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Contents
Introduction
3
The international system of units (SI units)
4
Fundamental SI units
4
Derived SI units
5
Scientific notation
6
Calculations using scientific notation
8
Using a scientific calculator
9
Advantages of scientific notation
10
Engineering notation
12
Multiples and sub-multiples
13
Significant figures
15
Rules to determine significant figures
15
Significant figures in calculations
16
Transposition of equations
18
Operating on equations
18
Rules for transposition
20
Summary
27
Answers
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Introduction
To understand the basic principles of electricity it is essential that you know
the units and quantities used in measuring electrical quantities such as
voltage, current, resistance and charge. You will also need to know the units
for other common physical quantities, including length, time, power, energy,
and so on.
To maintain the quality of goods and services in commerce and engineering,
it is important to be able to measure various physical quantities including
electrical quantities using standardised units. This section considers the units
of measurement and mathematical processes which you will need for this
and the remaining modules of the course.
At the end of this section, you should be able to:

identify the basic units of measurements

define the SI derived units for force, pressure, energy/work, temperature
and power

convert units to multiple and sub-multiple units

perform basic calculations of electrical and related mechanical quantities
given in any combination of units, multiple units or sub-multiple units

transpose a given equation for any variable in the equation.
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The international system of units
(SI units)
In 1960 the international authority on standards agreed to adopt the
Systeme International d’Unites, or the International System of Units,
the abbreviation of which is SI in all languages.
In your study of electrical principles you will find it necessary to know
the correct units to assign to electrical and other physical quantities.
Fundamental SI units
The SI units are divided into fundamental units and derived units.
Fundamental units are defined in terms of accurately measurable natural
phenomena, or in the case of the kilogram, the mass of a physical object
held at the bureau of weights and measures in Paris. Here for example is
how the metre is defined:
“The unit of length equal to the length of the path travelled by light in a
vacuum during the time interval of 1/299 792 458 of a second.”
A list of the fundamental (basic) SI units is provided in Table 1.
Table 1: Basic SI units
Physical unit
Unit name
Quantity symbol
length
metre
m
mass
kilogram
kg
time
second
s
electric current
ampere
A
temperature
kelvin
K
amount of substance
mole
mol
Note that the symbol for second is a lower case ‘s’, as distinct from an
uppercase S which stands for ‘Siemens’, the unit of conductivity.
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Derived SI units
The SI system has another much larger group of units termed derived SI
units. The derived are so called because they can be derived from the
fundamental units in various combinations.
For example, the SI derived unit of force is the newton (symbol, N) and
force is defined by the equation:
Force = mass × acceleration
Since the newton (N) is defined as that force that will give a mass of
1 kilogram (kg) an acceleration of 1 metre per second in each second (m/s2),
it follows that:
1 N = 1 kg m/s2
So you see that the unit of force can be derived for the fundamental units of
mass, length, and time.
Here is another example. Electrical current represents the rate of flow of
electric charge; therefore charge can be defined in terms of the fundamental
unit ampere and second.
1 coulomb = 1 ampere × 1 second
In other words, the coulomb is that quantity of charge that flows when one
ampere is maintained for one second.
The derived units important to you at this stage are listed in Table 2.
Memorise these units and their letter symbols.
Table 2: Some derived SI units
Physical
quantity
Quantity
symbol
Unit name
Unit symbol
force
F
newton
N
energy and work
W
joule
J
power
P
watt
W
charge
Q
coulomb
C
potential
difference and
voltage
V
volt
V
resistance
R
ohm
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Scientific notation
Scientific notation is a way of writing values in a standard way. This
notation allows us to represent very large or very small quantities without
using lots of zeros. For example, the large number:
123 000 000 000 000
can be written as:
1.23 × 1014
and the small number:
0.000 000 000 123
can be written as:
1.23 × 10-10
When using scientific notation, you must write the number in the form:
F  10e
where: F (known as the ‘coefficient’ or ‘mantissa’) is a number
between in the range between 1 and 10, but not including
10. That is: 1 ≤ F < 10.
The letter ‘e’ is the exponent, or power of 10.
The power (or exponent) is the number of places you move the decimal
point when converting to a power of 10. Moving the decimal point to the
left lowers the numerical value of the number (this corresponds to division).
In order to keep the value of the number the same you must raise the
positive power of the base 10.
Example 1
Convert 456 to scientific notation.
Step 1
First show the decimal point in the number.
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In this case, 456 is written as 456.0
Step 2
Move the decimal point until you get a number between 1 and 10. In this
case, the number 456 is larger than we want, so we must move the decimal
point to the left twice:
4
2

1

5
6
. 0
Step 3
Rewrite the number in scientific notation, noting these points:

The number is written at the front, with the new position of the
decimal point (that is, 4.56)

Then write × 10e , where the exponent e is the number of times the
decimal point was moved. If the movement was to the left, the
exponent is positive, if to the right, the exponent is negative. In this
case, we move the point twice to the left, so we have “× 102”
Therefore in this case, the number is 4.56 × 102
Let’s check this result by converting back to a ‘normal’ number.
4.56  102  4.56  10  10
 4.56  100
 456.0
 456
How can you remember if the exponent is positive of negative? Well, when
the decimal point moves to the left, you are making the number smaller.
Therefore, the exponent must bet bigger to compensate, so we have the
same value at each movement. This is illustrated below:
456 1
456 10

0
45.6 10
 45.6 10
1
 4.56  100

4.56  102
Example 2
0.0056  5.6 103
In this example, to obtain 5.6 from 0.0056, move the decimal point
three places to the right (0.0056). This means that the power of 10 required
to keep the value of the number the same is negative 3 (i.e. 10–3).
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1
, that is to say:
1000
5.6
5.6 103 
1000
 0.0056
Remember that 103 
Some more examples are shown in Table 3.
Table 3: Example of scientific notation
Decimal notation
Scientific notation
89
8.9 × 101
586
5.86 × 102
7500
7.5 × 103
89256
8.9256 × 104
0.08
8 × 10–2
0.00623
6.23 × 10–3
0.0008923
8.923 × 10–4
Calculations using scientific notation
When performing calculations using scientific notation you need to follow
these rules:
1
To multiply factors with base 10, add their powers.
For example:
(a) 22 000  3000  (2.2  10 4 )  (3  103 )
 2.2  3 1043
 6.6 107
(b) 0.02  0.003  (2  102 )  (3  103 )
 2  3 102 ( 3)
 6 105
2
To divide factors with base 10, subtract their powers. For example:
(a)
32 000 3.2 104

400
4  102
3.2

 104 2
4
 0.8 102
 8 101
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(b)
3
4000 4 103

0.02 2  102
4
  103( 2)
2
 2  105
To add or subtract numbers expressed in powers of 10, the powers of
10 must be the same. The coefficients are then added or subtracted
keeping the power of 10 the same. For example:
(a)
(b)
(c)
8 10   1.5 10 
3
3
8 10   1.5 10 
3
3
 7.5 10   9.2 10 
2
3

8  1.5  103

9.5 103

8  1.5  103

6.5 103

 7.5 10   92 10 

 7.5  92  102

99.5 102

9.95 103
2
2
Example 3
Evaluate the following using scientific notation.
56.8  403  82 600
75.8  37 000  0.04
Solution
 5.68 10    4.03 10   8.26 10   5.68  4.03  8.26  10
7.58  3.7  4
10
 7.58 10    3.7 10    4 10 
1
2
4
4
7
2
3
 1.69 107 3
 1.69 104
Using a scientific calculator
In using a scientific calculator to perform calculations involving scientific
and engineering notation it is essential that you understand the way the
calculator treats a number such as 104 (= 10 000).
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The calculator does not require the entry of the ‘10’, it assumes that when
you enter the exponent (4), it is always to a base of 10.
Therefore to enter the number 104 the following keys are pressed:
1
EXP
4
Examples
(a) (2 × 104) × (3 × 103) =?
Enter:
2
EXP

4
3
EXP
3
=
Answer: 6 000 000
(b)
15 103
?
4.3 106
Enter:
1
5
EXP
3


4
3
EXP
-
-
=
6
=
Answer: 348837209.3
(c) (4 (× 10–7) × 5 =?
Enter:
4



1
EXP
7
Answer: 6.283185307–6
Advantages of scientific notation
You will see from the previous calculations that the use of scientific
notation simplifies the writing of very large and very small numbers, and
numerical calculations using scientific notation are less cumbersome and
less prone to errors.
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Activity 1
1
Rewrite the following numbers in scientific notation:
(a) 800
__________________________________________________________________
(b) 8940
__________________________________________________________________
(c) 0.03
__________________________________________________________________
(d) 82 000 00
__________________________________________________________________
(e) 0.00009
__________________________________________________________________
2
Evaluate the following using scientific notation:
3800  0.005  0.000001
430 000 000  73
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Check your answers with those given at the end of the section.
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Engineering notation
Engineering notation is the same basic idea as scientific notation. The
difference is that the exponents used in engineering notation are multiples of
three.
A number written in engineering notation will take the form:
G × 10q.
Where:
The factor G is equal to or greater than 1 but less than 1000:
(1 < G < 1000).
The power (or exponent) q would be a multiple of 3 (that is, it can
have values of … –9, –6, –3, 0, 3, 6, 9 …).
For example:
532 000 in engineering notation would be:
532 × 103
and 0.0000245 A in engineering notation would be:
24.5 × 10–6 A
The advantage of engineering notation is that we need to deal with fewer
multiples and they are all factors of one thousand. This means that numbers
can be more easily compared, and it is a very easy to express the number
with an SI prefix such as ‘kilo’, ‘mega’ and so on.
For example
103 = 1000 (one thousand) = ‘kilo’
106 = 1 000 000 (one million) = ‘mega’
10-6 = 0.001 (one thousandth) = ‘micro’
This is discussed in detail in the next section.
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Multiples and sub-multiples
In the electrical industry, you will find that there are a variety of values
used, ranging from very large to very small. For example:

voltage is distributed around the state at 66 000 V or 132 000 V or even
higher

current, especially in electronic circuits, may be in the range of
five thousandths of an ampere to an even lower value of, say,
ten millionths of an ampere.

capacitances used in electronics may be in the order of picofarads (pF)
or 10-12 farads.
Metric prefixes can be used to replace powers of ten written in engineering
notation. The metric prefix is written ahead of the unit symbol.
For example, the prefix kilo, which means 1000, can be used instead of
‘thousands of volts’. Lets say we have a voltage V equal to 132 000 volts.
This can be expressed in any of the following ways:
V  132000 V
=1.32  103 V
=1.33 kilovolt
=1.33 kV
For engineering calculations, we normally use either engineering notation
(the second format), or the abbreviated SI unit prefix (the last format).
Always express your final answers using this last form (eg 1.33 kV).
Here’s another example. The prefix milli means one-thousandth, or 0.001.
Thus a current of 0.005 ampere can be expressed in any of the following
ways.
I  0.005A
 5.0 10 3 A
= 5 milliampere
 5 mA
If the corresponding power of 10 is greater than 0 (ie positive), the prefix is
known as a decimal multiple. If the corresponding power of 10 is less than
0 (ie negative), the prefix is known as a decimal sub-multiple. A list of
common multiples and sub-multiples is shown in Table 4.
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Table 4: Common multiples and sub-multiples
Factor
Prefix
Symbol
Example
109
giga
G
1 000 000 000 watt = 1 GW
6
meg
M
1 000 000 watt = 1 MW
3
kilo
K
66 000 volt = 66kV
–3
milli
m
0.012 ampere = 12 mA
–6
micro
µ
0.000005 ampere = 5 µA
–9
nano
n
0.000000018 farad = 18 nF
10
10
10
10
10
Activity 2
Express the following numbers in engineering notation:
1
570 000 m
_____________________________________________________________________
2
0.000273 A
_____________________________________________________________________
3
0.073 V
_____________________________________________________________________
4
1.2 × 108 W
_____________________________________________________________________
Check your answers with those given at the end of the section.
If you have Jenneson, refer to Section 1.3.4 for more information and
examples.
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Significant figures
The accuracy of every measurement is limited by the accuracy of the
measuring instrument. No measurement is ever exact.
Consider that the length of an object recorded as 18.6 cm. By convention,
this means that the length of the object was measured to the nearest tenth of
a centimetre and that its value lies between 18.55 cm and 18.65 cm.
Therefore, in engineering calculations, we do not simply copy down all the
digits shown on our calculator. This would give the impression that our
accuracy is much greater than we actually have.
We therefore limit the number of significant figures in our written
measurements and results. The value of 18.6 cm represents three significant
figures while the value of 18.60 cm represents four significant figures.
Rules to determine significant figures
1 All non-zero figures are significant.
2
For example, 264.89 has five significant figures.
All zeros between two non-zero integers are significant.
3
For example, 2060.05 has six significant figures.
Unless otherwise stated all zeros to the left of an ‘understood’ decimal
point, but to the right of a non-zero integer are not significant.
4
For example, 604 000 has three significant figures.
All zeros to the right of a decimal point but to the left of a non-zero
integer are not significant.
5
For example, 0.000852 has three significant figures.
All zeros to the right of a decimal point and to the right of a non-zero
integer are significant.
For example:
– 0.07080 has four significant figures
– 50.00 has four significant figures.
Note that sometimes we may want to say ‘1000 V’, where the first two zeros
are significant. This is another advantage of engineering notation – by
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writing ‘1.00 kV’, there is no ambiguity about the accuracy of the quantity –
it has three significant figures.
Significant figures in calculations
The rules to determine significant figures in calculations are set out below.

When adding or subtracting any set of numbers the result should
contain the same number of significant figures as the least accurate
component. For example, in the following addition:
1.052 cm (four significant figures)
8.24 cm (three significant figures)
28.1 cm (three significant figures)
37.392 cm
The answer is given as 37.4 cm (three significant figures).

When multiplying or dividing any set of numbers, the result should
contain the same number of significant figures as the least accurate
component. For example:
 0.047   160.0   0.0808
93.5
0.047 is the least accurate with two significant figures.
Answer: 0.081 with the two significant figures.
Rules for rounding off

When the digit dropped is less than 5, the last digit remains unchanged.
For example,
3.442 = 3.44 after rounding off at three significant figures.

When the digit dropped is equal to or greater than 5, one is added to the
last digit.
For example:
3.446 = 3.45 rounding off at three significant figures.
Intermediate results
If you have a string of separate calculations where each uses the previous
result, it is a good idea to retain more significant figures for the intermediate
numbers, because otherwise you will have an accumulation of rounding
errors. But when you express the final result, you should not write more
significant figures than is warranted by the accuracy of the initial data.
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Activity 3
Complete the following sentences by inserting the appropriate word, figure or symbol in
the spaces provided.
1
2000 mV equals ________ V.
2
47 × 104 m equals ________ km.
3
2 200 000 Ω equals ________ M.
4
11 × 103 volt equals ______ kV.
5
There are ________ ms in 0.625 s.
6
3.5 × 10–3 ampere equals 3.5_________.
Check your answers with those given at the end of the section.
When manipulating equations if any operation (addition, subtraction,
multiplication or division) is carried out upon one side of the equation and is
then also carried out upon the other, this will not affect the equality.
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Transposition of equations
A common operation in electrical calculations is the manipulation of
equations. For example, if we have V = IR, it may be necessary to express R
in terms of the other variables.
This process is called transposition.
Operating on equations
An equation is composed of expressions on either side of an equals sign and
has the following properties:

adding the same number to both sides will not destroy the equality
E 3  4
E 33  43
E 7

subtracting the same number from both sides will not destroy the
equality
P27
P22  72
P5

multiplying both sides by the same number will not destroy the equality
T
7
3
T
3  73
3
T  21

dividing both sides by the same number will not destroy the equality
3Z  7
3Z 15

3
3
Z 5
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Activity 4
Solve the following equations (find the value of the letter variable) by applying operations
to both sides of the equation.
1
4x – 5 = 3
_____________________________________________________________________
_____________________________________________________________________
2
w + 5 = 12
_____________________________________________________________________
_____________________________________________________________________
3
z
2
9
_____________________________________________________________________
_____________________________________________________________________
4
12k = 36
_____________________________________________________________________
_____________________________________________________________________
5
x
 3  15
4
_____________________________________________________________________
_____________________________________________________________________
Check your answers with those given at the end of the section.
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Rules for transposition
The following steps, combined with the basic rules of equality, will make
transposing equations a reasonably simple and logical process.
Even if an equation is complex, the first thing to remember is to carry out
the steps in the correct sequence. The acronym BODMAS (by, of, divide,
multiply, add, subtract) gives you the sequence to use when solving
complex problems. When carrying out transpositions, you reverse BODMAS
to SAMDOB and this gives you the correct sequence.
Step 1
Change the equation around so that the new subject (eg R) is on the left side,
for example:
from V  IR
to IR  V
Step 2
At each step of the SAMDOB sequence, you carry out the opposite operation
to both sides the equation. To isolate R divide both sides of the equation by
I, that is:
IR  V
through
IR V

I
I
V
to R 
I
Example 4
E2
Make E the subject in P 
R
Solution
Take the expression containing the subject to the left side of the equation.
E2
P
R
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Apply SAMDOB. There is no subtraction, no addition and no multiplication.
There is, however, division so multiply both sides by R. The equation now
becomes:
E2  R
 PR
R
The Rs on the left side of the equation cancel out. The equation now becomes:
E 2  PR
Take the square root of each side. The equation now becomes:
E 2  PR
The square root and the square cancel out. The equation now becomes:
E  PR
Activity 5
Transpose the expressions below to give expressions with the new subject as shown
(boxed).
1
R
V
I
_____________________________________________________________________
_____________________________________________________________________
2
A   R2
(Because you are asked to find R2 and not R, in this case R2 is treated as a single
quantity.)
_____________________________________________________________________
_____________________________________________________________________
3
BC D  K
_____________________________________________________________________
_____________________________________________________________________
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4
A
B
 Y (Be careful what you do first here!)
4
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
5
D C
L
5
_____________________________________________________________________
_____________________________________________________________________
Check your answers with those given at the end of the section.
Check your progress
1
Measure the length of a standard length of conduit.
(a) Write down the length in scientific format.
___________________________________________________________________
(b) Write down the length in millimetres (mm).
___________________________________________________________________
(c) Write down the length in mm in scientific format.
___________________________________________________________________
2
Convert the following to scientific notation.
(a) 17 540
___________________________________________________________________
(b) 1.55
___________________________________________________________________
(c) 0.0067
___________________________________________________________________
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(d) 33
___________________________________________________________________
3
Write the unit and symbol for each of the following physical quantities.
(a) length
__________________________________________________________________
(b) temperature
__________________________________________________________________
(c) luminous intensity
__________________________________________________________________
(d) mass
__________________________________________________________________
4
Give the physical quantities that each of the following units represent.
(a) joule
__________________________________________________________________
(b) newton
__________________________________________________________________
(c) watt
__________________________________________________________________
(d) ohm
__________________________________________________________________
5
Convert the following quantities to their base units.
(a) 220 k
__________________________________________________________________
(b) 3.6 mV
__________________________________________________________________
(c) 15 mA
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__________________________________________________________________
(d) 2.75 MW
__________________________________________________________________
6
Give the power of ten for each of the following prefixes.
(a) Mega
__________________________________________________________________
(b) kilo
__________________________________________________________________
(c) milli
__________________________________________________________________
(d) micro
__________________________________________________________________
7
Write the following in engineering notation.
(a) 220 000
__________________________________________________________________
(b) 0.0036
__________________________________________________________________
(c) 0.000 000 16
__________________________________________________________________
(d) 27 500 000
__________________________________________________________________
8
Using scientific notation round off the following to two significant figures.
(a) 235
__________________________________________________________________
(b) 0.000375
24
EEE042A: Appendix A Apply mathematical processes
 NSW DET 2017 2006/060/06/2017 LRR 3677
__________________________________________________________________
(c) 25.67
__________________________________________________________________
(d) 2.48
__________________________________________________________________
9
Use a scientific calculator to calculate the following.
(a) (3 × 103) × 600
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(b) (2.7 × 10–3) + (8.3 × 10–4)
__________________________________________________________________
(c) (6.5 × 10–2) (4.25 × 10–6)
__________________________________________________________________
(d) 106 – 104
__________________________________________________________________
10 Perform the following calculations and express the answers in scientific notation.
(a) 3000 × 250
__________________________________________________________________
(b) 0.0058 × 0.66
__________________________________________________________________
(c) 200  5 000 000
__________________________________________________________________
(d) 0.5 + 0.007 + 1.92
__________________________________________________________________
11 Perform the following calculations and express the answers in engineering notation.
(a) 3000 × 250
__________________________________________________________________
(b) 0.0058 × 0.66
__________________________________________________________________
(c) 200  5 000 000
__________________________________________________________________
(d) 0.5 + 0.007 + 1.92
__________________________________________________________________
Check your answers with those given at the end of this section.
26
EEE042A: Appendix A Apply mathematical processes
 NSW DET 2017 2006/060/06/2017 LRR 3677
Summary

The SI system of units is a metric system and is a worldwide standard
with very few exceptions.

There are six fundamental units in the SI system.

From these few fundamental units there are many derived units used in
science, engineering and commerce.

Use engineering notation or SI unit prefixes to express values in your
calculations.

Results of calculations should be expressed with a limited number of
significant figures, based upon the accuracy of the known quantities.
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Answers
1
(a) 8.00 × 102
(b) 8.94 × 103
(c) 3.5 × 10–2
(d) 8.2 × 107
(e) 9.2 × 10–5
2
 3.8 10    5 10   110   3.8  5 10
4.3  7.3 10
 4.3 10    7.3 10 
3
3
8
6
6
1
9
 0.605 1015
 6.05 1016
Activity 2
1
570 × 103 m or 570 km
2
273 × 10–6A or 273 µA
3
73 × 10–3 V or 73 mV
4
120 × 106 W or 120 MW
Activity 3
28
1
2V
2
470 km
3
2.2 M
4
11 kV
5
625 ms
6
mA.
EEE042A: Appendix A Apply mathematical processes
 NSW DET 2017 2006/060/06/2017 LRR 3677
Activity 4
1
4x  5  3
4x  3  5
4x  8
x2
2
w  5  12
w  12  5
w7
3
4
z
 2
9
z
9  29
9
z
 18
12k  36
k 3
5
x
3
4
x
4
x
4
x
4
4
x
 15
 15  3
 12
 12  4
 48
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Activity 5
1
V
I
IR  V
R
I
2
V
R
A   R2
R2 
3
A

BC  D  K
4
BC
 KD
B

A
B
4
B
4
B
5
KD
C
 Y
 YA
 4 Y  A 
DC
5
DC
 5L
C
 5L  D
 L
Check your progress
2
1 (a)
(b)
(c)
(d)
(a) 1.754 × 101
3
(b)
(c)
(d)
(a)
1.55 × 100
6.7 × 10–3
3.3 × 101
m
(b) K
(c) cd
30
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4
(d) kg
(a) work
(b) force
(c) power
(d) resistance
5
(a) 220 000 
6
(b) 0.0036 V
(c) 0.015 A
(d) 2 750 000 W
(a) 106
7
(b) 103
(c) 10–3
(d) 10–6
(a) 220 × 103
(b) 3.6 × 10–3
8
(c) 160 × 10–9
(d) 27 × 106
(a) 2.3 × 102
9
(b) 3.7 × 10–4
(c) 2.6 × 101
(d) 2.5 × 100
(a) 1 800 000
(b) 3.53 × 10–3
(c) 15294
(d) 990 000
10 (a) 7.5 × 105
(b) 3.828 × 10–3
(c) 4 × 10–6
(d) 2.427
11 (a) 750 × 103
(b) 3.828 × 10–3
(c) 40 × 10–6
(d) 2.427
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