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 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.  19.  20.  21.  22.  23.  24. General rule for adding fractions: 25.  26.  OR 27. General form for subtracting fractions 28. Subtract from  29. Reciprocal of a number x when multiplied by x yeilds _.
1
30. Reciprocal of any integer n (except 0) is _. 31. Reciprocal of is _.
n
32. Find the reciprocal of  33.  34. General rule for division 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. | | 54. Is every integer a rational number? Why?
Yes. Each integer can be written as a fraction with 1 in the denominator, like 4 = .
55. 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.
56. List the prime numbers between 50 and 60.
53 and 59 are the only primes between 50 and 60.
57. Is rational?
No. is close to both 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 times = 2. 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 as a single fraction. 62. Write as a single fraction. 63. 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)
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 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
• = 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. = = 25-332-3 = 223-1 80. = = = 81. =
32x2(3x-3)-1 = 32x23-1x3 = 31x5 = 3x5

OR = = 3x5
82. 4-6 = 83. =
32
84. x0 =
1
85. xnxm =
xn+m
86. (xn)m =
xnm
87. (xy)n =
xnyn
88. =
xm-n
89.   m = 90.   -n =
•   n
• 91. x-m = 92. 25 =
32
93. 52 =
25
94. 1100 =
1
95. 2-1 = 96. 2527 =
212 = 4096
97. (32)3 =
36 = 729
98. =
42 = 16
99. =
(2-1)-3 = 23 = 8
100. 19170 =
1
101. 6-2 = 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. =
35-1x5-2y2-1 = 34x3y= 81x3y
104. (2xy)2(2x2y)2 =
22x2y222x4y2 = 24x6y4 = 16x6y4
105. =
Only x's cancel. 106. =
(r2s)4(s2r2) = r8s4s-2r2 = r10s2
107. = 108.  Author: MarlieHopkins ID: 167544 Card Set: College Algebra Updated: 2012-08-29 22:59:46 Tags: Algebra Folders: Description: Algebra Show Answers: