# 7th Grade Pre-algebra by dffhrtcv3

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```									7th Grade Pre-algebra

Chapter 4 Notes
4.1 Factors and Monomials
Vocabulary

Factors: Two or more numbers that are
multiplied to for a product
Divisible: able to be divided without a
remainder
Monomial: a number, variable, or a product
of numbers and/or variables. Ex. 5, 4x,
3xy
Factors
Find all the factors of 36

Find all the factors of 72
4.2 Powers and Exponents
Vocabulary
Base: the number that is multiplied
Exponent: the ‘raised’ number, tells how many times the
base is used as a factor
Power: the number that can be expressed using an
exponent is called a power
34        exponent
base

Standard form: a number expresses without exponents
Expanded form: a number expressed using place values
and exponents
Remember Order of Operations
Read it, Write it, factor it
Read it                Write it   Factor it

2 to the first power

35

5∙5∙5∙5∙5

4 to the sixth power
Evaluate the expressions
3
1. 2

2. y2 + 5 if y = -3

3. 3(x + y)4 if x = -2 and y = 1
4.3 Prime Factorization
Vocabulary
Prime Number: a whole number that has exactly two
factors, 1 and itself
Composite Number: a whole number that has more than
two factors.

ZERO and ONE ARE NEITHER PRIME NOR COMPOSITE

Prime Factorization: when a composite number is
expressed as the product of prime factors
Factor tree: a way to find the prime factorization of a
number using branches of the numbers factors
Factor: to write a number as a product of its factors
Prime or Composite?
• 2

• 6

• 9

• 15

• 11
Prime Factorization
• Use a factor tree to find the prime
factorization of 280
You Try …
Find the prime factorization of 392
Factoring Monomials
Factor 8ab2

Factor -30x3y
You Try…
• Factor 64n3

• Factor -120r2st3
4.4 Greatest Common Factor
(GCF)
Vocabulary

Venn Diagram: shows the relationship
among sets of numbers or objects by
using overlapping circles
Greatest Common Factor: The greatest
factor of two or more numbers
Finding GCF
Method 1: List out factors

find the GCF of 12 and 20
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 20: 1, 2, 4, 5, 10, 20

GCF of 12 and 20 is 4
Find the GCF of 36 and 65
List the factors:

36: 1, 2, 3, 4, 6, 9, 12, 18, 36

64:    1, 2, 4, 8, 16, 32, 64

GCF is : 4
Method 2: Using prime factorization
Find the GCF of 30 and 24

Step 1: find prime factors using factor tree
30                 24
6 5                 6   4
2 3 2 2
3 2

2∙3∙5             2∙2∙2∙3
Step 2: List the common prime factors

Factors of 24
Factors of 30   Shared
Factors    2 2 2 3
2 3 5
2 3
Step 3: Find the product of the common factors

Shared Factors are 2 and 3

2∙3 = 6

GCF of 30 and 24 is 6
Use Prime factorization to find the
GCF of 252 and 126
252                     126

126     2               63    2

63       2                  21 3

9        7                  7    3

3       3
7∙3∙3∙2

2∙2∙3∙3∙7                   Common factors : 2 ∙ 3 ∙ 3 ∙ 7
GCF = 126
Finding GCF of Monomials
Find the GCF of 16xy2 and 30xy
16xy2                               30xy
4   4    x   y     y                6   5    x   y

2 2 2 2                            3 2

2∙2∙2∙2∙x∙y∙y                      5∙3∙2∙x∙y

Common Factors: 2 ∙ x ∙ y
GCF is 2xy
Finding GCF of Monomials
Find the GCF of 12x and 40x2
12x           40x2
8    5   x   x
3       4       x

4 2
2       2                        Common Factors: 2 ∙ 2 ∙ x
2   2              GCF is 4x
2∙2∙3∙x

5∙2∙2∙2∙ x∙x
Problem Solving
There are 208 boys and 240 girls participating in a
field day competition. What is the greatest
number of teams that can be formed if each
team has the same number of girls and each
team has the same number of boys?

HINT: FIND GCF of 208 and 240

4.5 Simplifying Algebraic Fractions
Vocabulary

Simplest form: when the numerator and
denominator of a fraction have a GCF of 1
Algebraic Fraction: A fraction with
variables in the numerator or denominator
Simplifying Fractions
Method 1: Divide the numerator and
denominator by their GCF

Simplify
Simplifying Fractions
Method 2: Factor the numerator and
denominator and divide by all common
factors.
Simplify

Simplify
You Try…
Simplify each fraction.
Problem Solving
There are 5280 feet in 1 mile. Eighty-eight
feet is what part of 1 mile?
4.6 Multiplying and Dividing
Monomials
How this work..
23 ∙ 24 =   2∙2∙2    ∙    2 ∙ 2 ∙ 2 ∙ 2 = 27

53 ∙ 52 =   53+2 =   55
How this work..
26 ÷ 23 =   2∙2∙2∙2∙2∙2   = 23
2∙2∙2

55 ÷ 52 =   55-2 =   53
You Try…
Multiplying and Dividing Monomials
Step 1: multiply or divide the coefficients
(not common bases)
Step 2: add or subtract the exponents of the
common bases

Find the product: 3n3 ∙ 2n4 =

Find the quotient: 15x9   =

3x6
You try…
1. 3x5 ∙ 2x3

2. -6b3 ∙ 4b2

3. 18y6 ÷ 6y4
Two step problems
Find each quotient or product
1. n3(n5)
n2

2.    k 3  m 2 
 k  m 
          
          
4.7 Negative Exponents
Power   Value

Copy the table.
26      64
Describe the pattern of the
powers in the right          25      32
column, then continue        24      16
the pattern by writing the
next two powers in the       23       8
table.                       22       4
Describe the pattern of the
values in the second         21       2
column, then complete        20
the second column.
2-1
Negative Exponents
• Method 1: x3   =   x3-5   =   x-2
x5

• Method 2: x3   =     x∙x∙x          = 1
x5       x∙x∙x∙x∙x          x2

Therefore…. x-2 =    1
x2
• To write an expression using only positive
exponents
– Flip the position of the term
• Move negative denominator values to numerator
• Move negative numerator values to denominator
– Keep the base the same
– Change the negative sign to positive
More examples
Write each expression using positive
exponents
-3
5     =

a-6 =

(-6)-4 =
Write each expression using a negative
exponent other than -1
4.8 Scientific Notation
Vocabulary

Scientific Notation: a number expressed as
a product of a factor and a power of 10.
The factor must be greater than or equal
to 1 and less than 10
Express Numbers in Standard Form

• 3.78 x 106
Positive exponent:
move decimal point to
the right the same
number of times of
• 5.1 x 10-5                  the exponent

Negative Exponents:
move the decimal
point to the left the
same number of
times of the exponent
Express Numbers in Scientific Notation

• 60,000,000

• 0.0049
You Try…
You Try…

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