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

Answer: 16 teams
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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