College Algebra

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Author:
MarlieHopkins
ID:
167544
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College Algebra
Updated:
2012-08-29 18:59:46
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Algebra
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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
  15. A fraction  is reduced if _.
    the numbers a and b have no common factors.
  16. A fraction   is _ the numbers a and b have no common factors.
    reduced
  17. Another way to say the fraction  is reduced is to say _.
    that it is in lowest terms.
  18. General rule for adding fractions:
  19.  OR

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

    4.2              
    •  is approximately -3.14159 (is the smallest),
    •   and is approximately .571429
    •  is approximately 1.7320508,
    •  = 3, and
    •  = 10 (is the largest)
  57. Are |3 - 1| and |1 - 3| equal?
    Yes. |3 - 1| = |2| = 2. |1 - 3| = |-2| = 2.
  58. 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  10, so the statement is not true for all a and b.
  59. Exponents
    Multiplying a number by itself several times
  60. Multiplying a number by itself several times
    exponent
  61. First law of exponents
    anam = an+m

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

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

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

    OR

     =   = 3x5
  75. 4-6 =
  76.  =
    32
  77. x0 =
    1
  78. xnxm =
    xn+m
  79. (xn)m =
    xnm
  80. (xy)n =
    xnyn
  81.  =
    xm-n
  82.   m
      =
  83.   -n
      =
    •   n
  84. x-m =
  85. 25 =
    32
  86. 52 =
    25
  87. 1100 =
    1
  88. 2-1 =
  89. 2527 =
    212 = 4096
  90. (32)3 =
    36 = 729
  91.  =
    42 = 16
  92.  =
    (2-1)-3 = 23 = 8
  93. 19170 =
    1
  94. 6-2 =
  95. 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.
  96.  =
    35-1x5-2y2-1 = 34x3y= 81x3y
  97. (2xy)2(2x2y)2 =
    22x2y222x4y2 = 24x6y4 = 16x6y4
  98.  =
    Only x's cancel.
  99.  =
    (r2s)4(s2r2) = r8s4s-2r2 = r10s2
  100.  =

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