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Level Six > Number and Algebra

Exponent Power

Keywords:
Purpose: 

Solving multiplication and division problems using powers is a powerful strategy that meaningfully practices students’ skills with the laws of exponents.

Achievement Objectives:

Achievement Objective: Extend powers to include integers and fractions.

Specific Learning Outcomes: 

In a problem solving context practice these power laws:

Equations

Activity: 

Session 1

Jill’s calculator is broken and she wants to work out fraction 16384 over 32
Her friend has prepared a table of powers of 2.

table

 

  1. From the table discuss why fraction 16384 over 32= 2 to the power of 14 over 2 to the power of 5 . Discuss why 2 to the power of 14 over 2 to the power of 5 = 2 to the power of 9
    If necessary drop back to fractions
    and discuss why this reduces to 28
    Discuss why 28 = 512 by looking up the table.
  2. Ask students to complete 8192 over 256 using the table and discuss the answers.
  3. Exercises:         Students work out exponent power.
  1. Discuss why 32 x 64 = 25 x 26 = 211 = 2048
  2. Exercises. Students work out

    512 x 32

    64 x 64

    4 x 2048

    4 x 16 x 32
    2 x 64 x 16 32 x 16 x 4  64 x 64 x 64 128 x 64 x 2

    Discuss why exponent power.  =2 to the power of 14 time 2 to the power of 5 over 2 to the power of 9  =2 to the power of 10  = 1024

    Exercises:         Students work out

    Fraction equations

Session 2

Apply lesson 1’s content and ideas to base 3.

exponent power table

  1. Discuss why  4782969 over 19683 time 27 =3 to the power of 14 over 3 to the power of 9 time 3 to the power of 3 = 3 to the power of 14 over 3 to the power of 12= 3 to the power of 2 = 9
    Student exercises. Work out

exponent power equations

  1. To work out 7392discuss why
    729power of 2 =(3 to the power of 6)to the power of 2 = 3 to the power of 6 time 2 = 3 to the power of 12= 531441

Student exercises. Work out from tables.

break down of equations

  1. Before calculators were allowed in the School Certificate Examination (in 1983) students had to use these power methods to do multiplications and divisions. Here is a small part of the tables students used. These methods are still used in some year 13 mathematics.
    Table of exponent examples
  1. Discuss why 2 x3          =10power of 0.3010 x 10 to the power 0.4771
    = 10to the power of 0.7781 = 6 (Note the slight rounding problem)

Student exercises. Show, using the base ten table, that the following are true

4 ÷ 2= 2           8 ÷ 2= 4           3 x 3 = 9         
10 ÷ 2 = 5        2 x 5= 10

Discuss how the table can be used to show      
5 x 4 = 20.

Session 3

Refer to previous power tables to find square roots, and cube roots.

  1. Discuss from the base 2 table how to find the square root of 4096
    4096    = Square x Square   so 2 to the power of 12  =  Square x Square
    Discuss why 2 to the power of 6 goes in the box so the square foot of 4096 equals 2 to the power of 6 equals 64
  2. Student exercises. Use the base 2 table to work out these answers.square root equations
  1. Student Exercises. Use the base 3 tables to work outsquare root equations

Session 4

Negative powers are introduced

exponent power equations

 

  1. Discuss why negative exponent equation.    Discuss why 2 to the power of 0 = 1.
  2. Discuss whyequation
    Discuss why equation
  3. Student Exercises. Work out exponent equations
  1. Discuss why exponent equation
     
  2. Student Exercises. Work out
    square root equations

Session 5

Relate powers to geometric sequences in an elementary way.

  1. Bacteria double every hour starting with 1 thousand bacteria.
    Discuss why the sequence of the bacteria is
    1, 2, 4, 8, 16, 32,...thousands or  2 to the power of 0,1,2,3,4..........
    Why is the number of bacteria after n hours equal to 2 to the power of n?
  2. At some time there are 64 thousand bacteria.  Later there are 16384 thousand bacteria.  How long did this increase take?
    Discuss why2 to the power of 14 minus 6  =2 to the power of 8  shows increasing from 2 to the power of 6 thousand to 2 to the power of 14 thousand must take 8 hours.
  3. Student Exercises. How long will it take to change from one number of bacteria to one other?
    8 thousand to 512 thousand
    4 thousand to 8192 thousand
    64 thousand to 8192 thousand 
    1/2 thousand to 512 thousand
    1/8 thousand to 64 thousand

Unfortunately for the bacteria the host takes an anti-biotic which halves the number of bacteria every hour.  Bert has 512 thousand bacteria per litre of his blood when he takes the antibiotic he will feel well when the level of live bacteria reaches 1/4 thousand bacteria per litre.  Discuss how long it will take before Bert is well.

Student Exercises. If bacterial levels divide by 2 every hour find how long it will take to change from one number to another.

 

8192 thousand to 1 thousand                64 thousand to 1/8 thousand

256 thousand to 1/32 thousand             1/8 thousand to 1/1024 thousand


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