image Upload 45 and 3.14, but it is not equal to either.
  58. Are the negative integers closed with respect to the operation of addition?
    Yes. Whenever you add two negative integers, the result is another negative integer.
  59. Is the product of two irrational numbers always irrational?
    No. Consider Image Upload 46 times Image Upload 47 = 2. Image Upload 48 is irrational, but 2 is not.
  60. Is it true that for any real numbers a, b, and c, a(b - c) = ab - ac?
    Yes. Subtracting a number is equivalent to adding its negative, so the distributive law applies here: a X (b + (-c) = a X b + a X (-c).
  61. Write Image Upload 49 as a single fraction.
    Image Upload 50
  62. Write Image Upload 51 as a single fraction.
    Image Upload 52
  63. Place these numbers on the number line. Which is largest? Which is smallest?

    4.2   Image Upload 53   Image Upload 54   Image Upload 55   Image Upload 56   Image Upload 57
    • Image Upload 58 is approximately -3.14159 (is the smallest),
    •  Image Upload 59 and is approximately .571429
    • Image Upload 60 is approximately 1.7320508,
    • Image Upload 61 = 3, and
    • Image Upload 62 = 10 (is the largest)
  64. Are |3 - 1| and |1 - 3| equal?
    Yes. |3 - 1| = |2| = 2. |1 - 3| = |-2| = 2.
  65. Is it true that for real numbers a and b, |a + b| = |a| + |b|? Provide an example where this is false.
    No. When the numbers have opposite signs, it is false. Consider a = 6 and b = -4. |6 + (-4)| = |6 - 4| = |2| = 2, while |6| + |-4| = 6 + 4 = 10. 2 Image Upload 63 10, so the statement is not true for all a and b.
  66. Exponents
    Multiplying a number by itself several times
  67. Multiplying a number by itself several times
    exponent
  68. First law of exponents
    anam = an+m

    Adding the powers of exponents
  69. The law of adding powers of exponents
    First law of exponents
  70. Second law of exponents
    (ab)c = abc

    Multiplying the powers of exponents
  71. Law of multiplying the powers of exponents
    Second law of exponents
  72. Third law of exponents
    acbc = (ab)c

    Exponent roots are paired off
  73. Law of pairing of the exponent roots
    Third law of exponents
  74. Last rule of exponents
    • Involves fractions
    • Image Upload 64 = an-m
  75. Law that involves fractions
    Last law of exponents
  76. Variable
    Letter that represents some unknown or undetermined number.
  77. Letter that represents some unknown or undetermined number.
    Variable
  78. (4ab)33a2b =
    43a3b33a2b = 433a3a2b3b = 64 X 3a5b4 = 192a5b4
  79. Image Upload 65 =
    Image Upload 66 = 25-332-3 = 223-1Image Upload 67
  80. Image Upload 68 =
    Image Upload 69 = Image Upload 70 = Image Upload 71
  81. Image Upload 72 =
    32x2(3x-3)-1 = 32x23-1x3 = 31x5 = 3x5

    OR

    Image Upload 73 =  Image Upload 74 = 3x5
  82. 4-6 =
    Image Upload 75
  83. Image Upload 76 =
    32
  84. x0 =
    1
  85. xnxm =
    xn+m
  86. (xn)m =
    xnm
  87. (xy)n =
    xnyn
  88. Image Upload 77 =
    xm-n
  89.   m
    Image Upload 78  =
    Image Upload 79
  90.   -n
    Image Upload 80  =
    •   n
    • Image Upload 81
  91. x-m =
    Image Upload 82
  92. 25 =
    32
  93. 52 =
    25
  94. 1100 =
    1
  95. 2-1 =
    Image Upload 83
  96. 2527 =
    212 = 4096
  97. (32)3 =
    36 = 729
  98. Image Upload 84 =
    42 = 16
  99. Image Upload 85 =
    (2-1)-3 = 23 = 8
  100. 19170 =
    1
  101. 6-2 =
    Image Upload 86
  102. Is there a difference between 2x-4 and (2x)4?
    Yes. In 2x-4, only the x is being raised to the 4th power. (2x)4, on the other hand, is 24x4= 16x4.
  103. Image Upload 87 =
    35-1x5-2y2-1 = 34x3y= 81x3y
  104. (2xy)2(2x2y)2 =
    22x2y222x4y2 = 24x6y4 = 16x6y4
  105. Image Upload 88 =
    Only x's cancel. Image Upload 89
  106. Image Upload 90 =
    (r2s)4(s2r2) = r8s4s-2r2 = r10s2
  107. Image Upload 91 =
    Image Upload 92
  108. Image Upload 93
    Image Upload 94
Author:
MarlieHopkins
ID:
167544
Card Set:
College Algebra
Updated:
2012-08-29 22:59:46
Tags:
Algebra
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Description:
Algebra
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