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Notes
8th Grade
Pre-Algebra
McDowell
Exponents
Exponents
9/11
Show repeated multiplication
baseexponent
Base
The number being multiplied
Exponent
The number of times to
multiply the base
Example
2³
2x2x2
4x2
8
Example
(-2)²
-2 x –2
4
-2²
-1 x 2²
-1 x 2 x 2
-1 x 4
-4
Examples
(12 – 3)²  (2² - 1²)
(-a)³ for a = -3
5(2h² – 4)³ for h = 3
Number Sets
Whole
Numbers
Natural
Numbers
9/14
0, 1, 2, 3, . . .
for short
Also known as the counting
numbers
1, 2, 3, 4, . . .
Integers
Positive and negative whole numbers
for short
. . . –2, -1, 0, 1, 2, . . .
Rational Numbers that can be written as
Numbers fractions
for short
½, ¾, -¼, 1.6, 8, -5.92
You Try
Copy and fill in the Venn Diagram
that compares Whole Numbers,
Natural Numbers, Integers, and
Rational Numbers
Whole #s
Prime
Numbers
Integers greater than one
with two positive factors
1 and the original number
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, . . .
Factor
Trees
Steps
A way to factor a number into its
prime factors
Is the number prime or composite?
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?
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
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
You Try
Find the prime factors of
1. 8
2. 15
3. 82
4. 124
5. 26
GCF
GCF
9/15
Greatest Common Factor
the largest factor two or
more numbers have in
common.
Steps to
Finding
GCF
1. Find the prime factors
of each number or
expression
2. Compare the factors
3. Pick out the prime
factors that match
4. Multiply them together
Example
Find the GCF of 126 and 150
126
2
150
63
3
75
2
21
15
5
5
3
7
The common factors are 2, 3
2x3
The GCF of 126 and 150 is 6
3
Exampl
e
Find the GCF of 24x4 and 16x3
24xxxx
2
16xxx
12
6
2
2
8
2
4
2
3
2
2
The common factors are 2, 2, 2, x, x, x
2(2)(2)xxx
The GCF is 8x3
You Try
Work Book
P 62
# 2 - 24 even
Simplifying Fractions
Simplest
form
When the numerator and
denominator have no
common factors
9/16
Simplifying
fractions
1. Find the GCF between
the numerator and
denominator
2. Divide both the numerator
and denominator of the
fraction by that GCF
Example
Simplify
28
52
28s Prime factors: 2, 2, 7
52s Prime factors: 2, 2, 13
28  4 = 7
52  4 13
Use a factor tree
to find the prime
factors of both
numbers and
then the GCF
GCF: 2 x 2
4
Example
Simplify
12a5b6
18a2b8 Use a factor tree
12s Prime factors: 2, 2, 3
18s Prime factors: 2, 3, 3
12  6 = 2aaaaabbbbbb
18  6 3aabbbbbbbb
2aaa
3bb
2a3
3b2
to find the prime
factors of both
numbers and
then the GCF
GCF: 2 x 3
6
You Try Write each fraction in simplest
form
1. 27
30
2. 15x2y
45xy3
Equivalent
fractions
Fractions that represent the
same amount
½ and 2/4 are equivalent
fractions
Making
Equivalen
t
Fractions
1. Pick a number
2. Multiply the numerator and
denominator by that same
number
5 x 3 = 15
8 x 3 24
You Try
Find 3 equivalent fractions
to
6
11
Are the
Fractions
equivalent?
1. Simplify each fraction
2. Compare the simplified
fraction
3. If they are the same then they
are equivalent
You try
Work Book
p 49 #1-17 odd
Least common Denominator
Common
Denominator
9/17
When fractions have the same
denominator
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!
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
5 x 3 = 15
each denominator
6 x 3 18
to 18
4x2=8
9 x 2 18
You try
What are the least
common denominators?
1. ¼ and 1/3
2. 5/7 and 13/12
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)
You try
Order the rational numbers
from least to greatest
1. 8/15, 6/13, 5/9, 4/7
2. -2/3, ½, 4/7, -4/5
Graph each group of rational
numbers on a number line
-1
0
1
Evaluating
fractions
Plug and chug
Substitute in the values for
the variables then chug
chug chug out the answer
in simplest form
Example
Evaluate
x(xy – 8) for x = 3 and y = 9
60
3(3•9 – 8)
60
Plug
3(27 – 8)
60
Chug
Remember Sally
3(19)  3
60  3
19
20
You try
Workbook
p 68
# 1-17 odd, 18
Exponents and Multiplication
9/18
The long
way
25 • 23
(2 • 2 • 2 • 2 • 2) • (2 • 2 • 2) expand
28
Convert back to
exponential form
The
short
way
25 • 23
Same bases so we can
add the exponents
25+3
Simplify
28
Multiplying Works for numbers and
variables
Powers
When same base powers are
With the
multiplied, just add the
Same base
exponents
Remember
baseexponent
Example
s
x2x2x2
x2+2+2
x6
32y5 • 34y10
32 • 34y5y10 Associative Property
32+4y5+10
36y15
Add exponents
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.)
Raising a power to a power
The long
way
(x2)3
x2 • x2 • x2
9/18
expand
(x • x) • (x • x) • (x • x)
x6
Convert back to
exponential form
The
short
way
(x2)3
x6
Multiply the exponents
You try
1. (x6)7
2. (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.
You try
Workbook
p 68
# 1-17 odd, 18
Exponent Rules
Exponents
Rules
9/21
Everything raised to the zero
power is 1(except zero)
x0 = 1for x  0
10980 = 1
(-23)0 = 1
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
Example
s
Simplify
a-3
1
= 3
a
1
5
=
y
y-5
b-10 = 22
2-2
b10
You Try
Simplify
1. a-12
2. 1
x-7
3. c-10
c2d-3
Division and Exponents
The long
way
9/21
x6
x9
expand
xxxxxx
xxxxxxxxx
Cross out pairs
1
x3
The
short
way
x6
x9
Subtract the exponents
x6-9
Simplify
x-3
Make all exponents
positive
1
x3
Top minus bottom
9 is bigger than 6 so it
makes sense that the x is
in the denominator
Examples
Simplify
45x4y7
9x6y3
You try
1. x5
x4
2. a10
a12
3. 16a2b4
8a5b2
Scientific Notation
Powers
Of
Ten
9/22
Factors
10
10x10
10x10x10
10x10x10x10
Product
10
100
1,000
10,000
Power
101
102
103
104
# of 0s
1
2
3
4
Factors
1
1
1
1
10 10x10 10x10x10 10x10x10x10
Product
0.1 0.01
0.001
0.0001
Power
10- 10-2
10-3
10-4
0
2
3
1
# of 0s
After the
decimal
1
Scientific
Notation
A short way to write really big
or really small numbers using
factors
Looks like:
2.4 x 104
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
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
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
Example
Convert between scientific notation
and expanded notation
4.6 x 106 Move the decimal 6
hops to the right
4.600000 Rewrite
4600000
You Try
Write in expanded notation
1. 2.3 x 10-3
2. 5.76 x 107
Answers
1. 0.0023
2. 57,600,000
Example
Convert between expanded notation
and scientific notation
13,700,000 Figure out how many hops
it takes to get a factor
between 1 and 10
1.3,700,000
1.3 x 107
Rewrite: the number
of hops is your
exponent
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
You Try
Write in scientific notation
1. 340,000,000
2. 0.000982
Answers
1. 3.4 x 108
2. 9.82 x 10-4