1. 8th Grade Pre-Algebra McDowell
Notes
8th Grade
Pre-Algebra
McDowell
Chapter 4

2. Exponents 9/11 Base Exponent Exponents Show repeated multiplication
The number being multiplied
Exponent
The number of times to multiply the base

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

7. Numbers that can be written as fractions
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 more than two positive factors
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, . . .
Composite
Numbers
Integers greater than one with more than two positive factors
4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24,

10. Factor Trees Steps A way to factor a number into its prime factors
Is the number prime or composite?
Steps
If prime: you’re 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. 126 150 2 63 75 2 15 21 5 3 5 3 3 7 The common factors are 2, 3 2 x 3
Find the GCF of 126 and 150
Example
126
150 2 63 75 2 15 21 5 3 5 3 3 7
The common factors are 2, 3
2 x 3
The GCF of 126 and 150 is 6

17. Example Find the GCF of 24x4 and 16x3 24xxxx 16xxx 2 12 8 2 4 6 2 2 2
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
# 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 28s Prime factors: 2, 2, 7
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
= 7 13

22. Example Simplify 12a5b6 18a2b8 12s Prime factors: 2, 2, 3
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
= 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
Fractions that represent the same amount
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
= 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
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!

32. Make each fraction have a common denominator 5/6, 4/9
Example
Make each fraction have a common denominator
5/6, 4/9
Find the LCM of 6 and 9
Multiply to change each denominator to 18
5 x 3
6 x 3
= 15 18
4 x 2
9 x 2
= 8 18

33. What are the least common denominators? ¼ and 1/3 5/7 and 13/12
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 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. Example Plug Chug Remember Sally Evaluate
x(xy – 8) for x = 3 and y = 9 60
Example
Plug
3(3•9 – 8) 60
Chug
Remember Sally
3(27 – 8) 60
3(19) 60 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 25 • 23 25+3 28 Same bases so we can add the exponents
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 x2+2+2 x6 32y5 • 34y10 32 • 34y5y10
Associative Property
32+4y5+10
Add exponents
36y15

43. You Try
x5x7
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
x2 • x2 • x2
expand
(x • x) • (x • x) • (x • x)
Convert back to exponential form x6

45. The short way
Multiply the exponents
(x2)3 x6

46. Exponent means “out of place” in Latin
You try
(x6)7
(x8)5
Exponent means “out of place” in Latin
Micheal Stifel named exponents—he 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 Exponents Rules
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
The long way
expand
x x x x x x
x x x x x x x x x
Cross out pairs 1 x3

53. The short way x6 x9 x6-9 x-3 1 x3 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
a2b4
8a5b2

56. Scientific Notation 9/22 Powers Of Ten Factors 10 10x10 10x10x10
Product
100
1,000
10,000
Power
101
102
103
104
# of 0s 1 2 3 4

57. Factors 1 10
10x10
10x10x10
10x10x10x10
Product
0.1
0.01
0.001
0.0001
Power
10-1
10-2
10-3
10-4
# of 0s
After the decimal 2 3

58. A short way to write really big or really small numbers using factors
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 < factor < 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. Move the decimal 6 hops to the right 4.6 x 106
Convert between scientific notation and expanded notation
Example
Move the decimal 6 hops to the right
4.6 x 106
Rewrite

63. You Try
Write in expanded notation
2.3 x 10-3
5.76 x 107
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. You Try
Write in scientific notation
340,000,000
Answers
3.4 x 108
9.82 x 10-4