Simplifying Exponents

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					Simplifying
Exponents
   Algebra I




               1
              Contents
• Multiplication Properties of Exponents ……….1 – 13
• Zero Exponent and Negative Exponents……14 – 24
• Division Properties of Exponents ……………….15 – 32
• Simplifying Expressions using Multiplication and
  Division Properties of Exponents…………………33 – 51
• Scientific Notation ………………………………………..52 - 61




                                                      2
Multiplication Properties
     of Exponents

• Product of Powers Property
• Power of a Power Property
• Power of a Product Property


                                3
     Product of Powers
         Property
• To multiply powers
  that have the same
  base, you add the
  exponents.


                               23
• Example:   a a  aaaaa  a  a
              2   3                     5



                                            4
     Practice Product of
      Powers Property:


•   Try:   x x
            5     4



•   Try:   n n n
            5         2   3


                              5
     Answers To Practice
         Problems
                            5 4
1.   Answer:   x x  x
                5    4
                                   x   9




2. Answer:     n5  n 2  n3  n5 23  n10




                                               6
     Power of a Power
        Property
• To find a power of a power, you
  multiply the exponents.

                                         2 2 2
• Example:     (a )  a  a  a  a
                 2 3      2    2    2
                                                   a   6



• Therefore,     (a 2 )3  a 23  a 6


                                                            7
Practice Using the Power
  of a Power Property
              4 4
1.   Try:   (p )

              4 5
2. Try:     (n )


                           8
     Answers to Practice
         Problems

1.   Answer:   ( p 4 ) 4  p 44  p16


                          45
2. Answer:     (n )  n
                  4 5
                                n   20




                                          9
    Power of a Product
        Property

• To find a power of a product, find
  the power of EACH factor and
  multiply.

• Example: (4 yz )  4  y  z  64 y z
                  3   3   3   3        3 3




                                             10
      Practice Power of a
       Product Property

1.   Try:             6
            ( 2mn )

                      4
2. Try:
            (abc)
                            11
     Answers to Practice
         Problems

1.   Answer:   (2mn )  2 m n  64 m n
                    6       6   6   6   6   6




2. Answer:     (abc)  a b c
                        4       4 4 4




                                                12
  Review Multiplication
 Properties of Exponents
• Product of Powers Property—To multiply powers
  that have the same base, ADD the exponents.

• Power of a Power Property—To find a power of a
  power, multiply the exponents.

• Power of a Product Property—To find a power of a
  product, find the power of each factor and
  multiply.



                                                     13
      Zero Exponents
• Any number, besides zero, to the
  zero power is 1.

• Example:   a 1
               0



• Example:   4 1
               0



                                     14
     Negative Exponents
• To make a negative
  exponent a positive
  exponent, write it as
  its reciprocal.
• In other words, when
  faced with a negative
  exponent—make it
  happy by “flipping” it.



                            15
     Negative Exponent
         Examples
• Example of Negative
  Exponent in the
  Numerator:
                               3     1
• The negative exponent    x         3
  is in the numerator—
  to make it positive, I
  “flipped” it to the
                                     x
  denominator.


                                          16
     Negative Exponents
          Example
• Negative Exponent in
  the Denominator:                  4
                              1     y
• The negative exponent        4
                                     y 4

  is in the denominator,     y      1
  so I “flipped” it to the
  numerator to make
  the exponent positive.


                                         17
Practice Making Negative
   Exponents Positive
                3
1.   Try:   d
             1
2. Try:       5
            z
                           18
     Answers to Negative
      Exponents Practice

1.   Answer:       3     1
               d         3
                         d
                         5
2. Answer:      1
                    
                      z
                        z 5
                 5
               z      1



                               19
  Rewrite the Expression
  with Positive Exponents
• Example:
                     3       2
              2x          y
• Look at EACH factor and decide if the factor belongs in the
  numerator or denominator.

• All three factors are in the numerator. The 2 has a positive
  exponent, so it remains in the numerator, the x has a
  negative exponent, so we “flip” it to the denominator. The y
  has a negative exponent, so we “flip” it to the denominator.

                              3   2     2
                        2x y            
                                          xy
                                                                 20
  Rewrite the Expression
  with Positive Exponents
             3    3 8
• Example:
                4 ab c
•    All the factors are in the numerator.
    Now look at each factor and decide if the
    exponent is positive or negative. If the
    exponent is negative, we will flip the
    factor to make the exponent positive.


                                                21
Rewriting the Expression
 with Positive Exponents
•   Example:       3       3 8
                 4 ab c
•   The 4 has a negative exponent so to make the exponent positive—
    flip it to the denominator.

•   The exponent of a is 1, and the exponent of b is 3—both positive
    exponents, so they will remain in the numerator.

•   The exponent of c is negative so we will flip c from the numerator
    to the denominator to make the exponent positive.
                                3                   3
                        ab      ab
                         3 8
                                   8
                        4 c    64 c                                      22
      Practice Rewriting the
     Expressions with Positive
            Exponents:


            1    2   3
1.   Try:   3 x        y z

2. Try:          2 3 4
            4a b c d
                                 23
              Answers

                  1       2       3            z
1.   Answer   3        x        y        z 
                                               3x 2 y 3


                                 4b 3 d
2. Answer     4a  2 b 3c  4 d  2 4
                                 a c



                                                          24
  Division Properties of
        Exponents

• Quotient of Powers Property

• Power of a Quotient Property



                                 25
    Quotient of Powers
        Property

• To divide powers
  that have the same
  base, subtract the
  exponents.
                5      53
• Example:     x     x
                 3
                        x 2

               x       1
                                26
     Practice Quotient of
       Powers Property
                  9
1.   Try:   a
            a3

             3
            y
2. Try:       4
            y

                            27
               Answers
                a9   a 9 3
1.   Answer:                a6
                a3    1


                y3     1     1
2. Answer:        4
                     4 3 
                y    y       y


                                   28
    Power of a Quotient
         Property
• To find a power of a
  quotient, find the
  power of the
  numerator and the
  power of the
  denominator and
  divide.
                         3
                  a
                                 3
                               a
• Example:                 
                  b
                                 3
                               b
                                     29
 Simplifying Expressions

                              3
              2m n
                 3    4
                          
• Simplify   
              3mn        
                          
                         


                                  30
 Simplifying Expressions
• First use the Power of a Quotient
  Property along with the Power of a Power
  Property
                3
   2m n3   4
                   3
                   2 m n33   43
                           2 m n    3   9    12
  
   3mn           3 3 3  3 3 3
                
                  3 mn   3 mn

                                              31
   Simplify Expressions

• Now use the
  Quotient of Power
  Property
   3   9 12           9 3 12 3       6    9
  2 mn     8m n                      8m n
   3 3 3
                                  
  3 mn       27                       27

                                                32
   Simplify Expressions

• Simplify
                                3
              2x y z
                3   4 2
                            
                           
              3x y z
                 4 2 3      
                           


                                     33
     Steps to Simplifying
        Expressions
1. Power of a Quotient Property along with
   Power of a Power Property to remove
   parenthesis
2. “Flip” negative exponents to make them
   positive exponents
3. Use Product of Powers Property
4. Use the Quotient of Powers Property



                                             34
       Power of a Quotient
     Property and Power of a
         Power Property
 • Use the power of a quotient property to remove
   parenthesis and since the expression has a power
   to a power, use the power of a power property.
                       3
 2 x 3 y 4 z  2            3
                             2 x    33
                                     y z    43    23
                           3 43 23 33
 3x 4 y 2 z  3
                              3 x y z
                  

                                                        35
               Continued

• Simplify powers

  3   33    43   23   3   9   12   6
 2 x       y z          2 x y z
    3 43 23 33
                       3 12 6 9
  3 x      y z         3 x y z



                                                  36
 “Flip” Negative Exponents
to make Positive Exponents

• Now make all of the exponents positive by
  looking at each factor and deciding if they
  belong in the numerator or denominator.
     3   9   12 6       3   9 6 12   6 9
   2 x y z          3 x z x y z
    3 12  6 9
                        3 12
  3 x y z               2 y

                                                37
      Product of Powers
          Property

• Now use the product of powers property
  to simplify the variables.


3   9 6 12   6 9    9 12   6 69        21 6 15
3 x z x y z 27 x y z                  27 x y z
     3 12
                  12
                                           12
    2 y         6y                       6y

                                                 38
    Quotient of Powers
        Property

• Now use the Quotient of Powers Property
  to simplify.

       21   6 15       21 15        21 15
  27 x y z   27 x z       27 x z
        12
                12  6
                              6
     6y       6y            6y

                                            39
 Simplify the Expression

• Simplify:

                    4
    5x y z 
         3   2 5
     2 3 4 
    2x y z 
             

                           40
   Step 1: Power of a
  Quotient Property and
Power of a Power Property

                4 12   8 20
              5 x y z
                4 8 12 16
              2 x y z

                                 41
 Step 2: “Flip” Negative
       Exponents


   4   12 16
  2 x z
 4 8 20 8 12
5 y z x y

                           42
  Step 3: Product of
   Powers Property

  4   12 16
 2 x z
 4 8 20 20
5 x y z

                       43
 Step 4: Quotient of
   Powers Property

      4
  16 x
      20 4
625 y z

                       44
 Simplifying Expressions
                                 2
• Given        4 xy  2 xy
                         2
                             
                     
                1 3 
                             
                             
             2 x y  3xy     
• Step 1: Power of a
  Quotient Property




                                      45
    Power of Quotient
        Property
• Result after Step 1:
                    2   2   4
      4 xy     2 x y
       1  3
               2 2 2
     2x y      3 x y

• Step 2: Flip Negative Exponents


                                    46
       “Flip” Negative
         Exponents
                 3       2    2    2
    4 xyxy 3 x y
            2 2 4
       2    2 x y
• Step 3: Make one large Fraction by using
  the product of Powers Property


                                             47
Make one Fraction by Using
Product of Powers Property


43 x y
    2   4   6

  3 2 4
 2 x y

                             48
Use Quotient of Powers
      Property

                   2     2
               9x y
                 2

                             49
Simplify the Expressions
                    3                   1
         3a 
            2
                           x 3
1. Try:  1 
         2x             2 
                           4a 
                            

                        3                   2
           2x  2
                             2x   5
                                        
2. Try:    4
           y       
                           
                              y        
                                        
                                     

                                                 50
             Answers
                       3               1
            3a 
               2
                              x 
                                 3
                                      27a 4 x 6
            1 
1. Answer:  2 x            2  
                              4a 
                                    2

                           3                   2
2. Answer:    2x  2
                                2x   5
                                                      2
              4
              y       
                              
                                 y        
                                                    4 10
                                                 x y

                                                             51
     Scientific Notation
• Scientific Notation uses powers of ten to express
  decimal numbers.

• For example:   2.39 10            5


• The positive exponent means that you move the
  decimal to the right 5 times.

•   So,   2.39 10  239 ,000
                     5




                                                      52
    Scientific Notation

• If the exponent of 10 is negative, you
  move the decimal to the left the
  amount of the exponent.

                     8
• Example:   2.65 10  0.0000000265

                                           53
     Practice Scientific
          Notation

Write the number in
   decimal form:

1.   4.9 10     6

              3
2.   1.23 10
                           54
          Answers

1.   4.9 10  4,900 ,000
            6



                3
2.   1.2310  0.00123

                            55
     Write a Number in
     Scientific Notation

• To write a number in scientific notation, move the
  decimal to make a number between 1 and 9.
  Multiply by 10 and write the exponent as the
  number of places you moved the decimal.
• A positive exponent represents a number larger
  than 1 and a negative exponent represents a
  number smaller than 1.



                                                       56
      Example of Writing a
      Number in Scientific
           Notation
1.   Write 88,000,000 in scientific notation

•    First place the decimal to make a number
     between 1 and 9.
•    Count the number of places you moved the
     decimal.
•    Write the number as a product of the decimal
     and 10 with an exponent that represents the
     number of decimal places you moved.
•    Positive exponent represents a number larger
     than 1.

                8.8 10              7
                                                    57
      Write 0.0422 in
     Scientific Notation
• Move the decimal to make a number between 1 and
  9 – between the 4 and 2
• Write the number as a product of the number you
  made and 10 to a power 4.2 X 10
• Now the exponent represents the number of
  places you moved the decimal, we moved the
  decimal 2 times. Since the number is less than 1
  the exponent is negative.

                                  2
               4.2 10                               58
        Operations with
       Scientific Notation
•   For example:   (2.3 10 3 )(1.8 10 5 )
•   Multiply 2.3 and 1.8
    = 4.14
•   Use the product of
    powers property 4.14 103 5
•   Write in scientific
                          2
    notation 4.14 10



                                               59
            Try These:
•    Write in scientific notation

1.   (4.1 10 )( 3  10 )
                  2        6


             5            1
2.   (6 10 )( 2.5 10 )

                                    60
           Answers

1.   (4.110 )(3 10 )  1.23 10
                2    6                    9



           5         1
2.   (6 10 )( 2.5 10 )  1.5 10   5




                                     61
           The End
• We have completed all the concepts
  of simplifying exponents. Now we
  just need to practice the concepts!




                                        62

				
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