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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:

Specific Learning Outcomes:

In a problem solving context practice these power laws:

Activity:

#### Session 1

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

1. From the table discuss why = . Discuss why =
If necessary drop back to
and discuss why this reduces to 28
Discuss why 28 = 512 by looking up the table.
2. Ask students to complete  using the table and discuss the answers.
3. Exercises:         Students work out
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  =  =  = 1024

Exercises:         Students work out

#### Session 2

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

1. Discuss why   = = =  = 9
Student exercises. Work out

1. To work out 7392discuss why
729 = = = 531441

Student exercises. Work out from tables.

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.
1. Discuss why 2 x3          =10 x 10
= 10 = 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
4096    = x    so   =  x
Discuss why  goes in the box so
2. Student exercises. Use the base 2 table to work out these answers.
1. Student Exercises. Use the base 3 tables to work out

#### Session 4

Negative powers are introduced

1. Discuss why .    Discuss why  = 1.
2. Discuss why
Discuss why
3. Student Exercises. Work out
1. Discuss why

2. Student Exercises. Work out

#### 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  ...
Why is the number of bacteria after n hours equal to ?
2. At some time there are 64 thousand bacteria.  Later there are 16384 thousand bacteria.  How long did this increase take?
Discuss why  =  shows increasing from thousand to  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

## Square and Cube Roots

The fact that squaring and square roots are inverses is explored geometrically and numerically.  Gauss’ method of determining square roots when only squares are available is developed.  Finally a powerful method of calculating square roots that produces answers to any desired accuracy quickly is shown.

## Moving and Shaking

This is a level 5 and 6 number activity from the Figure It Out theme series.

## Four 4s

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

## Hundreds

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

## The Power of Powers

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.