# College Algebra

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 Author: MarlieHopkins ID: 167544 Filename: College Algebra Updated: 2012-08-29 18:59:46 Tags: Algebra Folders: Description: Algebra Show Answers:

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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

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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