VOCABULARY
•
Exponent – Tells how many times a repeated factor is to be multiplied.
3
3
Base – The factor to be multiplied
3∙3∙3
27
Power – How many times to
multiply
WRITING EXPONENTS
The factor is the base. The number of times the factor appears is the exponent.
2×2×2
23
0.8 ∙ 0.8 ∙ 0.8 ∙ 0.8
0.84
𝑏∙𝑏∙𝑏∙𝑏∙𝑏∙𝑏
𝑏6
EVALUATING EXPONENTS
43
1. Write out what the exponent represents
2. Multiply. Write out each multiplication step
4∙4∙4
16 ∙ 4
64
EXAMPLES
1) 72
2) 63
3) 0.54
7∙7
6∙6∙6
0.5 ∙ 0.5 ∙ 0.5 ∙ 0.5
49
36 ∙ 6
0.25 ∙ 0.25
216
0.0625
4) 41
4
EVALUATING WITH SUBSTITUTING
Evaluate when n = 5
𝑛2 − 2
1) Rewrite the problem with the substitution.
𝑛2 − 2
𝑛=5
2) Substitute for the variable.
52 − 2
3) Evaluate using order of operations.
25 − 2
23
EXAMPLES
Evaluate each expression when n = 5.
1) (𝑛 − 3)4
2) 2𝑛2
(5 − 3)4
2 ∙ 52
24
2 ∙ 25
2∙2∙2∙2
4∙4
16
50
WORD PROBLEM
A certain cell doubles every hour. If you begin with one cell, at the end of 1 hour
there are 2 cells, at the end of 2 hours, there are 𝟐𝟐 or 4 cells, and so on. After 6
hours, how many cells will there be?
Hour 1 = 2
Hour 2 = 𝟐𝟐 = 4
Hour 3 = 𝟐𝟑 = 8
Hour 6 = 𝟐𝟔 = 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐
𝟒∙𝟒∙𝟒
𝟏𝟔 ∙ 𝟒
𝟔𝟒
64 cells in 6 hours.
DIVISIBILITY RULES
A number is divisible by . . . .
•
2
if the last digit is even (ends in 0, 2, 4, 6, 8)
•
3
if the sum of the digits is divisible by 3
•
4
if the number formed by the last two digits are divisible
by 4
•
5
if the last digit is a 0 or 5
•
6
if the number is divisible by both 2 and 3
•
9
if the sum of the digits is divisible by 9
•
10
if the last digit is a 0
EXAMPLES
Tell if 3,742 is divisible by 2, 3, 4, 5, 6, 9, or 10.
1) Test the 2s. Is the number even?
2) Test the 3s. Find the digital root.
3) Test the 4s. Look at the last two
numbers.
yes
3+7+4+2 = 16
Is 16 divisible by 3?
No
Is 42 divisible by 4?
No
4) Test the 5s. Does it end in 0 or 5?
No.
5) Test the 6s. Can it be divisible by 2
and 3?
Just 2 and not 3.
6) Test the 9s. Find the digital root.
No
3+7+4+2 = 16 = 1+6 = 7
No
7) Test the 10s. Does it end in 0?
No
3,742 is only divisible by 2
EXAMPLES
Tell if each number is divisible by 2, 3, 4, 5, 6, 9, 10 or none of these
1) 3,742
2
3
6
2) 5,310
2
5
9
3) 47,388
2
3
6
4) 9,999
3
9
10
VOCABULARY
•
Prime Number – A whole number with only two factors, 1 and itself
•
Composite Number – A whole number with more than two factors.
PRIME FACTORIZATION
Breaking down a number so that it is represented as a product of prime numbers
Use the factor tree method to find the prime factorization.
GREATEST COMMON FACTOR (GCF)
Two methods to finding the GCF
1. Factor Rainbow
2. Using Prime Factorizing
LEAST COMMON MULTIPLE (LCM)
Use the same methods of finding the GCF to find the LCM, with one major difference.
FINDING EQUIVALENT FRACTIONS
M U LT I P LY I N G
DIVIDING
Multiply both the
numerator and the
denominator
𝟐
𝟑
𝟐× 𝟒
𝟑× 𝟒
𝟖
𝟏𝟐
Divide both the
numerator and the
denominator by the
GCF
𝟏𝟐
𝟏𝟓
𝟏𝟐 ÷ 𝟑
𝟏𝟓 ÷ 𝟑
𝟒
𝟓
SIMPLEST FORM IS WHEN THE GCF IS 1 (YOU
CAN’T DIVIDE ANYMORE)
12
Write 42 in simplest form
1) Find the GCF of 12 and 42
12 – 1 2 3 4 6 12
42 – 1 2 3 6 7 14 21 42
GCF = 6
2) Divide the numerator and denominator
2
12 ÷ 6
=
7
42 ÷ 6
EXAMPLES
Write each fraction in simplest form
1)
12
22
2)
14
28
12 – 1 2 3 4 6 12
14 – 1 2 7 14
22 – 1 2 11 22
28 – 1 2 4 7 14 28
GCF = 2
GCF = 14
12 ÷ 2
6
=
22 ÷ 2
11
14 ÷ 14
28 ÷ 14
=
1
2
FINDING THE LEAST COMMON DENOMINATOR
Least Common Denominator (LCD) – The common denominator (or multiple) of two or
more fractions.
The LCD is the same thing as the LCM but now used with fractions.
3
4
5
6
REWRITE AND AS FRACTIONS WITH THE
SAME DENOMINATOR
1) Find the LCM of the denominator
4 – 4 8 12 16 20
6 – 6 12
LCM = 12
3
=
4 12
2) Use the LCM as the new
denominator for each fraction
5
=
6 12
3 ×3
9
=
4 ×3
12
3) “What you do to the bottom,
you do to the top”
9
12
10
and 12
10
5 ×2
=
12
6 ×2
EXAMPLES
Use the LCD to write each set of fractions with the same denominator.
1)
5 5
,
6 20
2)
20 – 20 40 60
LCM: 60
6 4
,
10 30
3)
4 5 7
, ,
3 8 9
30 - 30
9 – 9 18 27 36 45 54 63 72
LCM: 30
LCM: 72
5
=
6 60
6
=
10 30
4
=
3 72
5
=
20 60
4
=
30 30
5
=
8 72
50
60
18
30
7
=
9 72
15
60
4
30
96 45 56
, ,
72 72 72
VOCABULARY
Improper Fraction – A fraction where the numerator is larger than the denominator.
They can be rewritten as mixed numbers.
Ex:
𝟒
𝟑
𝟏𝟎𝟎
𝟏𝟎
𝟏𝟕
𝟔
Mixed Number – A number that is made of a whole number and a fraction.
Ex:
1
𝟏
𝟑
𝟐
𝟓
𝟔
𝟏𝟕
𝟏𝟑
𝟏𝟓
IMPROPER FRACTIONS AND MIXED NUMBERS
IMPROPER FRACTION TO
MIXED NUMBER
MIXED NUMBERS TO
IMPROPER FRACTIONS
Divide the denominator
into the numerator.
The quotient is the whole
number.
The remainder is the new
numerator.
The divisor is the new
denominator
Multiply the whole
number and the
denominator.
Add the numerator.
The sum is the new
numerator.
Use the same
denominator.
IMPROPER FRACTION TO MIXED NUMBER
1)
7
3
2)
7 ÷ 3 = 2 𝑟1
2
1
3
22
3
3)
22 ÷ 3 = 7 𝑟1
7
17
8
17 ÷ 8 = 2 𝑟1
1
3
2
1
8
MIXED NUMER TO IMPROPER FRACTION
4
1) 9 7
3
2) 10 5
3
3) 1 4
7 × 9 = 63
5 × 10 = 50
1×4=4
63 + 4 = 67
50 + 3 = 53
4+3=7
67
7
53
5
7
4
COMPARING FRACTIONS
1. Compare whole numbers first.
2. Find the LCD for all fractions.
3. Compare or order from least to greatest.
Compare. Write <, >, or =.
1)
5
8
3
4
2)
LCD: 8
5
8
5 6
<
8 8
3 6
=
4 8
7
9
2
3
LCD: 9
7 2 6
=
3
9
9
7 6
>
9 9
4
3) 3 5
3
34
LCD: 20
4 16
=
5 20
3
16
15
>3
20
20
3 15
=
4 20
ORDERING FRACTIONS
1.
Rewrite all mixed numbers as improper fractions
2.
Find the LCD for all fractions
3.
Compare
4.
Order from least to greatest
5.
Rewrite using original fractions and mixed numbers
6.
Place the fractions on a number line
Arrange in order from least to greatest and place them on a number line.
7 3 3 17
, ,1 ,
8 2 4 8
3 17 3 7
, ,
8 2 8
1) 1 4 ,
1
3 7
=
4 4
LCD: 8
7
4
=
14
8
3
2
=
12
8
7 12 14 17
, , ,
8 8 8 8
EXAMPLES
Arrange in order from least to greatest.
1)
5 3 1
, ,
7 8 2
2)
4 1 5
, ,
9 3 6
1
11
3) 5 4 , 5 16 , 6
LCD: 56
LCD: 18
LCD: 16
5 40
=
7 56
4
8
=
9 18
1
4
=
4 16
3 21
=
8 56
1
6
=
3 18
5
1 28
=
2 56
5 15
=
6 18
1 11
5 ,5 ,6
4 16
21 28 40
, ,
56 56 56
6 8 15
, ,
18 18 18
3 1 5
, ,
8 2 7
1 4 5
, ,
3 9 6
4
11
,5 ,6
16 16
CONVERTING FRACTIONS AND DECIMALS
FRACTION TO DECIMAL
DECIMAL TO FRACTION
• Divide the numerator by
the denominator
• If there is a whole
number, place the whole
number in front of the
decimal
• If the decimal repeats,
round to the nearest
hundredths
• Place the decimal
digits over the place
value
• Rewrite in simplest
form
• If there is a whole
number, place the
whole number in front
of the fraction
EXAMPLES
Write each fraction as a decimal.
1)
3
4
2)
3 ÷ 4 = 0.75
2
3
3
3) 4 20
2 ÷ 3 = 0.6666
0.67
0.75
3 ÷ 20 = 0.15
4.15
Write each decimal as a fraction in simplest form.
1) 0.6
2) 0.08
Tenths place = 10
6
3
=
10 5
3
5
Hundredths place = 100
2
8
=
100 25
2
25
3) 5.11
Hundredths place
= 100
11
100
11
5
100