Notes

1
Notes

• 8th Grade
• Pre-Algebra
• McDowell

Chapter 4
2
Exponents 9/11

• Exponents

Show repeated multiplication
baseexponent
Base
The number being multiplied
The number of times to multiply the base
Exponent
3

• Example

• 2³

2 x 2 x 2
4 x 2
8
4

• Example

• (-2)²

-2 x 2
4
-2²
-1 x 2²
-1 x 2 x 2
-1 x 4
-4
5

• Examples

• (12 3)² ? (2² - 1²)

(-a)³ for a -3
5(2h² 4)³ for h 3
6
Number Sets 9/14

• Whole
• Numbers

0, 1, 2, 3, . . .
Natural Numbers
for short
Also known as the counting numbers
1, 2, 3, 4, . . .
7

• Integers

• Positive and negative whole numbers

for short
. . . 2, -1, 0, 1, 2, . . .
Rational Numbers
Numbers that can be written as fractions
for short
½, ¾, -¼, 1.6, 8, -5.92
8

• You Try

• Copy and fill in the Venn Diagram that compares
  Whole Numbers, Natural Numbers, Integers, and
  Rational Numbers

Whole s
9

• Integers greater than one with two positive
  factors
• 1 and the original number

• Prime
• Numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .
Integers greater than one with more than two
positive factors
Composite Numbers
4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24,
. . .
10

• Factor
• Trees

• A way to factor a number into its prime factors

Is the number prime or composite?
Steps
If prime youre done
If Composite
Is the number even or odd?
If even divide by 2
If odd divide by 3, 5, 7, 11, 13 or another
prime number
Write down the prime factor and the new number
Is the new number prime or composite?
11

• Example

• Find the prime factors of
• 99

prime or composite
even or odd
divide by 3
3
33
prime or composite
even or odd
divide by 3
3
11
prime or composite
The prime factors of 99 3, 3, 11
12

• Example

• Find the prime factors of
• 12

prime or composite
even or odd
divide by 2
2
6
prime or composite
even or odd
divide by 2
2
3
prime or composite
The prime factors of 12 2, 2, 3
13

• You Try

• Find the prime factors of
• 8
• 2. 15
• 3. 82
• 4. 124
• 5. 26

14
GCF 9/15

• GCF

Greatest Common Factor
the largest factor two or more numbers have in
common.
15

• 1. Find the prime factors of each number or
  expression

• Steps to
• Finding
• GCF

2. Compare the factors
3. Pick out the prime factors that match
4. Multiply them together
16

• Find the GCF of 126 and 150

• Example

150
126
2
63
75
2
15
5
21
3
5
3
3
7
The common factors are 2, 3
2 x 3
The GCF of 126 and 150 is 6
17

• Find the GCF of 24x4 and 16x3

• Example

16xxx
24xxxx
2
12
8
2
4
2
6
2
2
2
2
3
The common factors are 2, 2, 2, x, x, x
2(2)(2)xxx
The GCF is 8x3
18

• You Try

• Work Book
• P 62
• 2 - 24 even

19
Simplifying Fractions 9/16

• Simplest form

When the numerator and denominator have no common
factors
20

• Simplifying fractions

• 1. Find the GCF between the numerator and
  denominator

2. Divide both the numerator and denominator of
the fraction by that GCF
21

• Example

• Simplify 28
• 52

Use a factor tree to find the prime factors of
both numbers and then the GCF
28s Prime factors 2, 2, 7 52s Prime factors 2,
2, 13
GCF 2 x 2 4
28 52
? 4 ? 4
7 13
22

• Example

• Simplify 12a5b6
• 18a2b8

Use a factor tree to find the prime factors of
both numbers and then the GCF
12s Prime factors 2, 2, 3 18s Prime factors 2,
3, 3
12 18
? 6 ? 6
2aaaaabbbbbb 3aabbbbbbbb
GCF 2 x 3 6
2aaa 3bb
2a3 3b2
23

• You Try

• Write each fraction in simplest form
• 27
• 30
• 15x2y
• 45xy3

24
Fractions that represent the same amount
Equivalent fractions
½ and 2/4 are equivalent fractions
25

• Making
• Equivalent
• Fractions

• 1. Pick a number

2. Multiply the numerator and denominator by
that same number
5 8
x 3 x 3
15 24
26

• You Try

• Find 3 equivalent fractions to
• 6
• 11

27

• Are the
• Fractions
• equivalent?

• 1. Simplify each fraction

2. Compare the simplified fraction
3. If they are the same then they are equivalent
28

• You try

• Work Book
• p 49 1-17 odd

29
Least common Denominator 9/17

• Common
• Denominator

When fractions have the same denominator
30

• Steps to
• Making
• Common
• Denominators

• 1. Find the LCM of all the denominators

2. Turn the denominator of each fraction into
that LCM using multiplication
Remember what ever you multiply by on the
bottom, you have to multiply by on the top!
31
(No Transcript)
32

• Example

• Make each fraction have a common denominator
• 5/6, 4/9

Find the LCM of 6 and 9
6 12 18 24 30 36 42 48
9 18 27 36 45 64 73 82
Multiply to change each denominator to 18
5 x 3 6 x 3
15 18
8 18
4 x 2 9 x 2
33

• You try

• What are the least common denominators?
• ¼ and 1/3
• 5/7 and 13/12

34

• Comparing
• And
• Ordering
• fractions

• Manipulate the fractions so each has the same
  denominator

Compare/order the fractions using the numerators
(the denominators are the same)
35

• You try

• Order the rational numbers from least to greatest
• 8/15, 6/13, 5/9, 4/7
• -2/3, ½, 4/7, -4/5

Graph each group of rational numbers on a number
line
-1
0
1
36

• Evaluating fractions

• Plug and chug

Substitute in the values for the variables then
chug chug chug out the answer in simplest form
37

• Evaluate
• x(xy 8) for x 3 and y 9
• 60

• Example

Plug
3(39 8) 60
Chug Remember Sally
3(27 8) 60
3(19) 60

• 3
• 3

19 20
38

• You try

• Workbook
• p 68
• 1-17 odd, 18

39
Exponents and Multiplication 9/18

• The long way

25 23
(2 2 2 2 2) (2 2 2)
expand
Convert back to exponential form
28
40

• The short way

Same bases so we can add the exponents
25 23
253
Simplify
28
41

• Multiplying
• Powers
• With the
• Same base

• Works for numbers and variables

When same base powers are multiplied, just add
the exponents
Remember baseexponent
42

• Examples

• x2x2x2

x222
x6
32y5 34y10
32 34y5y10
Associative Property
324y510
Add exponents
36y15
43

• You Try

1. x5x7
2. 74a8 7a11

A Parisian mathematician, Nicolas Chuquet, who is
credited with the first use exponents and with
naming large numbers (billion, trillion, etc.)
44
Raising a power to a power 9/18

• The long way

(x2)3
expand
x2 x2 x2
(x x) (x x) (x x)
Convert back to exponential form
x6
45

• The short way

Multiply the exponents
(x2)3
x6
46

• You try

1. (x6)7
2. (x8)5

Exponent means out of place in Latin
Micheal Stifel named exponentshe was German, a
monk, a mathematics professor. He was once
arrested for predicting the end of the world once
it was proven he was wrong.
47

• You try

• Workbook
• p 68
• 1-17 odd, 18

48
Exponent Rules 9/21

• Everything raised to the zero power is 1(except
  zero)

• Exponents
• Rules

x0 1for x ? 0 10980 1 (-23)0 1
49

• Exponent
• Rules

• Negative exponents mean the exponential is on the
  wrong side of the fraction bar

Make that power happy by moving it to the other
side of the fraction bar
x-2 1 x2
50

• Examples

• Simplify

1 a3
a-3
1 y-5
y5
b-10 2-2
22 b10
51

• You Try

• Simplify

• a-12
• 1
• x-7
• 3. c-10
• c2d-3

52
Division and Exponents 9/21
x6 x9

• The long way

expand
Cross out pairs
x x x x x x x x x x x x x x x
1 x3
53

• The short way

x6 x9
Subtract the exponents Top minus bottom
x6-9
Simplify
x-3
Make all exponents positive
1 x3
9 is bigger than 6 so it makes sense that the x
is in the denominator
54

• Examples

• Simplify
• 45x4y7
• 9x6y3

55

• You try

• 1. x5
• x4
• 2. a10
• a12
• 3. 16a2b4
• 8a5b2

56
Scientific Notation 9/22

• Powers
• Of
• Ten

Factors 10 10x10 10x10x10 10x10x10x10
Product 10 100 1,000 10,000
Power 101 102 103 104
of 0s 1 2 3 4
57
Factors 1 10 1 10x10 1 10x10x10 1 10x10x10x10
Product 0.1 0.01 0.001 0.0001
Power 10-1 10-2 10-3 10-4
of 0s After the decimal 0 1 2 3
58

• Scientific
• Notation

A short way to write really big or really small
numbers using factors
Looks like 2.4 x 104
59
One factor will always be a power of ten 10n
The other factor will be less than 10 but greater
than one 1 lt factor lt 10 And will usually have a
decimal
60
The first factor tells us what the number looks
like
The exponent on the ten tells us how many places
to move the decimal point
61
A positive exponent moves the decimal to the right
Makes the number bigger
A negative exponent moves the decimal to the left
Makes the number smaller
62
Convert between scientific notation and expanded
notation
Example
Move the decimal 6 hops to the right
4.6 x 106
4.600000
Rewrite
4600000
63

• Write in expanded notation
• 2.3 x 10-3
• 5.76 x 107

You Try

• Answers
• 0.0023
• 57,600,000

64
Convert between expanded notation and scientific
notation
Example
13,700,000
Figure out how many hops it takes to get a factor
between 1 and 10
1.3,700,000
Rewrite the number of hops is your exponent
1.3 x 107
65
If you hop left the exponent will be
positive---the number is bigger than 0
If you hop right the exponent will be
negative---the number is less than zero
66

• Write in scientific notation
• 340,000,000
• 0.000982

You Try

• Answers
• 3.4 x 108
• 9.82 x 10-4