Download Chapter 4 Notes

Chapter 4 Notes
4-1 Divisibility and Factors
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Divisibility Rules for 2, 5, and 10
An integer is divisible by
– 2 if it ends in 0, 2, 4, 6, or 8
– 5 if it ends in 0 or 5
– 10 if it ends in 0
Examples
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Is the first number divisible by the
second?
567 by 2
1015 by 5
111,120 by 10
Examples - Answers
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Is the first number divisible by the
second?
567 by 2
no; it ends in an
odd number
1015 by 5
yes; it ends in a 5
111,120 by 10
yes; it ends in a 0
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Divisibility rules for 3 and 9
An integer is divisible by
– 3 if the sum of its digits is divisible by 3
– 9 if the sum of its digits is divisible by 9
Examples - Answers
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Is the first number divisible by the
second?
567 by 3
yes; 5+6+7=18
and 18 is divisible by 3
1015 by 9
no; 1+0+1+5=7
and 7 is NOT divisible by 9
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Divisibility rules for 4, 6, 8
An integer is divisible by
– 4 if the number formed by the last 2
digits is divisible by 4
– 6 if the number is divisible by BOTH 2
and 3
– 8 if the number formed by the last 3
digits is divisible by 8
Examples
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If the first number divisible by the second?
532 by 4 yes; 32 is divisible by 4
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342 by 6 yes; 3+4+2= 9 and is divisible by
3 also 2 is even which is divisible by 2
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5,832 by 8
8
yes; 832 is divisible by
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Divisibility rules for 7
There are no rules for 7. You just
need to work it out!!!
Finding Factors
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Factor: a number is a factor of
another number if it divides into that
number with a remainder of 0
Examples:
21 --> 1, 3, 7, 21
31 --> 1, 31
24 --> 1, 2, 3, 4, 6, 8, 12, 24
4-2 Exponents
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Exponents - show repeated multiplication
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26 = 2 x 2 x 2 x 2 x 2 x 2 = 64
Base -->2
Exponent --> 6
Value of expression --> 64
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a2 = a x a
b4= b x b x b x b
Write the expression using an
exponent.
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(-5)(-5)(-5)
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-2 x a x b x a
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6x6x6
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4xdxdxcxc
Write the expression using an
exponent. Answers
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(-5)(-5)(-5) = (-5)3
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-2 x a x b x a = -2 x a x a x b = -2a2b
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6 x 6 x 6 = 63
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4 x d x d x c x c = 4c2d2
Examples
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(-3)(-3)(-3)(-3)
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-7 x a x a x b
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Simplify:
104
(-5)4
-54
Examples - Answers
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(-3)(-3)(-3)(-3) = (-3)4
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-7 x a x a x b = (-7)a2b
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Simplify:
104 = 10 x 10 x 10 x 10 = 10,000
(-5)4 = (-5) x (-5) x (-5) x (-5) = 625
-54 = - (5 x 5 x 5 x 5) = -625
Using the Order of
Operations
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Simplify 4(3 + 2)2
=4(5)2
=4 x 5 x 5 or 4 x 25
= 100
Examples
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49 - (4 x 2)2
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2(9 - 4)2
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(-4)(-6)2(2)
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(12 - 3)2 - (22 - 12)
Examples - Answers
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49 - (4 x 2)2 = -15
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2(9 - 4)2 = 50
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(-4)(-6)2(2) = 288
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(12 - 3)2 - (22 - 12) = 78
Evaluate each expression
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c3 + 4, for c = -6
•
3(2m + 5)2, for m = 2
Evaluate each expression Answers
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•
c3 + 4, for c = -6
=(-6)3 + 4
= -216 + 4
= -212
3(2m + 5)2, for m = 2
=3(2x2 + 5)2
=3(9)2
= 3(81)
= 243
4-3 GCF and LCM
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GCF = Greatest Common Factor
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LCM = Least Common Multiple
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Prime Factorization = factor tree
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Prime number = has exactly 2 factors - 1
and itself
Composite number = has more than 2
factors
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Write the prime factorization
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825
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34
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360
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186
Write the prime factorization
(PF) - Answers
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825
PF = 3 * 52 * 11
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34
PF = 2 * 17
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360
PF = 23 * 32 * 5
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186
PF = 2 * 3 * 31
Relatively prime
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Two numbers are relatively prime if
their GCF is 1
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Examples:
8, 17: Yes, because their GCF is 1
7, 35: No, because their GCF is 7
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Find each GCF and LCM
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42, 60
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8, 16, 20
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180a2, 210a
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a3b, a2b2
Find each GCF and LCM Answers
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42, 60
GCF:6, LCM: 420
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8, 16, 20
GCF: 4, LCM: 160
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180a2, 210a GCF: 30a, LCM:1260a2
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a3b, a2b2
GCF: a2b, LCM:a3b2
4-4 Simplifying Fractions
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Finding Equivalent Fractions =
multiply or divide the numerator and
denominator by the same number
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Look at examples on page 196
Writing Factions in Simplest
Form
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Simplest form = when the numerator
and denominator have no common
factors except 1
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Look at examples on page 197
4-6 Rational Numbers
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Look at diagram on p. 205
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Rational number = any number you can
write as a fraction, with denominator NOT
being a zero
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All integers are rational number because
they can be written as a fraction
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Example: 5 can be written as 5/1
Writing Equivalent Fractions
with Rational Numbers
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1/2
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-(4/5)
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5/8
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-(12/27)
Writing Equivalent Fractions
with Rational Numbers Answers
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1/2
2/4, 4/8, 12/24
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-(4/5)
-(16/20), -(8/10)
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5/8
10/16, 150/240
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-(12/27)
-(24/54), -(36/81)
Evaluate each expression
1. b + a
3a
2. b + 7
2a
a=(-2) b=(-3)
Evaluate each expression Answers
1.
b+a
3a
a=(-2) b=(-3)
(-5)/(-6) or 5/6
2. b + 7
2a
4/(-4) or -4/4 = -1
Graphing a Rational Number
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Graph each rational number on a
number line
Examples: 1/2
-(8/10)
-0.2
4-9 Scientific Notation
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Scientific Notation = is a way to write
numbers using powers of 10
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It lets you know the size of a number
without having to count zeros
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Example: 7,500,000,000,000
7.5 x 1012
(The first number must be greater than 1 but
less than 10. The second number is a power
of 10.)
Writing in Scientific Notation
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0.000079 = 7.9 x 10-5
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89,000 = 8.9 x 104
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Examples:
0.00000005
54,500,000
Writing in Scientific Notation Answers
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0.000079 = 7.9 x 10-5
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89,000 = 8.9 x 104
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Examples:
0.00000005 = 5.0 x 10-8
54,500,000 = 5.45 x 107
Writing in Standard Notation
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8.9 x 105 = 890,000
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2.71 x 10-6 = 0.00000271
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Examples:
3.21 x 107
5.9 x 10-8
Writing in Standard Notation Answers
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8.9 x 105 = 890,000
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2.71 x 10-6 = 0.00000271
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Examples:
3.21 x 107 = 32,100,000
5.9 x 10-8 = 0.000000059
Multiplying with Scientific
Notation
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(2.3 x 106)(5 x 103) =
(2.3 x 5) x (106 x 103) =
11.5 x (106 x 103) =
11.5 x 109
=
1.15 x 1010
Examples: (5 x 106)(6 x 102)
(9 x 10-3)(7 x 108)
(4.3 x 103)(2 x 10-8)
Multiplying with Scientific
Notation - Answers
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(2.3 x 106)(5 x 103) =
2.3 x 5 x 106 x 103 =
11.5 x 106 x 103 =
11.5 x 109
=
1.15 x 1010
Examples: (5 x 106)(6 x 102) = 3.0 x 109
(9 x 10-3)(7 x 108) = 6.3 x 106
(4.3 x 103)(2 x 10-8) = 8.6 x 10-5