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TERMINOLOGYTERMINOLOGTERMINOLOGY
1Basic Arithmetic
Absolute value: The distance o a number rom zero onthe number line. Hence it is the magnitude or value o anumber without the sign
Directed numbers: The set o integers or wholenumbers 3, 2, 1, 0, 1, 2, 3,f f- - -
Exponent: Power or index o a number. For example 23has a base number o 2 and an exponent o 3
Index: The power o a base number showing howmany times this number is multiplied by itsel
e.g. 2 2 2 2.3 # #= The index is 3
Indices: More than one index (plural)
Recurring decimal: A repeating decimal that does notterminate e.g. 0.777777 is a recurring decimal that canbe written as a raction. More than one digit can recure.g. 0.14141414 ...
Scientific notation: Sometimes called standard notation.A standard orm to write very large or very small numbersas a product o a number between 1 and 10 and a powero 10 e.g. 765 000 000 is 7.65 108# in scientifc notation
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3Chapter 1 Basic Arithmetic
INTRODUCTION
THIS CHAPTER GIVES A review o basic arithmetic skills, including knowing the
correct order o operations, rounding o, and working with ractions, decimals
and percentages. Work on signifcant fgures, scientifc notation and indices isalso included, as are the concepts o absolute values. Basic calculator skills are
also covered in this chapter.
Real Numbers
Types of numbers
Irrational
numbers
Unreal or imaginary
numbers
Integers
Rational
numbers
Real numbers
Integers are whole numbers that may be positive, negative or zero.
e.g. , , ,4 7 0 11- -
Rational numbers can be written in the orm o a ractionb
a
where a and b are integers, .b 0! e.g. , . , . ,143
3 7 0 5 5
-
Irrational numbers cannot be written in the orm o a ractionb
a(that
is, they are not rational) e.g. ,2 r
EXAMPLE
Which o these numbers are rational and which are irrational?
, . , , , , .3 1 353
94
2 65 r
-
Solution
34
andr
are irrational as they cannot be written as ractions (r is irrational).
. , .1 3 131
91
32 65 2
20
13and
= = - = - so they are all rational.
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4 Maths In Focus Mathematics Preliminary Course
Order of operations
1. Brackets: do calculations inside grouping symbols first. (For example,
a fraction line, square root sign or absolute value sign can act as a
grouping symbol.)2. Multiply or divide from left to right.
3. Add or subtract from left to right.
EXAMPLE
Evaluate .40 3 5 4- +] g
Solution
40 3(5 4) 40 3 9
40 27
13
#- + = -
= -
=
PROBLEM
What is wrong with this calculation?
Evaluate1 2
19 4
+
-
- +Press19 4 1 2 19 4 1 2'+- ='17
What is the correct answer?
BRACKETS KEYS
Use ( and ) to open and close brackets. Always use them in pairs.
For example, to evaluate 40 5 43- +] gpress 40 3 ( 5 4 )
31
#
=
- + =
To evaluate1.69 2.77
5.67 3.49
+
-correct to 1 decimal place
press ( ( 5.67 3.49 ) ( 1.69 2.77 ) )': - + =
0.7
correct to 1decimal place
=
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5Chapter 1 Basic Arithmetic
Rounding off
Rounding o is oten done in everyday lie. A quick look at a newspaper will
give plenty o examples. For example in the sports section, a newspaper may
report that 50 000 ans attended a ootball match.An accurate number is not always necessary. There may have been exactly
49 976 people at the ootball game, but 50 000 gives an idea o the size o the
crowd.
EXAMPLES
1. Round o 24 629 to the nearest thousand.
Solution
This number is between 24 000 and 25 000, but it is closer to 25 000.
24 629 25 000` = to the nearest thousand
CONTINUED
MEMORY KEYS
Use STO to store a number in memory.
There are several memories that you can use at the same timeany letter rom
A to F, or X, Y and M on the keypad.To store the number 50 in, say, A press 50 STO A
To recall this number, press ALPHA A =
To clear all memories press SHIFT CLR
X-1 KEY
Use this key to fnd the reciprocal o x. For example, to evaluate
7.6 2.1
1
#-
0.063= -
press ( ( ) 7.6 2.1 ) x1
#- =-
(correct to 3 decimalplaces)
Different calculators use
different keys so check
the instructions for your
calculator.
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6 Maths In Focus Mathematics Preliminary Course
2. Write 850 to the nearest hundred.
Solution
This number is exactly halway between 800 and 900. When a number is
halway, we round it o to the larger number.
850 900` = to the nearest hundred
In this course you will need to round o decimals, especially when using
trigonometry or logarithms.
To round a number o to a certain number o decimal places, look at the
next digit to the right. I this digit is 5 or more, add 1 to the digit beore it and
drop all the other digits ater it. I the digit to the right is less than 5, leave thedigit beore it and drop all the digits to the right.
EXAMPLES
1. Round o 0.6825371 correct to 1 decimal place.
Solution
.. .
0 68253710 6825371 0 7 correct to1 decimal place` =
#
2. Round o 0.6825371 correct to 2 decimal places.
Solution
.
. .
0 6825371
0 6825371 0 68 correct to 2 decimal places` =#
3. Evaluate . .3 56 2 1'
correct to 2 decimal places.
Solution
. . . 5
.
3 56 2 1 1 69 238095
1 70 correct to 2 decimal places
' =
=
#
Drop off the 2 and all digits
to the right as 2 is smaller
than 5.
Add 1 to the 6 as the 8 is
greater than 5.
Check this on your
calculator. Add 1 to the
69 as 5 is too large to just
drop off.
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7Chapter 1 Basic Arithmetic
While using a fxed number o decimal places on the display, the
calculator still keeps track internally o the ull number o decimal places.
EXAMPLE
Calculate . . . .3 25 1 72 5 97 7 32#' + correct to 2 decimal places.
Solution
. . . . . . .
. .
.
3 25 1 72 5 97 7 32 1 889534884 5 97 7 32
11 28052326 7 32
18 60052326
18.60 correct to 2 decimal places
' # #+ = +
= +
=
=
I the FIX key is set to 2 decimal places, then the display will show
2 decimal places at each step.
3.25 1.72 5.97 7.32 1.89 5.97 7.32
. .
.
11 28 7 32
18 60
' # #+ = +
= +
=
I you then set the calculator back to normal, the display will show theull answer o 18.60052326.
Dont round off at
each step of a series of
calculations.
The calculator does not round o at each step. I it did, the answer might
not be as accurate. This is an important point, since some students round
o each step in calculations and then wonder why they do not get the same
answer as other students and the textbook.
1.1 Exercises
FIX KEY
Use MODE or SET UP to fx the number o decimal places (see the
instructions or your calculator). This will cause all answers to have a fxed number
o decimal places until the calculator is turned o or switched back to normal.
1. State which numbers are rational
and which are irrational.
(a) 169
0.546(b)
(c) 17-
(d)3
r
(e) .0 34
() 218
(g) 2 2
(h)271
17.4%(i)
(j)5
1
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8 Maths In Focus Mathematics Preliminary Course
2. Evaluate
(a) 20 8 4'-
(b) 3 7 2 5# #-
(c) 4 27 3 6# ' '] g (d) 17 3 2#+ -
(e) . .1 9 2 3 1#-
()1 3
14 7'- +
(g) 253
51
32
#-
(h)
65
143
81
-
(i)
41 81
85
65
'
+
(j)1
41
21
351
107
-
-
3. Evaluate correct to 2 decimal
places.
(a) 2.36 4.2 0.3'+
(b) . . .2 36 4 2 0 3'+] g (c) 12.7 3.95 5.7# '
(d) 8.2 0.4 4.1 0.54' #+
(e) . . . .3 2 6 5 1 3 2 7#- +] ]g g()
4.7 1.31
+
(g)4.51 3.28
1
+
(h)5.2 3.60.9 1.4
-
+
(i)1.23 3.15
5.33 2.87
-
+
(j) 1.7 8.9 3.942 2 2
+ -
4. Round o 1289 to the nearest
hundred.
5. Write 947 to the nearest ten.
6. Round o 3200 to the nearest
thousand.
7. A crowd o 10 739 spectators
attended a tennis match.
Write this fgure to the nearest
thousand.
8. A school has 623 students. What
is this to the nearest hundred?
9. A bank made loans to the value
o $7 635 718 last year. Round this
o to the nearest million.
10. A company made a proft o
$34 562 991.39 last year. Write
this to the nearest hundred
thousand.
11. The distance between two cities
is 843.72 km. What is this to the
nearest kilometre?
12. Write 0.72548 correct to
2 decimal places.
13. Round o 32.569148 to the
nearest unit.
14. Round o 3.24819 to 3 decimal
places.
15. Evaluate 2.45 1.72# correct to
2 decimal places.
16. Evaluate 8.7 5' correct to
1 decimal place.
17. I pies are on special at 3 or
$2.38, fnd the cost o each pie.
18. Evaluate 7.48 correct to
2 decimal places.
19. Evaluate8
6.4 2.3+
correct to
1 decimal place.
20. Find the length o each piece
o material, to 1 decimal place,
i 25 m o material is cut into
7 equal pieces.
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9Chapter 1 Basic Arithmetic
DID YOU KNOW?
In building, engineering and other industries where accurate measurements are used, the
number o decimal places used indicates how accurate the measurements are.
For example, i a 2.431 m length o timber is cut into 8 equal parts, according to the
calculator each part should be 0.303875 m. However, a machine could not cut this accurately.
A length o 2.431 m shows that the measurement o the timber is only accurate to the nearest
mm (2.431 m is 2431 mm). The cut pieces can also only be accurate to the nearest mm (0.304 m
or 304 mm).
The error in measurement is related to rounding o, as the error is hal the smallest
measurement. In the above example, the measurement error is hal a millimetre. The length o
timber could be anywhere between 2430.5 mm and 2431.5 mm.
Directed Numbers
Many students use the calculator with work on directed numbers (numbers
that can be positive or negative). Directed numbers occur in algebra and
other topics, where you will need to remember how to use them. A good
understanding o directed numbers will make your algebra skills much better.
-^ h KEYUse this key to enter negative numbers. For example,
press ( ) 3- =
21. How much will 7.5 m2 o tiles
cost, at $37.59 per m2?
22. Divide 12.9 grams o salt into
7 equal portions, to 1 decimal
place.
23. The cost o 9 peaches is $5.72.
How much would 5 peaches cost?
24. Evaluate correct to 2 decimal
places.
(a) 17.3 4.33 2.16#-
(b) . . . .8 72 5 68 4 9 3 98# #-
(c)5.6 4.35
3.5 9.8
+
+
(d)7.63 5.12
15.9 6.3 7.8
-
+ -
(e)
6.87 3.21
1
-
25. Evaluate.
. ..
5 39
9 68 5 479 912
-- ] g
correct to 1 decimal place.
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10 Maths In Focus Mathematics Preliminary Course
Adding and subtracting
To add: move to the right along the number line
To subtract: move to the let along the number line
AddSubtract
-4 -3 -2 -1 0 1 2 3 4
Same signs
Dierent signs
= +
+ + = +
- =
= -
+ - = -
- + = -
- +
EXAMPLES
Evaluate
1. 4 3- +
Solution
Start at 4- and move 3 places to the right.
-4 -3 -2 -1 0 1 2 3 4
4 3 1- + = -
2. 1 2- -
Solution
Start at 1- and move 2 places to the let.
-4 -3 -2 -1 0 1 2 3 4
1 2 3- - = -
Multiplying and dividing
To multiply or divide, ollow these rules. This rule also works i there are two
signs together without a number in between e.g. 32 - -
You can also do these on a
calculator, or you may have
a different way of working
these out.
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11Chapter 1 Basic Arithmetic
EXAMPLES
Evaluate
1. 2 7#-
Solution
Dierent signs ( 2 7and- + ) give a negative answer.
2 7 14#- = -
2. 12 4'- -
Solution
Same signs ( 12 4and- - ) give a positive answer.
12 4 3'- - =
3. 1 3- - -
Solution
The signs together are the same (both negative) so give a positive answer.
1 3
2
= - +
=
1 3- - -
1. 2 3- +
2. 7 4- -
3. 8 7# -
4. 37 -- ] g5. 28 7' -
6. . .4 9 3 7- +
7. . .2 14 5 37- -
8. . .4 8 7 4# -
9. . .1 7 4 87- -] g10.
53
132
- -
11. 5 3 4#-
12. 2 7 3#- + -
13. 4 3 2#- -
14. 1 2- - -
15. 7 2+ -
16. 2 1- -] g17. 2 15 5'- +
18. 2 6 5# #- -
19. 28 7 5#'- - -
20. 3 2-] g
1.2 Exercises
Evaluate
Start at 1- and move 3
places to the right.
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12 Maths In Focus Mathematics Preliminary Course
Fractions, Decimals and Percentages
EXAMPLES
1. Write 0.45 as a raction in its simplest orm.
Solution
.0 4510045
55
20
9
'=
=
2. Convert83 to a decimal.
Solution
.
.
.
8 3 0000 375
8
30 375So =
g
3. Change 35.5% to a raction.
Solution
. %.
35 5100
35 5
22
20071
#=
=
4. Write 0.436 as a percentage.
Solution
. . %
. %
0 436 0 436 100
43 6
#=
=
5. Write 20 g as a raction o 1 kg in its simplest orm.
Solution
1 1000kg g=
1
20
1000
20
501
kg
g
g
g=
=
Multiply by 100% to
change a fraction or
decimal to a percentage.
Conversions
You can do all theseconversions on your
calculator using the
ac
bor S D+ key.
83 means3 8.'
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13Chapter 1 Basic Arithmetic
Sometimes decimals repeat, or recur.
Example
. 0.31
0 33333333 3
f= =
There are dierent methods that can be used to change a recurring
decimal into a raction. Here is one way o doing it. Later you will discover
another method when studying series. (See HSC Course book, Chapter 8.)
EXAMPLES
1. Write .0 4
as a rational number.
Solution
. ( )
. ( )
( ) ( ):
n
n
n
n
0 44444 1
10 4 44444 2
2 1 9 4
94
Let
Then
f
f
=
=
- =
=
2. Change .1 329
to a raction.
Solution
. ( )
. ( )
( ) ( ): .
.
n
n
n
n
1 3292929 1
100 132 9292929 2
2 1 99 131 6
99131 6
1010
990
1316
1495
163
Let
Then
#
f
f
=
=
- =
=
=
=
A rational number is
any number that can be
written as a fraction.
Check this on your
calculator by dividing
4 by 9.
Try multiplying n by 10.
Why doesnt this work?
6. Find the percentage o people who preer to drink Lemon Fuzzy, i 24
out o every 30 people preer it.
Solution
% %3024
1100 80# =
CONTINUED
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14 Maths In Focus Mathematics Preliminary Course
1. Write each decimal as a raction
in its lowest terms.0.64(a)
0.051(b)
5.05(c)
11.8(d)
2. Change each raction into a
decimal.
(a)52
(b) 1
8
7
(c)12
5
(d)117
3. Convert each percentage to a
raction in its simplest orm.
2%(a)
37.5%(b)
0.1%(c)
109.7%(d)
4. Write each percentage as a decimal.
27%(a)
109%(b)
0.3%(c)
6.23%(d)
5. Write each raction as a
percentage.
(a)207
(b)31
(c) 2
15
4
(d)1000
1
6. Write each decimal as a
percentage.
1.24(a)
0.7(b)
0.405(c)
1.2794(d)
7. Write each percentage as a
decimal and as a raction.
52%(a)
7%(b)
16.8%(c)
109%(d)
43.4%(e)
() %1241
8. Write these ractions as recurring
decimals.
(a)65
(b)7
99
(c)99
13
(d)61
(e)32
1.3 Exercises
Another method
Let .
. ( )
. ( )
( ) ( ):
n
n
n
n
n
1 3292929
10 13 2929292 1
1000 1329 292929 2
2 1 990 1316
990
1316
1495
163
Then
and
f
f
f
=
=
=
- =
=
=
This method avoids decimals
in the fraction at the end.
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15Chapter 1 Basic Arithmetic
Investigation
Explore patterns in recurring decimals by dividing numbers by 3, 6, 9, 11,
and so on.
Can you predict what the recurring decimal will be i a raction has 3 in
the denominator? What about 9 in the denominator? What about 11?
Can you predict what raction certain recurring decimals will be? What
denominator would 1 digit recurring give? What denominator would you
have or 2 digits recurring?
Operations with fractions, decimals and percentages
You will need to know how to work with ractions without using a calculator,
as they occur in other areas such as algebra, trigonometry and surds.
()33
5
(g)71
(h) 1112
9. Express as ractions in lowest
terms.
(a) .0 8
(b) .0 2
(c) .1 5
(d) .3 7
(e) .0 67
() .0 54
(g) .0 15
(h) .0 216
(i) .0 219
(j) .1 074
10. Evaluate and express as a decimal.
(a)3 6
5
+
(b) 8 3 5'-
(c)12 34 7
+
+
(d) 199
31-
(e)7 413 6
+
+
11. Evaluate and write as a raction.
(a) . . .7 5 4 1 7 9' +] g(b)
4.5 1.3
15.7 8.9
-
-
(c)12.3 8.9 7.6
6.3 1.7
- +
+
(d). .
.
11 5 9 7
4 3
-
(e)8100
64
12. Angel scored 17 out o 23 in a
class test. What was her score as apercentage, to the nearest unit?
13. A survey showed that 31 out o
40 people watched the news on
Monday night. What percentage
o people watched the news?
14. What percentage o 2 kg is 350 g?
15. Write 25 minutes as a percentage
o an hour.
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16 Maths In Focus Mathematics Preliminary Course
DID YOU KNOW?
Some countries use a comma or the decimal pointor example, 0,45 or 0.45.
This is the reason that our large numbers now have spaces instead o commas between
digitsor example, 15 000 rather than 15,000.
EXAMPLES
1. Evaluate 1 .52
43
-
Solution
152
43
57
43
2028
2015
20
13
- = -
= -
=
2. Evaluate 221
3' .
Solution
221
325
1
3
25
31
56
' '
#
=
=
=
3. Evaluate . .0 056 100#
Solution
. .0 056 100 5 6# =
Move the decimal point
2 places to the right.
The examples on ractions show how to add, subtract, multiply or divide
ractions both with and without the calculator. The decimal examples will
help with some simple multiplying and the percentage examples will be useul
in Chapter 8 o the HSC Course book when doing compound interest.
Most students use their calculators or decimal calculations. However, it
is important or you to know how to operate with decimals. Sometimes thecalculator can give a wrong answer i the wrong key is pressed. I you can
estimate the size o the answer, you can work out i it makes sense or not. You
can also save time by doing simple calculations in your head.
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17Chapter 1 Basic Arithmetic
4. Evaluate . . .0 02 0 3#
Solution
. . .0 02 0 3 0 006# =
5. Evaluate10
8.753.
Solution
. .8 753 10 0 8753' =
6. The price o a $75 tennis racquet increased by %.521
Find the new
price.
Solution
% $ . $
$ .
5 75 0 055 75
4 13
o` #=
=
% . % $ . $
$ .
521
0 055 10521
75 1 055 75
79 13
21
or o #= =
=
So the price increases by $4.13 to $79.13.
7. The price o a book increased by 12%. I it now costs $18.00, what did
it cost beore the price rise?
Solution
The new price is 112% (old price 100%, plus 12%)
1%$ .
100%$ .
$16.07
112
18 00
112
18 00
1100
`
#
=
=
=
So the old price was $16.07.
1.4 Exercises
1. Write 18 minutes as a raction o
2 hours in its lowest terms.
2. Write 350 mL as a raction o
1 litre in its simplest orm.
3. Evaluate
(a)53
41
+
(b) 352
2107
-
(c)43
152
#
(d)73
4'
(e) 153
232
'
Multiply the numbers
and count the number
of decimal places in
the question.
Move the decimal
point 1 place to
the left.
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18 Maths In Focus Mathematics Preliminary Course
4. Find53
o $912.60.
5. Find75
o 1 kg, in grams correct
to 1 decimal place.
6. Trinh spends 31 o her day
sleeping,247
at work and121
eating. What raction o the day
is let?
7. I get $150.00 a week or a casual
job. I I spend101
on bus ares,
152
on lunches and31
on outings,
how much money is let over orsavings?
8. John grew by20017
o his height
this year. I he was 165 cm tall
last year, what is his height now,
to the nearest cm?
9. Evaluate
(a) 8.9 3+
(b) 9 3.7-
(c) .1 9 10# (d) .0 032 100#
(e) .0 7 5#
() . .0 8 0 3#
(g) . .0 02 0 009#
(h) .5 72 1000#
(i)1008.74
(j) . .3 76 0 1#
10. Find 7% o $750.
11. Find 6.5% o 845 mL.
12. What is 12.5% o 9217 g?
13. Find 3.7% o $289.45.
14. I Kaye makes a proft o $5 by
selling a bike or $85, fnd the
proft as a percentage o the
selling price.
15. Increase 350 g by 15%.
16. Decrease 45 m by %.821
17. The cost o a calculator is now
$32. I it has increased by 3.5%,
how much was the old cost?
18. A tree now measures 3.5 m, which
is 8.3% more than its previous
years height. How high was the
tree then, to 1 decimal place?
19. This month there has been a
4.9% increase in stolen cars. I
546 cars were stolen last month,
how many were stolen this
month?
20. Georges computer cost $3500. I
it has depreciated by 17.2%, what
is the computer worth now?
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19Chapter 1 Basic Arithmetic
Powers and Roots
A power(or index) o a number shows how many times a number is
multiplied by itsel.
PROBLEM
I both the hour hand and minute hand start at the same position at
12 oclock, when is the frst time, correct to a raction o a minute, that
the two hands will be together again?
EXAMPLES
1. 4 4 4 4 643 # #= =
2. 2 2 2 2 2 2 325 # # # #= =
In 43 the 4 is called the base
number and the 3 is called
the index or power.
A root o a number is the inverse o the power.
EXAMPLES
1. 36 6= since 6 362 =
2. 8 23 = since 2 83 =
3. 64 26 = since 2 646 =
DID YOU KNOW?
Many ormulae use indices (powers and roots).
For example the compound interest ormula that you will study in Chapter 8 o the HSC
Course book is 1A P rn
= +^ h Geometry uses ormulae involving indices, such as
3
4V r3r= . Do you know what this
ormula is or?
In Chapter 7, the ormula or the distance between 2 points on a number plane is
d x x y y ( ) ( )2 1
2
2 1
2= - + -
See i you can fnd other ormulae involving indices.
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20 Maths In Focus Mathematics Preliminary Course
Proof
( )( )( )
a aa
a
a a aa a a m
na a a m n
a
1
timestimes
times
m nn
m
m n
'
# # #
# # #
# # #
f
f
f
=
=
=-
=-
Index laws
There are some general laws that simpliy calculations with indices.
a a am n m n# = +
Proof
( ) ( )a a a a a a a a
a a a
a
m n
m n
m n
m n
times times
times
# # # # # # # #
# # #
f f
f
=
=
=+
+
1 2 34444 4444 1 2 34444 4444
1 2 34444 4444
These laws work for any m
and n, including fractions and
negative numbers.
a a am n m n' = -
a=( )am n mn
Proof
( ) ( )
( )
a a a a a n
a n
a
times
times
m n m m m m
m m m m
mn
# # # #f=
=
=
f+ + + +
POWER AND ROOT KEYS
Use the x2
and x3
keys or squares and cubes.
Use the xy or ^ key to fnd powers o numbers.
Use the key or square roots.
Use the 3 key or cube roots.
Use the x or other roots.
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21Chapter 1 Basic Arithmetic
( )ab a bn n n=
Proof
( ) ( )
( ) ( )
ab ab ab ab ab n
a a a b b b
a b
timesn
n n
n ntimes times
# # # #
# # # # # # #
f
f f
=
=
=
1 2 34444 4444 1 2 34444 4444
b
a
b
an
n
n
=c m
Proof
( )
( )( )
b
a
b
a
b
a
b
a
b
an
b b b b
a a a a nn
b
a
times
timestimes
n
n
n
# # # #
# # # #
# # # #
f
f
f
=
=
=
c m
EXAMPLES
Simpliy
1. m m m9 7 2# '
Solution
m m m m
m
9 7 2 9 7 2
14
# ' =
=
+ -
2.
3
( )y2
4
Solution
( ) ( )y y
y
y
2 2
2
8
4 3 3 4 3
3 4 3
12
=
=
=
#
CONTINUED
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22 Maths In Focus Mathematics Preliminary Course
1. Evaluate without using a
calculator.
(a) 5 23 2#
(b) 3 84 2+
(c)41 3c m
(d) 273
(e) 164
2. Evaluate correct to 1 decimal
place.
(a) 3.72
(b) 1.061.5
(c) 2.3 0.2-
(d) 193
(e) . . .34 8 1 2 43 13 #-
()0.99 5.61
13
+
3. Simpliy(a) a a a6 9 2# #
(b) y y y3 8 5# #-
(c) a a1 3#- -
(d) 2 2w w#1 1
(e) x x6 '
() p p3 7' -
(g)y
y5
11
(h) ( )x7 3
(i) (2 )x5 2
(j) (3 )y 2 4-
(k) a a a3 5 7# '
(l)y
x9
2 5f p (m)
w
w w3
6 7#
(n)( )
p
p p9
2 3 4#
(o)x
x x2
6 7'
(p)( )
a b
a b4 9
2 2 6
#
#
(q)( ) ( )
x y
x y1 4
2 3 3 2
#
#
-
-
4. Simpliy
(a) x x5 9
# (b) a a1 6#- -
(c)m
m3
7
(d) k k k13 6 9# '
(e) a a a5 4 7# #- -
() 5 5x x#2 3
(g)m n
m n4 2
5 4
#
#
1.5 Exercises
3.( )
y
y y5
6 3 4#
-
Solution
( )
y
y y
y
y y
y
y
y
y
y
( )
5
6 3 4
5
18 4
5
18 4
5
14
9
# #=
=
=
=
- -
+ -
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23Chapter 1 Basic Arithmetic
(h)2 2
p
p p2
#
1 1
(i) (3 )x11 2
(j)( )
x
x3
4 6
5. Simpliy
(a) 5( )pq3
(b)b
a 8c m (c)
4
b
a4
3d n (7(d) a5b)2
(e)(2 )
m
m4
7 3
() ( )xyxy xy
3 2 4
#
(g)3
4
( )
( )
k
k
6
23
8
(h) yy
28
5 712
#_ i (i)
a
a a11
6 4 3#
-e o (j)
x y
xy58 3
9 3
#
f p 6. Evaluate a3b2 when 2a = and
43
b = .
7. I32
x = and91
,y= fnd the value
oxy
x y5
3 2
.
8. I21
,31
a b= = and41
,c=
evaluatec
a b4
2 3as a raction.
9. (a) Simpliya b
a b8 7
11 8
.
Hence evaluate(b)a b
a b8 7
11 8
when
52
a = and85
b = as a raction.
10. (a) Simpliyp q r
p q r4 6 2
5 8 4
.
(b) Hence evaluatep q r
p q r4 6 2
5 8 4
as a
raction when
8
7,
3
2p q= = and
43
r= .
11. Evaluate ( )a4 3 when6.a
32
=
1
c m
12. Evaluateb
a b4
3 6
when a21
= and
b32
= .
13. Evaluatex y
x y5 5
4 7
when x 31= and
y92
= .
14. Evaluatek
k9
5
-
-
when .k31
=
15. Evaluate( )a b
a b3 2 2
4 6
when a43
= and
b91
= .
16. Evaluatea b
a b5 2
6 3
#
#as a raction
when a91
= and b43
= .
17. Evaluatea b
a b3
2 7
as a raction in
index orm when a52 4
= c m andb
85 3
= c m .
18. Evaluate( )
( )
a b c
a b c2 4 3
3 2 4
as a raction
when ,a31
= b76
= and c97
= .
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24 Maths In Focus Mathematics Preliminary Course
Proof
x x x
x
x xx
x
x
1
1
n n n n
n nn
n
0
0
'
'
`
=
=
=
=
=
-
Negative and zero indices
Class Investigation
Explore zero and negative indices by looking at these questions.
For example simpliy x x3 5' using (i) index laws and (ii) cancelling.
(i) x x x3 5 2' = - by index laws
(ii)x
xx x x x x
x x x
x
1
5
3
2
# # # #
# #=
=
xx
1So 2
2=
-
Now simpliy these questions by (i) index laws and (ii) cancelling.
(a) x x2 3'
(b) x x2 4'
(c) x x2 5'
(d) x x3 6'
(e) x x3 3'
() x x2 2'
(g) x x2'
(h) x x5 6'
(i) x x4 7'
(j) x x3'
Use your results to complete:
x
x
0
n
=
=-
x 10 =
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25Chapter 1 Basic Arithmetic
1x
xn
n=-
Proofx x x
x
x xx
x
x
xx
1
1
n n
n
nn
n
nn
0 0
00
'
'
`
=
=
=
=
=
-
-
-
EXAMPLES
1. Simpliy .abc
ab c4
5 0e o Solution
1abc
ab c4
5 0
=e o
2. Evaluate .2 3-
Solution
22
1
81
3
3=
=
-
3. Write in index orm.
(a)1
x2
(b)3
x5
(c)51x
(d)x 1
1+
CONTINUED
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26 Maths In Focus Mathematics Preliminary Course
1. Evaluate as a raction or whole
number.(a) 3 3-
(b) 4 1-
(c) 7 3-
(d) 10 4-
(e) 2 8-
6() 0
(g) 2 5-
(h) 3 4-
(i) 7 1-
(j) 9 2-
(k) 2 6-
(l) 3 2-
4(m) 0
(n) 6 2-
(o) 5 3-
(p) 10 5-
(q) 2 7-
(r) 20
(s) 8 2-
(t) 4 3-
2. Evaluate
(a) 20
(b)21 4-c m
(c)32 1-c m
(d)6
5 2-c m
(e)3
2
x y
x y 0
-
+f p ()
51 3-c m
(g)43
1-
c m
(h)71 2-c m
(i)32 3-c m
(j)21 5-c m
(k)73 1-c m
1.6 Exercises
Solution
(a)1
xx
2
2=
-
(b) x x
x
33
1
3
5 5
5
#=
=-
(c)x x
x
51
51 1
51 1
#=
=-
(d)( )x x
x
11
1
1
1
1
1
+=
+
= +-] g
4. Write a3 without the negative index.
Solution
aa
133
=-
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27Chapter 1 Basic Arithmetic
(l)9
8 0c m
(m)76 2-c m
(n)
10
9 2-c m
(o)11
6 0c m
(p)41 2
-
-c m (q)
52 3
-
-c m (r) 3
72 1
-
-c m (s)
8
3 0-c m
(t) 1 412
-
-
c m
3. Change into index orm.
(a)1
m3
(b)1x
(c)1
p7
(d)1
d9
(e)1
k5
()1
x2
(g)2
x4
(h)3
y2
(i)2
1
z6
(j)53t8
(k)72x
(l)2
5
m6
(m)3
2
y7
(n)(3 4)
1
x 2+
(o)( )
1
a b 8+
(p)
2
1
x-
(q)( )p5 1
13
+
(r)(4 9)
2
t 5-
(s)( )x4 1
111
+
(t)9( 3 )
5
a b 7+
4. Write without negative indices.
(a) t 5-
(b) x 6-
(c) y 3-
(d) n 8-
(e) w 10-
() x2 1-
(g) 3m 4-
(h) 5x 7-
(i) 2x 3-] g (j) n4
1-
] g (k) x 1 6+ -] g (l) y z8 1+ -^ h (m) 3k 2- -] g (n) 3 2x y 9+ -^ h (o)
1x
5-b l
(p) y1 10-c m
(q)2p
1-
d n
(r)1
a b
2
+
-c m
(s) x y
x y 1
-
+-e o
(t)32
x y
w z 7
+
--e o
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28 Maths In Focus Mathematics Preliminary Course
Proof
n
n
a a
a a
a a
by index lawsn
n n
n`
=
=
=
1
1
` ^^j hh
Fractional indices
Class Investigation
Explore ractional indices by looking at these questions.
For example simpliy (i) 2x21` j and (ii) .x 2^ h
2( ) x xx
i by index laws2
1=
=
1` ^j h
2
2
( ) x x
x x x
x x
ii
So
2
22
`
=
= =
=
1
1
^` ^
hj h
Now simpliy these questions.
(a) 2x21^ h
(b) x2
(c) 3x31` j
(d) 3x31^ h
(e) x33^ h
() x33
(g)4
x41` j
(h) 4x41
^ h (i) x4
4^ h (j) x44
Use your results to complete:
nx =1
na an=1
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29Chapter 1 Basic Arithmetic
EXAMPLES
1. Evaluate
(a)2
491
(b) 3271
Solution
(a)2
49 497
=
=
1
(b)3
27 273
3=
=
1
2. Write x3 2- in index orm.
Solution
2( )x x3 2 3 2- = -1
3. Write 7( )a b+1
without ractional indices.
Solution
7( )a b a b7+ = +1
Proof
n n
n n
a a
a
a
a
m
n m
m
mn
=
=
a =
=
m
m
1
1
`^^
jhh
Putting the ractional and negative indices together gives this rule.
- na
a
1n
=
1
Here are some urther rules.
n
( )
a a
a
mn
n m
=
=
m
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30 Maths In Focus Mathematics Preliminary Course
b
aabn n
=
-c bm l
EXAMPLES
1. Evaluate
(a)3
8
4
(b)-
31251
(c)32 3-c m
Solution
(a) 3 ( ) ( )
8 8 8
2
16
or3 4 43
4
=
=
=
4
(b)-
3
3
125
125
1
125
1
51
3
=
=
=
1
1
Proof
b
a
b
a
b
a
b
a
a
b
ab
ab
1
1
1
1
n
n
n
n
n
n
n
n
n
n
n
'
#
=
=
=
=
=
=
-c c
b
m m
l
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31Chapter 1 Basic Arithmetic
(c)32
2
3
827
3 8
3
3 3
=
=
=
-c cm m
2. Write in index orm.
(a) x5
(b)( )x4 1
12 23
-
Solution
(a) 2x x5 =5
(b)
-
3
3
( ) ( )
( )
x x
x
4 1
1
4 1
1
4 1
2 23 2
2
-
=
-
= -
2
2
3. Write-
5r3
without the negative and ractional indices.
Solution
-5
5
r
r
r
1
135
=
=
3
3
DID YOU KNOW?
Nicole Oresme (132382) was the frst mathematician to use ractional indices.
John Wallis (16161703) was the frst person to explain the signifcance o zero, negative
and ractional indices. He also introduced the symbol 3 or infnity.Do an Internet search on these mathematicians and fnd out more about their work and
backgrounds. You could use keywords such as indices and infnity as well as their names to fnd
this inormation.
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32 Maths In Focus Mathematics Preliminary Course
1. Evaluate
(a) 2811
(b) 3271
(c) 2161
(d) 381
(e) 2491
() 310001
(g) 4161
(h) 2641
(i) 3641
(j) 711
(k) 4811
(l) 5321
(m) 801
(n) 31251
(o) 33431
(p) 71281
(q) 42561
(r) 293
(s)-
381
(t)-
3642
2. Evaluate correct to 2 decimal
places.
(a) 4231
(b) 45.84
(c) 1.24 4.327 +
(d)12.9
15
(e). .
. .
1 5 3 7
3 6 1 48
+
-
(). .
. .
8 79 1 4
5 9 3 74 #
-
3. Write without ractional indices.
(a) 3y1
(b) 3y2
(c) 2x-
1
(d) 2( )x2 5+1
(e)-
2( )x3 1-1
() 3( )q r6 +1
(g)-
5( )x 7+2
4. Write in index orm.
(a) t
(b) y5
(c) x3
(d) 9 x3 -
(e) s4 1+
()2 3
1
t+
(g)(5 )
1
x y 3-
(h) ( )x3 1 5+
(i)( 2)
1x 23 -
(j)2 7
1
y+
(k)4
5
x3 +
(l)y3 1
22
-
(m)5 ( 2)
3
x2 34 +
5. Write in index orm and simpliy.
(a) x x
(b) xx
(c)x
x3
(d)x
x3
2
(e) x x4
1.7 Exercises
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33Chapter 1 Basic Arithmetic
6. Expand and simplify, and write in
index form.
(a) ( )x x 2+
(b) ( )( )a b a b3 3 3 3+ -
(c)1
pp
2
+
f p
(d) (1
)xx
2+
(e)( )
x
x x x3 13
2- +
7. Write without fractional or
negative indices.
(a)-
3( )a b2-1
(b) 3( )y 3--
2
(c)-
7( )a4 6 1+4
(d)
-4( )x y
3
+
5
(e)
-9( )x
7
6 3 8+2
Scientific notation (standard form)
Very large or very small numbers are usually written in scientific notation to
make them easier to read. What could be done to make the figures in the box
below easier to read?
DID YOU KNOW?
The Bay o Fundy, Canada, has the largest tidal changes in the world. About 100 000 000 000
tons o water are moved with each tide change.
The dinosaurs dwelt on Earth or 185 000 000 years until they died out 65 000 000 years ago.
The width o one plant cell is about 0.000 06 m.
In 2005, the total storage capacity o dams in Australia was 83 853 000 000 000 litres and
households in Australia used 2 108 000 000 000 litres o water.
A number in scientific notation is written as a number between 1 and 10
multiplied by a power of 10.
EXAMPLES
1. Write 320 000 000 in scientific notation.
Solution
.320 000 000 3 2 108#=
2. Write .7 1 10 5# - as a decimal number.
Solution
. .
.
7 1 10 7 1 10
0 000 071
5 5# '=
=
-
Write the number
between 1 and 10
and count the decimal
places moved.
Count 5 places to
the left.
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34 Maths In Focus Mathematics Preliminary Course
SIGNIFICANT FIGURES
The concept of significant figures is related to rounding off. When we look
at very large (or very small) numbers, some of the smaller digits are not
significant.
For example, in a football crowd of 49 976, the 6 people are not really
significant in terms of a crowd of about 50 000! Even the 76 people are not
significant.
When a company makes a profit of $5 012 342.87, the amount of87 cents is not exactly a significant sum! Nor is the sum of $342.87.
To round off to a certain number of significant figures, we count from the
first non-zero digit.
In any number, non-zero digits are always significant. Zeros are not
significant, except between two non-zero digits or at the end of a decimal
number.
Even though zeros may not be significant, they are still necessary. For
example 31, 310, 3100, 31 000 and 310 000 all have 2 significant figures but
are very different numbers!
Scientific notation uses the significant figures in a number.
SCIENTIFIC NOTATION KEY
Use the EXP or 10 x# key to put numbers in scientifc notation.
For example, to evaluate 3.1 10 2.5 10 ,4 2
# ' #-
press 3.1 EXP 4 2.5 EXP ( ) 2
1240 000
' =-
=
DID YOU KNOW?
Engineering notation is similar to scientifc notation, except the powers o 10 are always
multiples o 3. For example,
3.5 103
#
15.4 10 6# -
EXAMPLES
. ( )
. . ( )
. . ( )
12 000 1 2 10 2
0 000 043 5 4 35 10 3
0 020 7 2 07 10 3
significant figures
significant figures
significant figures
4
5
2
#
#
#
=
=
=
-
-
When rounding off to significant figures, use the usual rules for rounding off.
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35Chapter 1 Basic Arithmetic
EXAMPLES
1. Round o 4 592 170 to 3 signifcant fgures.
Solution
4 592 170 4 590 000= to 3 signifcant fgures
2. Round o 0.248 391 to 2 signifcant fgures.
Solution
. .0 248 391 0 25= to 2 signifcant fgures
3. Round o 1.396 794 to 3 signifcant fgures.
Solution
. .1 396 794 1 40= to 3 signifcant fgures
1. Write in scientifc notation.
3 800(a)
1 230 000(b)61 900(c)
12 000 000(d)
8 670 000 000(e)
416 000()
900(g)
13 760(h)
20 000 000(i)
80 000(j)
2. Write in scientifc notation.
0.057(a)0.000 055(b)
0.004(c)
0.000 62(d)
0.000 002(e)
0.000 000 08()
0.000 007 6(g)
0.23(h)
0.008 5(i)
0.000 000 000 07(j)
3. Write as a decimal number.
(a) .3 6 104#
(b) .2 78 10
7#
(c) .9 25 103#
(d) .6 33 106#
(e) 4 105#
() .7 23 10 2# -
(g) .9 7 10 5# -
(h) .3 8 10 8# -
(i) 7 10 6# -
(j) 5 10 4# -
4. Round these numbers to
2 signifcant fgures.235 980(a)
9 234 605(b)
10 742(c)
0.364 258(d)
1.293 542(e)
8.973 498 011()
15.694(g)
322.78(h)
2904.686(i)
9.0741(j)
1.8 Exercises
Remember to putthe 0s in!
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36 Maths In Focus Mathematics Preliminary Course
5. Evaluate correct to 3 signifcant
fgures.
(a) . .14 6 0 453#
(b) .4 8 7'
(c) 4. . .47 2 59 1 46#+
(d) . .3 47 2 71-
6. Evaluate . . ,4 5 10 2 9 104 5# # #
giving your answer in scientifc
notation.
7. Calculate.
.
1 34 10
8 72 107
3
#
#-
and write
your answer in standard ormcorrect to 3 signifcant fgures.
Investigation
A logarithm is an index. It is a way o fnding the power (or index) to
which a base number is raised. For example, when solving ,3 9x = the
solution is .x 2=
The 3 is called the base number and the x is the index or power.
You will learn about logarithms in the HSC course.
Ia yx = then log y xa
=
The expression log1.7
49 means the power o 7 that gives 49.
The solution is 2 since .7 492 =
The expression log2.2
16 means the power o 2 that gives 16.
The solution is 4 since .2 164 =
Can you evaluate these logarithms?
log1.3
27
log2.5
25
log3.10
10 000
log4.2
64
log5.4
4
log6.7
7
log7.3
1
log8.4
2
9.31
log3
10.41
log2
The a is called the base
number and the x is the
index or power.
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37Chapter 1 Basic Arithmetic
Absolute Value
Negative numbers are used in maths and science, to show opposite directions.
For example, temperatures can be positive or negative.
But sometimes it is not appropriate to use negative numbers.For example, solving 9c2 = gives two solutions, c 3!= .
However when solving 9,c2 = using Pythagoras theorem, we only use
the positive answer, 3,c = as this gives the length o the side o a triangle. The
negative answer doesnt make sense.
We dont use negative numbers in other situations, such as speed. In
science we would talk about a vehicle travelling at 60k/h going in a negative
direction, but we would not commonly use this when talking about the speed
o our cars!
Absolute value defnitions
We write the absolute value ox as x
xx x
x
0when
when x 01
$=
-
)
EXAMPLES
1. Evaluate .4
Solution
4 4 04 since $=
We can also defne
x as the distance
o x rom 0 on the
number line. We will
use this in Chapter 3.
CONTINUED
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38 Maths In Focus Mathematics Preliminary Course
2. Evaluate .3-
Solution
3 3 3 0
3
since 1- = - - -
=
] g
The absolute value has some properties shown below.
Properties o absolute value
a 9= = =
| | | | | | | | | | | |
| | | || | | |
| | | | | | | |
| | | | | | | |
| | | | | | | | | | | | | | | | | |
ab a b
aa a
a a
a b b a
a b a b
2 3 2 3 6
3 35 5 57 7 7
2 3 3 2 1
2 3 2 3 3 4 3 4
e.g.
e.g.e.g.e.g.
e.g.
e.g. but
2 2 2 2
2 2
# # #
1#
= - = - =
- -
= = =
- = - = =
- = - - = - =
+ + + = + - + - +
] g
EXAMPLES
1. Evaluate 2 1 32
- - + - .
Solution
2 1 3 2 1 3
2 1 9
10
22- - + - = - +
= - +
=
2. Show that a b a b#+ + when a 2= - and 3b = .
Solution
a b
2 3
1
1
LHS = +
= - +
=
=
LHS means Let Hand Side.
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39Chapter 1 Basic Arithmetic
a b
2 3
2 3
5
RHS = +
= - +
= +
=
a b a b
1 5Since 1#+ +
3. Write expressions or 2 4x - without the absolute value signs.
Solution
1
x x x
x
x
x x x
x x
x
2 4 2 4 2 4 0
2 4
2
2 4 2 4 2 4 0
2 4 2 4
2
when
i.e.
when
i.e.
1
1
$
$
$
- = - -
- = - - -
= - +] g
Class Discussion
Are these statements true? I so, are there some values or which the
expression is undefned (values ox or ythat the expression cannot
have)?
1.x
x1=
2. 2 2x x=
3. 2 2x x=
4. x y x y + = +
5. x x2 2=
6. x x3 3=
7. x x1 1+ = +
8.x
x
3 2
3 21
-
-=
9.x
x1
2=
10. x 0$
Discuss absolute value and its defnition in relation to these statements.
RHS means Right Hand Side.
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40 Maths In Focus Mathematics Preliminary Course
1. Evaluate
(a) 7
(b) 5-
(c) 6- (d) 0
(e) 2
() 11-
(g) 2 3-
(h) 3 8-
(i) 52
-
(j) 5 3-
2. Evaluate
(a) 3 2+ -
(b) 3 4- - (c) 5 3- +
(d) 2 7# -
(e) 3 1- + -
() 5 2 62
#- -
(g) 2 5 1#- + -
(h) 3 4-
(i) 2 3 3 4- - -
(j) 5 7 4 2- + -
3. Evaluate a b- i
(a) 5 2a band= =
(b) 1 2a band= - =
(c) 2 3a band= - = -
(d) 4 7a band= =
(e) .a b1 2and= - = -
4. Write an expression or
(a) a a 0when 2
(b) 0a awhen 1
(c) 0a awhen =
(d) 0a a3 when2
(e) 0a a3 when 1
() 0a a3 when =
(g) a a1 1when 2+ -
(h) 1a a1 when 1+ -
(i) 2x x2 when 2-
(j) 2x x2 when 1- .
5. Show that a b a b#+ +
when
(a) 2 4a band= =
(b) 1 2a band= - = -
(c) 2 3a band= - =
(d) 4 5a band= - =
(e) .a b7 3and= - = -
6. Show that x x2 = when
(a) 5x =
(b) x 2= - (c) x 3= -
(d) 4x =
(e) .x 9= -
7. Use the defnition o absolute
value to write each expression
without the absolute value signs
(a) x 5+
(b) 3b -
(c) 4a +
(d) 2 6y- (e) 3 9x +
() 4 x-
(g) k2 1+
(h) 5 2x -
(i) a b+
(j) p q-
8. Find values ox or which .x 3=
9. Simpliy nn
where .n 0!
10. Simpliy2
2
x
x
-
-and state which
value x cannot be.
1.9 Exercises
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41Chapter 1 Basic Arithmetic
1. Convert
0.45 to a raction(a)
14% to a decimal(b)
(c)85
to a decimal
78.5% to a raction(d)
0.012 to a percentage(e)
()1511
to a percentage
2. Evaluate as a raction.(a) 7 2-
(b) 5 1-
(c) 29-
1
3. Evaluate correct to 3 signifcant fgures.
(a) . .4 5 7 62 2+
(b) 4.30.3
(c)5.7
23
(d)..
3 8 101 3 10
6
9
#
#
(e)-
362
4. Evaluate
(a) | | | |3 2- -
(b) |4 5 |-
(c) 7 4 8#+
(d) [( ) ( ) ]3 2 5 1 4 8# '+ - -
(e) 4 3 9- + -
() 12- - -
(g) 24 6'- -
5. Simpliy
(a) x x x5 7 3# '
(b) (5 )y3 2
(c)( )
a b
a b9
5 4 7
(d)3
2x63d n
(e)
a b
ab5 6
4 0
e o
6. Evaluate
(a) 153
87
-
(b)76
332
#
(c) 943
'
(d)52
2101
+
(e) 1565#
7. Evaluate
(a) 4-
(b) 2361
(c) 5 2 32- -
(d) 4 3- as raction
(e) 382
() 2 1- -
(g) 249-
1
as a raction
(h) 4161
(i) 3 0-] g (j) 4 7 2 32- - - -
8. Simpliy
(a) a a14 9'
(b) x y5 36_ i
(c) p p p6 5 2# '
(d) 2b9 4
^ h (e)
(2 )
x y
x y10
7 3 2
9. Write in index orm.
(a) n
(b)1
x5
(c)1
x y+
(d) x 14 +
Test Yourself 1
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42 Maths In Focus Mathematics Preliminary Course
(e) a b7 +
() 2x
(g)2
1
x3
(h) x43
(i) (5 3)x 97 +
(j)1
m34
10. Write without ractional or negative
indices.
(a) a 5-
(b) 4n1
(c) 2( )x 1+1
(d) ( )x y 1- -
(e) (4 7)t 4- -
() 5( )a b+1
(g) 3x-
1
(h) 4b3
(i)3
( )x2 3+
4
(j)
-2x3
11. Show that a b a b#+ + when 5a =
and 3b = - .
12. Evaluate a2b4 when259
a = and 132
b = .
13. I31
a4
= c m and43
,b = evaluate ab3 as a
raction.
14. Increase 650 mL by 6%.
15. Johan spends31
o his 24-hour day
sleeping and41
at work.
How many hours does Johan spend(a)
at work?
What raction o his day is spent at(b)
work or sleeping?
I he spends 3 hours watching TV,(c)
what raction o the day is this?What percentage o the day does he(d)
spend sleeping?
16. The price o a car increased by 12%. I
the car cost $34 500 previously, what is
its new price?
17. Rachel scored 56 out o 80 or a maths
test. What percentage did she score?
18. Evaluate ,2118 and write your answer in
scientifc notation correct to 1 decimalplace.
19. Write in index orm.
(a) x
(b)1y
(c) 3x6 +
(d)(2 3)
1
x 11-
(e) y73
20. Write in scientifc notation.
0.000 013(a)
123 000 000 000(b)
21. Convert to a raction.
(a) .0 7
(b) .0 124
22. Write without the negative index.
(a) x 3-
(b) ( )a2 51
+
-
(c)b
a 5-c m
23. The number o people attending a
ootball match increased by 4% rom last
week. I there were 15 080 people at the
match this week, how many attended
last week?
24. Show that | |a b a b#+ + when
2a = - and 5.b = -
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43Chapter 1 Basic Arithmetic
1. Simpliy 843
332
4 1 .52
87
'+ -c cm m
2. Simpliy .53
125
180149
307
+ + -
3. Arrange in increasing order o size: 51%,
0.502, . ,0 5
.9951
4. Mark spends31 o his day sleeping,
121
o the day eating and201
o the day
watching TV. What percentage o the day
is let?
5. Write-
3642
as a rational number.
6. Express . .3 2 0 01425 ' in scientifc
notation correct to 3 signifcant fgures.
7. Vinh scored 17 2
1
out o 20 or a mathstest, 19 out o 23 or English and 55
21
out o 70 or physics. Find his average
score as a percentage, to the nearest
whole percentage.
8. Write .1 3274
as a rational number.
9. The distance rom the Earth to the moon
is .3 84 105# km. How long would it take
a rocket travelling at .2 13 10 km h4# to
reach the moon, to the nearest hour?
10. Evaluate. . .
. .0 2 5 4 1 3
8 3 4 13
'
#
+correct to
3 signifcant fgures.
11. Show that ( ) ( ) .2 2 1 2 2 2 1k k k1 1- + = -+ +
12. Find the value ob c
a3 2
in index orm i
.,a b c52
31
53
and4 3 2
= = - =c c cm m m
13. Which o the ollowing are rational
numbers: , . , , , . , ,3 0 34 2 3 1 5 0
7
3r- ?
14. The percentage o salt in 1 L o water is
10%. I 500 mL o water is added to this
mixture, what percentage o salt is there
now?
15. Simpliy| |
x
x
1
12
-
+
or .x 1!!
16. Evaluate2.4 3.31
4.3 2.93 2
1.3
6
+
-correct to
2 decimal places.
17. Write 15 g as a percentage o 2.5 kg.
18. Evaluate . .2 3 5 7 10.1 8 2#+ - correct to
3 signifcant fgures.
19. Evaluate( . )
. .
6 9 10
3 4 10 1 7 105 3
3 2
#
# #- +- -
and
express your answer in scientifc notation
correct to 3 signifcant fgures.
20. Prove | | | | | |a b a b#+ +
or all real a, b.
Challenge Exercise 1