Download Part 1 Index form - NSW Department of Education

Mathematics Stage 5
NS5.1.1 Rational numbers
Part 1
Index form
Number: 44538
Title: NS5.1.1 Rational numbers
This publication is copyright New South Wales Department of Education and Training (DET), however it may contain
material from other sources which is not owned by DET. We would like to acknowledge the following people and
organisations whose material has been used:
Extracts from Mathematics Syllabus Years 7-10 © Board of Studies, NSW 2002
Unit overview pp iii-v,
Part 1-4 p 3
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Centre for Learning Innovation (CLI)
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© State of New South Wales, Department of Education and Training 2006.
Contents – Part 1
Introduction – Part 1 ..........................................................3
Indicators ...................................................................................3
Preliminary quiz – Part 1 ...................................................5
Index form .........................................................................7
Describing index form ...............................................................8
Calculating squares and cubes ................................................9
Evaluating higher powers........................................................11
Spreadsheets and powers...............................................13
Zero index .......................................................................17
Negative indices..............................................................21
Further negative indices..........................................................24
Suggested answers – Part 1 ...........................................27
Exercises – Part 1 ...........................................................29
Part 1
Index form
1
2
NS5.1.1 Rational numbers
Introduction – Part 1
In this section you will explore how numbers can be expressed in index
form. You will use calculators and spreadsheets to calculate arithmetic
expressions written in this form.
Indicators
By the end of Part 1, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
•
describing numbers written in index form using terms such as base,
power, index or exponent
•
evaluating numbers expressed as powers of positive whole numbers
•
establishing the meaning of the zero index
•
translating numbers into index form (integral indices) and vice versa.
By the end of Part 1, you will have been given the opportunity to work
mathematically by:
•
solving numerical problems involving indices.
Source:
Part 1
Index form
Extracts from outcomes of the Mathematics Years 7–10 syllabus
<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_
710_syllabus.pdf > (accessed 04 November 2003).
© Board of Studies NSW, 2002.
3
4
NS5.1.1 Rational numbers
Preliminary quiz – Part 1
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz – Part 1
Try these.
1
2
3
4
Without a calculator, calculate the following.
a
14 × 14 = ___________________________________________
b
5 × 5 × 5 = __________________________________________
c
2 × 2 × 2 × 2 × 2 × 2 = __________________________________
Square the following numbers. You may use a calculator if needed.
a
7 _________________________________________________
b
16 ________________________________________________
c
35 ________________________________________________
Use your calculator to calculate the following.
a
192 = ______________________________________________
b
123 ________________________________________________
c
133 ________________________________________________
What value should replace the triangle, ∆?
a
∆2 = 36 ____________________________________________
b
4∆ = 64 _____________________________________________
c
∆4 = 81 ____________________________________________
Check your response by going to the suggested answers section.
Part 1
Index form
5
6
NS5.1.1 Rational numbers
Index form
Sometimes you need to multiply a number by itself, such as
2 × 2, 5 × 5, 12 × 12, or 27 × 27 . These products can be written using
mathematical shorthand: 22, 52, 122, and 272.
For instance, 100 is the product of ten multiplying itself: 100 = 10 × 10 .
You can also write 100 = 10 2 .
You read 102 as ‘ten squared’. You can
also say ‘ten to the power of two’.
102 means 10 × 10, not 10 × 2.
And 26 means 2 × 2 × 2 × 2 × 2 × 2,
not 2 × 6.
As another example, 64 can be written as six two’s multiplied together:
64 = 2 × 2 × 2 × 2 × 2 × 2 .
Using index shorthand, 64 = 2 6 .
You read 26 as ‘two to the power of six’.
Use your calculator to verify that 2 × 2 × 2 × 2 × 2 × 2 is indeed 64.
Writing numbers like this is writing them in index form. The plural of
‘index’ is ‘indices’. Another name for index form is power form or
power notation.
Part 1
Index form
7
The index (or power) tells you how many times the number appears in
the multiplication.
4
4
4
Can you see why the words ‘squared’ and ‘cubed’ are used when you are
reading 42 and 43?
4
4
42 = 4 × 4
43 = 4 × 4 × 4
Describing index form
There are two parts to a number written in index form.
The base indicates the number that is multiplied together.
The power, index, or exponent
(you can use any of these words)
tells you the number of times that
base is multiplied by itself.
power, index or exponent
35 base
So in 35, three appears 5 times in the multiplications: 3× 3× 3× 3× 3.
If you use a calculator to work this out you would get 243.
8
NS5.1.1 Rational numbers
Activity – Index form
Try these.
1
For the number written in index form, identify the base and the
power.
a
34 _________________________________________________
Check your responses by going to the suggested answers section.
You may already be familiar with the use of squares and cubes written in
index form. The next section should deepen your understanding of this.
Calculating squares and cubes
Sometimes it is easy to calculate squares mentally.
For instance, 32 means 3× 3 which you can quickly do to obtain an
answer of 9. You shouldn’t need a calculator for this.
Other numbers may be more difficult to square. For instance, 272 means
27 × 27 . You could calculate this using long multiplication, but an easier
way is to use a calculator.
Locate the square key on your calculator.
2
On this calculator it is shown as x .
Depending on your calculator you may be
able to access it directly, as with the
calculator shown here, or via the SHIFT , INV ,
or 2nd F key.
cube
key
square
key
Using this calculator, to evaluate 272 you
2
would press 27 x
=
. On some
calculators you may not need to press
to obtain the square.
Part 1
Index form
=
Source
Casio Computer Co., Ltd
9
Check to see that you can do this calculation on your own calculator.
The answer is 729. Also verify the answer by multiplying 27 by itself:
27 × 27 .
Activity – Index form
Try these.
2
Use your calculator to find the square using the calculator square
key. Then verify that you obtain the correct answer by multiplying
the number by itself.
a
532 = ______________________________________________
Check your responses by going to the suggested answers section.
You should follow a similar procedure to calculate cubes.
Many calculators have a cube key: x3 . You can use it to evaluate cubes.
For instance, 143 = 14 ×14 ×14
,
= 2744
but using a calculator with a cube key you could press 14
x3
=
.
You can see that index keys, like the square and cube keys, can save you
time because you don’t have to repeat multiplications.
Activity – Index form
Try these.
3
Use your calculator to find the cube. Verify you obtain the same
answer by multiplying the number by itself twice.
a
253 = ______________________________________________
Check your responses by going to the suggested answers section.
10
NS5.1.1 Rational numbers
In the next section you will look at calculating higher powers.
But for the moment, practise evaluating squares and cubes.
Evaluating higher powers
Calculators generally have square keys x2 and cube keys x3 as squares
and cubes are fairly commonly calculated. But your calculator can also
calculate higher powers.
But your calculator can also calculate higher powers.
Locate the index key on your
calculator. On the calculator shown
,
here it is represented by the caret,
but on other calculators it may appear
x
y
as x or y . Depending on your
calculator you may be able to access it
directly, as is shown here, or via the
SHIFT , INV , or 2nd F key.
[The caret,
, is used because in
many computer programs.
This symbol means ‘raised to the
power of’.]
power
key
Source
Casio Computer Co.,Ltd
Use your calculator to evaluate 56 by pressing 5
6 = , or the
appropriate index key for your calculator. The answer is 15 625.
Check with the instruction manual, or your teacher, if the answer you
obtain is different.
Now practise this process using your calculator by completing this
following activity.
Part 1
Index form
11
Activity – Index form
Try these.
4
Use your calculator to find the basic numeral. Verify that you obtain
the same answer using repeated multiplication.
a
34 = _______________________________________________
Check your responses by going to the suggested answers section.
Can you see why a power button is provided? Because there are so many
different powers it would be impossible to provide keys for all
possibilities. So the common ones, x2 and possibly x3 are often on the
calculator and a general index power key is used for the others.
Go to the exercises section and complete Exercise 1.1 – Index form
12
NS5.1.1 Rational numbers
Spreadsheets and powers
You can use a spreadsheet program to calculate powers of numbers.
In this exercise you will draw up a table of squares, cubes, fourth and
fifth powers of numbers from 1 to 10.
1
Open a computer spreadsheet document, such as Microsoft Excel®.
2
Type the headings as shown in row 1.
Your screen should look similar to this.
While you can type in the numbers from 1 to 10 in column A, an easier
way is to let the computer do it for you.
3
Type in the numbers 1 and 2, as shown in the first diagram.
Highlight both cells A2 and A3.
When you move the pointer to the
bottom, right-hand corner of your
selection it changes to a +.
Hold down the mouse and drag this point
to A11. Release.
The numbers 1 to 10 should now appear.
The program notices the pattern 1, 2, … and continues it on.
Part 1
Index form
13
The method described here is particularly useful if you need a list of
many numbers that contain a pattern.
4
Click in cell B2 and type =A2^2. Press Return.
The command is understood
whether you type a capital A or a
lower case a.
The instruction you entered tells the computer a formula (=) is entered.
This formula takes the value in cell A2 and raises it to the power (^) of 2.
The result is recorded in the current cell.
5
With the mouse, grab the handle at the bottom, right-hand corner of
this cell and drag this point to B11. Release.
The formula is repeated for each of the values in column A and the
square of each number is calculated in column B. .
6
In cell C2 type =A2^3. Press Return.
Can you see that this instruction commands the computer to calculate
cubes of the numbers in column A?
7
With the mouse, grab the handle at the bottom, right-hand corner of
this cell and drag this point to C11. Release.
You should now have the cubes of numbers from 1 to 10.
8
14
Repeat the process to obtain fourth powers and fifth powers.
NS5.1.1 Rational numbers
Your table should now look like the one displayed below.
To calculate powers in spreadsheets, you need to use ^ in your formula.
Spreadsheets provide you with a powerful tool for performing lots of
calculations quickly.
Check your understanding of these spreadsheet skills by completing the
following exercise.
Go to the exercises section and complete Exercise 1.2 – Spreadsheets and
powers.
Part 1
Index form
15
16
NS5.1.1 Rational numbers
Zero index
So far you have considered numbers where the index power is a
positive integer.
An integer is a whole number. 1, 2, 3 and 15
are integers.
1
are not integers as these are
2
not whole numbers.
But 2.3 and 4
So it is natural to ask if there is a meaning for zero, negative and
fractional indices.
Here you will consider whether there is a meaning for a zero (0) index.
Consider this table:
20
21
22
23
24
25
?
2
4
8
16
32
You know that 21 = 2. Any number raised to the power of 1 equals itself.
22 = 2 × 2 = 4
2 =2×2×2 =8
3
2 4 = 2 × 2 × 2 × 2 = 16
Can you see the pattern? As you move to the right in the table you are
multiplying by 2:
2
4
×2
Part 1
Index form
8
×2
16
×2
32
×2
×2
17
So as you move to the left, you should be dividing by 2:
20
21
22
23
24
25
1
2
4
8
16
32
÷2
÷2
÷2
÷2
÷2
÷2
Using this pattern can you see that 20 = 1?
But what about 30, or 40, or any other number raised to the power
of zero?
Activity – Zero index
Try these.
1
a
Starting at 31 in the table, complete the values to the right.
(Use your calculator to help you.)
30
b
31
32
33
34
35
36
By what value is each number in the table multiplied to get the
next number to its right? ______________________________
c
By what value is each number in the table divided to get the next
number to its left? ____________________________________
2
d
Use this pattern to complete: 30 = ______________________ .
a
Starting at 41 in the table, complete the values to the right.
40
b
41
42
43
44
45
46
By what value is each number in the table multiplied to get the
next number to its right? ______________________________
c
By what value is each number in the table divided to get the next
number to its left? ____________________________________
d
18
Use this pattern to complete: 40 = ______________________ .
NS5.1.1 Rational numbers
Check your response by going to the suggested answers section.
So far you have seen that 20 = 1, 30 = 1, and 40 = 1. You are probably
starting to suspect that any number raised to the power of zero is one,
regardless of what that number is.
Also you should now see that the value you are multiplying or dividing
by is the same as the base of the number you are expressing in
index form.
•
With powers of 2: 20, 21, 22, 23, and so on you are multiplying or
dividing by 2 as you move across the table.
•
With powers of 3: 30, 31, 32, 33, and so on you are multiplying or
dividing by 3 as you move across the table.
•
With powers of 4: 40, 41, 42, 43, and so on you are multiplying or
dividing by 4 as you move across the table.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Show how you can determine the value for 70.
Solution
As 71 = 7, to find 70 you need to divide 7 by 7.
70
71
72
1
7
49
÷7
÷7
÷7
Hence 70 = 1.
Any number divided by itself equals one.
Check that you understand this learning by doing the activity.
Part 1
Index form
19
Activity – Zero index
Try this.
3
Show how you can determine the value for 100.
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
From this you should be able to conclude:
any positive whole number raised to the power
number 0 = 1
of zero always equals one.
Another way to look at the zero index is to consider the pattern in this
list, worked for a base of two.
20 = 1
21 = 1× 2
2
2 = 1× 2 × 2
2 3 = 1× 2 × 2 × 2
4
2 = 1× 2 × 2 × 2 × 2
and so on.
Can you see how to obtain 70 using a list like this?
Don’t confuse the zero index with the degree sign.
Both look very similar: 300 or 30°.
You can usually distinguish which one you are using
from the context.
Go to the exercises section and complete Exercise 1.3 – Zero index.
20
NS5.1.1 Rational numbers
Negative indices
In a previous session you established the meaning of the zero index by
looking at patterns.
You looked at a table like this one,
20
21
22
23
24
25
?
2
4
8
16
32
and arrived at the pattern that to move to the right you multiply by two:
2
4
×2
8
×2
16
×2
32
×2
×2
Therefore, as you move to the left you divide by two:
20
21
22
23
24
25
1
2
4
8
16
32
÷2
÷2
÷2
÷2
÷2
÷2
Using these patterns you established that 20 = 1. In fact, any number
raised to the power of zero equals one.
Here you will consider whether there is a meaning for negative indices.
To do this you need to continue the pattern to the left to complete
this table.
2–4
2–3
2–2
2–1
20
21
22
23
24
25
?
?
?
?
1
2
4
8
16
32
Can you see the indices (powers) follow the sequence on a number line?
–6
Part 1
Index form
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
21
To move to the left in the table, divide by two.
2–4
2–3
2–2
2–1
20
21
1
16
1
8
1
4
1
2
1
2
÷2
÷2
÷2
÷2
÷2
÷2
Use your calculator to check that the fractions are correct.
While you could write the fractions as decimals, leaving them as
fractions makes it easier to visualise.
1
2
1
1
÷2=
4
2
1
1
÷2=
and so on.
8
4
1÷ 2 =
Recall that the
fraction key on
the calculator is
given by ab⁄c .
Remember, when dividing by fractions
you multiply by the reciprocal.
1
For example,
÷2
2
1 2
= ÷
2 1
1 1
= ×
2 2
1
=
4
Now you can complete the table.
2–4
2–3
2–2
2–1
20
21
22
23
24
25
1
16
1
8
1
4
1
2
1
2
4
8
16
32
Now it is time to see if you have understood this learning by doing
the following activity.
22
NS5.1.1 Rational numbers
Activity – Negative indices
Try these.
1
Starting at 30 in the table, complete the values to the right.
(Use your calculator to help you.)
a
3–4
3–3
3–2
3–1
30
31
32
33
34
35
b
By what value is each number in the table multiplied to get the
next number to its right? _______________________________
c
By what value is each number in the table divided to get the next
number to its left? ____________________________________
d
Use this pattern to complete the remainder of the table.
Check your response by going to the suggested answers section.
You should now realise that:
•
a positive integer raised to the power of a positive index gives a
value that is greater than one
•
a positive integer raised to the power of zero equals one
•
a positive integer raised to the power of a negative index gives a
value that lies between zero and one.
One thing you should keep in mind:
positive numbers written with negative
indices are not negative numbers. They
are still positive numbers.
1
For instance 2−1 =
2
and a half is a positive number.
Part 1
Index form
23
Further negative indices
In the previous section you established that negative indices
produce fractions.
Here is the table produced for some powers of 2:
2–5
2–4
2–3
2–2
2–1
20
21
22
23
24
25
1
32
1
16
1
8
1
4
1
2
1
2
4
8
16
32
Look at the values for 21 and 2–1.
What do you notice?
Look at the values for 22 and 2–2.
What do you notice?
Is the same pattern true for 23 and 2–3?
What about 24 and 2–4?
You should be able to see that if 21 = 2, then 2–1 =
Likewise, 22 = 4 and so 2–2 =
1
.
2
1
.
4
Activity – Further negative indices
Try these.
2
The table shows some powers of 3.
3–5
3–4
3–3
3–2
3–1
30
31
32
33
34
35
1
243
1
81
1
27
1
9
1
3
1
3
9
27
81
243
a
24
Is 3−1 =
1
? ________________________________________
31
NS5.1.1 Rational numbers
3
b
Is 3−2 =
1
? ________________________________________
32
c
Is 3−3 =
1
? _________________________________________
33
d
If 3−10 =
1
, what is the value of ∆ ? _____________________
3∆
e
If 3∆ =
1
, what is the value of ∆ ? ______________________
325
Given that 75 = 16 807, write the fraction represented by 7 −5 ?
_______________________________________________________
Check your response by going to the suggested answers section.
By now you should realise that you can write the reciprocals of powers
using negative indices.
You simply turn the term ‘upside down’ and remove the
minus sign.
So, for instance, 3–4 gives the same answer as
And 2–7 gives the same result as
a−b =
1
ab
1
.
34
1
.
27
Go to the exercises section and complete Exercise 1.4 – Negative indices.
Part 1
Index form
25
26
NS5.1.1 Rational numbers
Suggested answers – Part 1
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz – Part 1
1
a
196
b
125
c
64
2
a
49
b
256
c
1225
3
a
361
b
1728
c
2197
4
a
6
b
3
c
3
Activity – Index form
1
a
base, 3; power, 4
2
a
2809
3
a
15 625
4
a
81
Activity – Zero index
1
a
30
b
2
3
32
33
34
35
36
3
9
27
81
243
729
c
3
d
1
a
40
b
3
31
4
41
42
43
44
45
46
4
16
64
256
1024
4096
c
4
d
1
Since 101 = 10, you need to divide this answer by 10 to obtain 100.
Now 10 ÷ 10 = 1, and so 100 = 1.
Part 1
Index form
27
Activity – Negative indices
1
b
3
c
3
3–4
3–3
3–2
3–1
30
31
32
33
34
35
1
81
1
27
1
9
1
3
1
3
9
27
81
243
2
3
28
a, d (below)
a
e
yes
∆ = –25
b
yes
c
yes
d
∆ = 10
1
16 807
NS5.1.1 Rational numbers
Exercises – Part 1
Exercises 1.1 to 1.4
Name
___________________________
Teacher
___________________________
Exercise 1.1 – Index form
1
Explain what is wrong with Georgie’s statement.
Since 22 means 2 × 2, then 23 must
mean 2 × 3.
_______________________________________________________
_______________________________________________________
_______________________________________________________
2
For a number written as 183, which number is the base and which is
the index? ______________________________________________
3
Without using a calculator, show how you would find the answer to:
a
43
________________________________________________________________________
___________________________________________________
4
Part 1
Index form
Use your calculator to evaluate the following.
a
162 = ______________________________________________
b
283 = ______________________________________________
29
5
Harry made the following comment. Explain how you could show
whether or not he is correct.
I figure that 23 must give me the same
answer as 32. Am I correct?
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
6
Which value is larger: 113 or 312? __________________________
By how much is it larger? _________________________________
7
2
Ivan and Ali were asked to find the answer to 2 × 5 .
These are their responses.
2 × 5 is 10. And 102 is 100.
So 2 × 52 must be 100.
2 × 52 = 102
= 100
We first square 5 to get 25.
Now 2 × 25 is 50.
2 × 52 = 2 × 25
= 50
Who is correct? (You may use your calculator to help you with this
question.) ______________________________________________
8
30
Use your calculator to evaluate the following.
a
75 = _______________________________________________
b
210 = ______________________________________________
NS5.1.1 Rational numbers
9
You can use your calculator to calculate 152 by either multiplying
15 × 15 or using the square key, x2 . Describe how you can use the
power key to evaluate 152.
_______________________________________________________
10 Describe three ways to calculate 63 on the calculator.
a
___________________________________________________
b
___________________________________________________
c
___________________________________________________
Which method requires fewer key strokes? ____________________
11 How can you most efficiently find 4 × 4 × 4 × 4 × 4 × 4 on the
calculator? Write down the answer.___________________________
______________________________________________________
12 Use your calculator to find the following.
4
a
3× 2 = ____________________________________________
b
(3 × 2)4 = __________________________________________
c
3 × 2 = ____________________________________________
d
3 × 2 = ___________________________________________
4
4
4
13 Use your power key to calculate the following.
a
31 = ________________________________________________
b
51 = ________________________________________________
c
121 = _______________________________________________
What is the value when any number is raised to the power of 1?
_______________________________________________________
Part 1
Index form
31
14 Complete the following table.
21
22
4
23
24
25
26
27
28
29
210
32
What pattern do you notice? _______________________________
32
NS5.1.1 Rational numbers
Exercise 1.2 – Spreadsheets and powers
Use the skills you have developed with spreadsheets to produce a table
showing the powers from 1 to 6 for numbers up to 20.
Your table should be similar to this.
Make a copy of your table and send it to your teacher.
Part 1
Index form
33
Exercise 1.3 – Zero index
1
a
Starting at 51 in the table, complete the values to the right.
50
2
3
51
52
53
54
55
56
b
By what value is each number in the table multiplied to get the
next number to its right? ______________________________
c
By what value is each number in the table divided to get the next
number to its left? ____________________________________
d
Use this pattern to complete: 50 = ______________________ .
Write down the value of:
a
80 = _______________________________________________
b
10 = _______________________________________________
Calculate the following. You may use a calculator if you wish, but
try working them out manually first.
a
0
3× 12 = ___________________________________________
___________________________________________________
b
3 × 12 = ___________________________________________
0
___________________________________________________
c
30 × 12 0 = ___________________________________________
___________________________________________________
d
(3 × 12)0 = __________________________________________
___________________________________________________
34
NS5.1.1 Rational numbers
4
Give the answer to each of the following, leaving it in index form.
Example: 6 0 × 6 8 = 1× 6 8
= 68
a
12 × 12 = __________________________________________
5
0
___________________________________________________
b
15 × 15 × 15 = ______________________________________
0
0
1
___________________________________________________
Part 1
Index form
35
Exercise 1.4 – Negative indices
1
Starting at 40 in the table, complete the values to the right.
(Use your calculator to help you.)
a
4–4
4–3
4–2
4–1
40
41
42
43
44
45
b
By what value is each number in the table multiplied to get the
next number to its right? ______________________________
c
By what value is each number in the table divided to get the next
number to its left? ____________________________________
d
Use this pattern to complete the remainder of the table.
Leave answers less than one as fractions.
2
Complete these tables. (You may use your calculator to help you,
giving answers less than 1 as fractions.)
a
5–4
5–3
5–2
5–1
50
51
52
53
54
55
10–4
10–3
10–2
10–1
100
101
102
103
104
105
b
3
4
Write the fractions represented by the following, then use your
calculator to check if you are correct.
a
6–1 = _______________________________________________
b
12–1 = ______________________________________________
a
Given 65 = 7776, write down the fraction shown by 6 −5 .
___________________________________________________
b
36
If 8 −3 =
1
, what is 83? ______________________________
512
NS5.1.1 Rational numbers
5
a
Given 81 = 8, what fraction is 8 −1 ? _______________________
b
What is the effect of raising a number to the power of –1?
___________________________________________________
Part 1
Index form
37