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					 7-4 Division Properties of Exponents
  7-4 Division Properties of Exponents




                   Warm Up
                   Lesson Presentation
                   Lesson Quiz




Holt Algebra 1 1
 Holt Algebra
 7-4 Division Properties of Exponents

    Warm Up
    Simplify.

    1. (x2)3             2.
    3.                   4.
    5.                   6.

    Write in Scientific Notation.
    7.
    8.

Holt Algebra 1
 7-4 Division Properties of Exponents

                 Objective
   Use division properties of exponents to
   evaluate and simplify expressions.




Holt Algebra 1
 7-4 Division Properties of Exponents

      A quotient of powers with the same base
      can be found by writing the powers in a
      factored form and dividing out common
      factors.




      Notice the relationship between the
      exponents in the original quotient and the
      exponent in the final answer: 5 – 3 = 2.

Holt Algebra 1
 7-4 Division Properties of Exponents




Holt Algebra 1
 7-4 Division Properties of Exponents
           Example 1: Finding Quotients of Powers

   Simplify.


     A.                           B.




Holt Algebra 1
 7-4 Division Properties of Exponents
           Example 1: Finding Quotients of Powers

   Simplify.

   C.                     D.




Holt Algebra 1
 7-4 Division Properties of Exponents



   Helpful Hint


   Both          and 729 are considered to be simplified.




Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 1

   Simplify.

    a.                           b.




Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 1

   Simplify.

     c.                       d.




Holt Algebra 1
 7-4 Division Properties of Exponents
  Example 2: Dividing Numbers in Scientific Notation
   Simplify                     and write the
   answer in scientific notation



                            Write as a product of quotients.
                            Simplify each quotient.
                            Simplify the exponent.
                            Write 0.5 in scientific notation
                              as 5 x 10 .
                            The second two terms have
                              the same base, so add the
                              exponents.
                            Simplify the exponent.
Holt Algebra 1
 7-4 Division Properties of Exponents


      Writing Math
      You can “split up” a quotient of products into a
      product of quotients:


      Example:




Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 2
   Simplify                      and write the
   answer in scientific notation.



                             Write as a product of quotients.
                             Simplify each quotient.
                             Simplify the exponent.
                             Write 1.1 in scientific notation
                               as 11 x 10 .
                             The second two terms have
                               the same base, so add the
                               exponents.
                             Simplify the exponent.
Holt Algebra 1
 7-4 Division Properties of Exponents
                   Example 3: Application
   The Colorado Department of Education spent
   about            dollars in fiscal year 2004-05
   on public schools. There were about
   students enrolled in public school. What was
   the average spending per student? Write your
   answer in standard form.
        To find the average spending per student, divide
        the total debt by the number of students.



                                   Write as a product of
                                     quotients.
Holt Algebra 1
 7-4 Division Properties of Exponents
                    Example 3 Continued
   The Colorado Department of Education spent
   about            dollars in fiscal year 2004-05
   on public schools. There were about
   students enrolled in public school. What was
   the average spending per student? Write your
   answer in standard form.
        To find the average spending per student, divide
        the total debt by the number of students.
                          Simplify each quotient.
                        Simplify the exponent.
                        Write in standard form.
        The average spending per student is $5,800.
Holt Algebra 1
 7-4 Division Properties of Exponents
                  Check It Out! Example 3

   In 1990, the United States public debt was
   about           dollars. The population of the
   United States was about           people. What
   was the average debt per person? Write your
   answer in standard form.
         To find the average debt per person, divide the
         total debt by the number of people.



                                    Write as a product of
                                      quotients.

Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 3 Continued

    In 1990, the United States public debt was
    about           dollars. The population of the
    United States was about           people. What
    was the average debt per person? Write your
    answer in standard form.
           To find the average debt per person, divide the
           total debt by the number of people.

                                 Simplify each quotient.
                                 Simplify the exponent.
                                Write in standard form.
           The average debt per person was $12,800.
Holt Algebra 1
 7-4 Division Properties of Exponents

      A power of a quotient can be found by first
      writing the numerator and denominator as
      powers.




      Notice that the exponents in the final answer
      are the same as the exponent in the original
      expression.

Holt Algebra 1
 7-4 Division Properties of Exponents




Holt Algebra 1
 7-4 Division Properties of Exponents
    Example 4A: Finding Positive Powers of Quotient

   Simplify.




                          Use the Power of a Quotient
                            Property.

                          Simplify.




Holt Algebra 1
 7-4 Division Properties of Exponents
    Example 4B: Finding Positive Powers of Quotient

   Simplify.


                           Use the Power of a Product
                             Property.
                           Use the Power of a Product
                             Property:


                           Simplify and use the Power
                             of a Power Property:


Holt Algebra 1
 7-4 Division Properties of Exponents
    Example 4C: Finding Positive Powers of Quotient
   Simplify.




                           Use the Power of a Product
                             Property.
                           Use the Power of a Product
                             Property:

                           Use the Power of a Product
                             Property:

Holt Algebra 1
 7-4 Division Properties of Exponents
                 Example 4C Continued
   Simplify.


                       Use the Power of a Product
                         Property:




Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 4a

   Simplify.




                               Use the Power of a Quotient
                                 Property.


                               Simplify.




Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 4b
   Simplify.




Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 4c
   Simplify.




Holt Algebra 1
 7-4 Division Properties of Exponents

       Remember that   . What if x is a fraction?


                       Write the fraction as division.
                       Use the Power of a Quotient
                         Property.

                       Multiply by the reciprocal.

                       Simplify.

                       Use the Power of a Quotient
                         Property.
 Therefore,
Holt Algebra 1
 7-4 Division Properties of Exponents




Holt Algebra 1
 7-4 Division Properties of Exponents
  Example 5A: Finding Negative Powers of Quotients

   Simplify.



                        Rewrite with a positive exponent.

                        Use the Powers of a Quotient
                          Property .


                                 and


Holt Algebra 1
 7-4 Division Properties of Exponents
  Example 5B: Finding Negative Powers of Quotients

   Simplify.




Holt Algebra 1
 7-4 Division Properties of Exponents
  Example 5C: Finding Negative Powers of Quotients

   Simplify.



                           Rewrite each fraction with a
                             positive exponent.
                           Use the Power of a Quotient
                              Property.
                           Use the Power of a Product
                              Property:
                           (3)2 (2n)3 = 32  23n3
                           and (2)2  (6m)3 = 22  63m3
Holt Algebra 1
 7-4 Division Properties of Exponents
  Example 5C: Finding Negative Powers of Quotients

   Simplify.



                                   Square and cube terms.

                     1
                 1       2
                                   Divide out common
                                     factors.
                 1
                         24
                              12

                                   Simplify.

Holt Algebra 1
 7-4 Division Properties of Exponents



   Helpful Hint
   Whenever all of the factors in the numerator or
   the denominator divide out, replace them with 1.




Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 5a

   Simplify.



                                Rewrite with a positive
                                  exponent.

                                Use the power of a Quotient
                                  Property.

                                93=729 and 43 = 64.


Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 5b

   Simplify.


                              Rewrite with a positive
                                exponent.

                              Use the Power of a Quotient
                                Property.

                              Use the Power of a Power
                                Property: (b2c3)4= b2•4c3•4 =
                                b8c12 and (2a)4= 24a4= 16a4.
Holt Algebra 1
 7-4 Division Properties of Exponents
                 Check It Out! Example 5c
   Simplify.




                                Rewrite each fraction with a
                                  positive exponent.
                                Use the Power of a
                                  Quotient Property.
                                Use the Power of a Product
                                  Property: (3)2= 9.
                                Add exponents and divide
                                  out common terms.
Holt Algebra 1
 7-4 Division Properties of Exponents
                 Lesson Quiz: Part I

    Simplify.
    1.                  2.


    3.                  4.


    5.




Holt Algebra 1
 7-4 Division Properties of Exponents
                     Lesson Quiz: Part II

    Simplify.

       6. Simplify (3  1012) ÷ (5  105) and write the
          answer in scientific notation. 6  106

        7. The Republic of Botswana has an area of 6 
           105 square kilometers. Its population is about
           1.62  106. What is the population density of
           Botswana? Write your answer in standard
           form. 2.7 people/km2



Holt Algebra 1