1
Basic Arithmetic
TERMINOLOGY
Absolute value: The distance of a number from zero on
the number line. Hence it is the magnitude or value of a
number without the sign
Directed numbers: The set of integers or whole
numbers f -3, -2, -1, 0, 1, 2, 3, f
Exponent: Power or index of a number. For example 23
has a base number of 2 and an exponent of 3
Index: The power of a base number showing how
many times this number is multiplied by itself
e.g. 2 3 = 2 # 2 # 2. The index is 3
Indices: More than one index (plural)
Recurring decimal: A repeating decimal that does not
terminate e.g. 0.777777 … is a recurring decimal that can
be written as a fraction. More than one digit can recur
e.g. 0.14141414 ...
Scientific notation: Sometimes called standard notation.
A standard form to write very large or very small numbers
as a product of a number between 1 and 10 and a power
of 10 e.g. 765 000 000 is 7.65 # 10 8 in scientific notation
Chapter 1 Basic Arithmetic
INTRODUCTION
THIS CHAPTER GIVES A review of basic arithmetic skills, including knowing the
correct order of operations, rounding off, and working with fractions, decimals
and percentages. Work on significant figures, scientific notation and indices is
also included, as are the concepts of absolute values. Basic calculator skills are
also covered in this chapter.
Real Numbers
Types of numbers
Unreal or imaginary
numbers
Real numbers
Rational
numbers
Irrational
numbers
Integers
Integers are whole numbers that may be positive, negative or zero.
e.g. - 4, 7, 0, -11
a
Rational numbers can be written in the form of a fraction
b
•
3
where a and b are integers, b ! 0. e.g. 1 , 3.7, 0. 5, - 5
4
a
Irrational numbers cannot be written in the form of a fraction (that
b
is, they are not rational) e.g. 2 , r
EXAMPLE
Which of these numbers are rational and which are irrational?
• 3
r
3 , 1. 3, , 9 , , - 2.65
4
5
Solution
r
are irrational as they cannot be written as fractions (r is irrational).
4
•
3
13
1
1. 3 = 1 , 9 = and - 2.65 = - 2
so they are all rational.
3
1
20
3 and
3
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Maths In Focus Mathematics Extension 1 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
BRACKETS KEYS
Use ( and ) to open and close brackets. Always use them in pairs.
For example, to evaluate 40 - 3 ] 5 + 4 g
press 40 - 3 #
( 5 + 4 ) =
= 13
5.67 - 3.49
correct to 1 decimal place
1.69 + 2.77
To evaluate
press :
(
( 5.67 - 3.49 )
'
( 1.69 + 2.77 )
)
=
= 0.7
correct to 1 decimal place
PROBLEM
What is wrong with this calculation?
19 - 4
1+2
Press 19 - 4 ' 1 + 2 = 19 - 4 '1 + 2
Evaluate
What is the correct answer?
17
Chapter 1 Basic Arithmetic
MEMORY KEYS
Use STO to store a number in memory.
There are several memories that you can use at the same time—any letter from
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 find the reciprocal of x. For example, to evaluate
1
- 7.6 # 2.1
-1
=
press ( (-) 7.6 # 2.1 ) x
= - 0.063 (correct to 3 decimal places)
Rounding off
Rounding off is often done in everyday life. A quick look at a newspaper will
give plenty of examples. For example in the sports section, a newspaper may
report that 50 000 fans attended a football match.
An accurate number is not always necessary. There may have been exactly
49 976 people at the football game, but 50 000 gives an idea of the size of the
crowd.
EXAMPLES
1. Round off 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
Different calculators use
different keys so check
the instructions for your
calculator.
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Maths In Focus Mathematics Extension 1 Preliminary Course
2. Write 850 to the nearest hundred.
Solution
This number is exactly halfway between 800 and 900. When a number is
halfway, we round it off to the larger number.
` 850 = 900 to the nearest hundred
In this course you will need to round off decimals, especially when using
trigonometry or logarithms.
To round a number off to a certain number of decimal places, look at the
next digit to the right. If this digit is 5 or more, add 1 to the digit before it and
drop all the other digits after it. If the digit to the right is less than 5, leave the
digit before it and drop all the digits to the right.
EXAMPLES
1. Round off 0.6825371 correct to 1 decimal place.
Add 1 to the 6 as the 8 is
greater than 5.
Solution
0.6825371
#
` 0.6825371 = 0.7 correct to 1 decimal place
2. Round off 0.6825371 correct to 2 decimal places.
Drop off the 2 and all digits
to the right as 2 is smaller
than 5.
Solution
0.6825371
#
` 0.6825371 = 0.68 correct to 2 decimal places
3. Evaluate 3.56 ' 2.1 correct to 2 decimal places.
Check this on your
calculator. Add 1 to the
69 as 5 is too large to just
drop off.
Solution
3.56 ' 2.1 = 1.69 #
5238095
= 1.70 correct to 2 decimal places
Chapter 1 Basic Arithmetic
FIX KEY
Use MODE or SET UP to fix the number of decimal places (see the
instructions for your calculator). This will cause all answers to have a fixed number
of decimal places until the calculator is turned off or switched back to normal.
While using a fixed number of decimal places on the display, the
calculator still keeps track internally of the full number of 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
If 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
If you then set the calculator back to normal, the display will show the
full answer of 18.60052326.
The calculator does not round off at each step. If it did, the answer might
not be as accurate. This is an important point, since some students round
off each step in calculations and then wonder why they do not get the same
answer as other students and the textbook.
1.1 Exercises
1.
State which numbers are rational
and which are irrational.
(a) 169
(b) 0.546
(c) -17
r
(d)
3
•
(e) 0.34
(f)
218
(g) 2 2
1
(h)
27
(i) 17.4%
1
(j)
5
Don’t round off at
each step of a series of
calculations.
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Maths In Focus Mathematics Extension 1 Preliminary Course
2.
3.
Evaluate
(a) 20 - 8 ' 4
(b) 3 # 7 - 2 # 5
(c) 4 # ] 27 ' 3 g ' 6
(d) 17 + 3 # - 2
(e) 1.9 - 2 # 3.1
14 ' 7
(f)
-1 + 3
3
1 2
(g) 2 - #
5
5 3
3
1
1 4
8
(h)
5
6
5
5
'
8
6
(i)
1
1
+
4
8
1
7
3 5
10
(j)
1
1
1 4
2
7.
A crowd of 10 739 spectators
attended a tennis match.
Write this figure 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
of $7 635 718 last year. Round this
off to the nearest million.
Evaluate correct to 2 decimal
places.
(a) 2.36 + 4.2 ' 0.3
(b) ] 2.36 + 4.2 g ' 0.3
(c) 12.7 # 3.95 ' 5.7
(d) 8.2 ' 0.4 + 4.1# 0.54
(e) ] 3.2 - 6.5 g # ] 1.3 + 2.7 g
1
(f)
4.7 + 1.3
1
(g)
4.51 + 3.28
13. Round off 32.569148 to the
nearest unit.
0.9 + 1.4
(h)
5.2 - 3.6
5.33 + 2.87
(i)
1.23 - 3.15
(j)
4.
1.7 2 + 8.9 2 - 3.94 2
Round off 1289 to the nearest
hundred.
5.
Write 947 to the nearest ten.
6.
Round off 3200 to the nearest
thousand.
10. A company made a profit of
$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.
14. Round off 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. If pies are on special at 3 for
$2.38, find the cost of each pie.
18. Evaluate 7.48 correct to
2 decimal places.
6.4 + 2.3
correct to
8
1 decimal place.
19. Evaluate
20. Find the length of each piece
of material, to 1 decimal place,
if 25 m of material is cut into
7 equal pieces.
Chapter 1 Basic Arithmetic
21. How much will 7.5 m 2 of tiles
cost, at $37.59 per m2?
3.5 + 9.8
5.6 + 4.35
15.9 + 6.3 - 7.8
(d)
7.63 - 5.12
1
(e)
6.87 - 3.21
(c)
22. Divide 12.9 grams of salt into
7 equal portions, to 1 decimal
place.
23. The cost of 9 peaches is $5.72.
How much would 5 peaches cost?
9.91 - ] 9.68 - 5.47 g
5.39 2
correct to 1 decimal place.
25. Evaluate
24. Evaluate correct to 2 decimal
places.
(a) 17.3 - 4.33 # 2.16
(b) 8.72 # 5.68 - 4.9 # 3.98
DID YOU KNOW?
In building, engineering and other industries where accurate measurements are used, the
number of decimal places used indicates how accurate the measurements are.
For example, if a 2.431 m length of 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 of 2.431 m shows that the measurement of 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 off, as the error is half the smallest
measurement. In the above example, the measurement error is half a millimetre. The length of
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 of directed numbers will make your algebra skills much better.
^ - h KEY
Use this key to enter negative numbers. For example,
press (-) 3
=
9
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Maths In Focus Mathematics Extension 1 Preliminary Course
Adding and subtracting
To add: move to the right along the number line
To subtract: move to the left along the number line
-4
-3
-2
-1
0
1
Subtract
2
3
4
Add
EXAMPLES
You can also do these on a
calculator, or you may have
a different way of working
these out.
Evaluate
1. - 4 + 3
Solution
Start at - 4 and move 3 places to the right.
-4
-3
-2
-1
0
1
2
3
4
2
3
4
- 4 + 3 = -1
2. -1 - 2
Solution
Start at -1 and move 2 places to the left.
-4
-3
-2
-1
0
1
-1 - 2 = -3
Multiplying and dividing
To multiply or divide, follow these rules. This rule also works if there are two
signs together without a number in between e.g. 2 - -3
Same signs = +
+ + =+
- - =+
Different signs = + - =- + =-
Chapter 1 Basic Arithmetic
11
EXAMPLES
Evaluate
1. - 2 #7
Solution
Different signs (- 2 and + 7) give a negative answer.
- 2 # 7 = -14
2. -12 ' - 4
Solution
Same signs (-12 and - 4) 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 = -1 + 3
=2
1.2 Exercises
Evaluate
1.
-2 + 3
11. 5 - 3 # 4
2.
-7 - 4
12. - 2 + 7 # - 3
3.
8 # -7
13. 4 - 3 # - 2
4.
7 - ]-3 g
14. -1 - -2
5.
28 ' -7
15. 7 + - 2
6.
- 4 . 9 + 3 .7
16. 2 - ] -1 g
7.
- 2.14 - 5.37
17. - 2 + 15 ' 5
8.
4.8 # -7.4
18. - 2 # 6 # - 5
9.
1.7 - ] - 4.87 g
19. - 28 ' -7 # - 5
10. -
3
2
-1
5
3
20. ] - 3 g2
Start at -1 and move 3
places to the right.
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Maths In Focus Mathematics Extension 1 Preliminary Course
Fractions, Decimals and Percentages
Conversions
You can do all these
conversions on your
calculator using the
b
a or S + D key.
c
EXAMPLES
1. Write 0.45 as a fraction in its simplest form.
Solution
45
5
'
5
100
9
=
20
0.45 =
3
means 3 ' 8.
8
2. Convert
3
to a decimal.
8
Solution
0.375
8 g 3.000
3
So
= 0.375
8
3. Change 35.5% to a fraction.
Solution
35.5 2
#
100
2
71
=
200
35.5% =
4. Write 0.436 as a percentage.
Solution
Multiply by 100% to
change a fraction or
decimal to a percentage.
0.436 = 0.436 #100%
= 43.6%
5. Write 20 g as a fraction of 1 kg in its simplest form.
Solution
1 kg = 1000 g
20 g
20 g
=
1000 g
1 kg
1
=
50
Chapter 1 Basic Arithmetic
13
6. Find the percentage of people who prefer to drink Lemon Fuzzy, if 24
out of every 30 people prefer it.
Solution
24 100%
#
= 80%
30
1
Sometimes decimals repeat, or recur.
Example
•
1
= 0.33333333 f = 0. 3
3
There are different methods that can be used to change a recurring
decimal into a fraction. Here is one way of doing it. Later you will discover
another method when studying series. (See HSC Course book, Chapter 8.)
EXAMPLES
A rational number is
any number that can be
written as a fraction.
•
1. Write 0. 4 as a rational number.
Solution
Let
n = 0.44444 f
Then
10n = 4.44444 f
(2) - (1): 9n = 4
4
n=
9
( 1)
( 2)
Check this on your
calculator by dividing
4 by 9.
• •
2. Change 1.329 to a fraction.
Solution
n = 1.3292929 f
Let
Then 100n = 132.9292929 f
(2) - (1): 99n = 131.6
131.6
10
n=
#
99
10
1316
=
990
163
=1
495
( 1)
( 2)
CONTINUED
Try multiplying n by 10.
Why doesn’t this work?
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Maths In Focus Mathematics Extension 1 Preliminary Course
Another method
Let
n = 1.3292929 f
Then
10n = 13.2929292 f
and
1000n = 1329.292929 f
(2) - (1): 990n = 1316
1316
n=
990
163
=1
495
This method avoids decimals
in the fraction at the end.
(1 )
(2 )
1.3 Exercises
1.
2.
3.
Write each decimal as a fraction
in its lowest terms.
(a) 0.64
(b) 0.051
(c) 5.05
(d) 11.8
Change each fraction into a
decimal.
2
(a)
5
7
(b) 1
8
5
(c)
12
7
(d)
11
Convert each percentage to a
fraction in its simplest form.
(a) 2%
(b) 37.5%
(c) 0.1%
(d) 109.7%
4.
Write each percentage as a decimal.
(a) 27%
(b) 109%
(c) 0.3%
(d) 6.23%
5.
Write each fraction as a
percentage.
7
20
1
(b)
3
(a)
4
15
1
(d)
1000
(c) 2
6.
Write each decimal as a
percentage.
(a) 1.24
(b) 0.7
(c) 0.405
(d) 1.2794
7.
Write each percentage as a
decimal and as a fraction.
(a) 52%
(b) 7%
(c) 16.8%
(d) 109%
(e) 43.4%
1
(f) 12 %
4
8.
Write these fractions as recurring
decimals.
5
(a)
6
7
(b)
99
13
(c)
99
1
(d)
6
2
(e)
3
Chapter 1 Basic Arithmetic
5
33
1
(g)
7
2
(h) 1
11
31
99
13 + 6
(e)
7+4
(d) 1 -
(f)
9.
Express as fractions in lowest
terms.
•
(a) 0. 8
(b)
(c)
(d)
(e)
(f)
(g)
•
0. 2
•
1. 5
•
3. 7
• •
0. 67
• •
0. 54
•
0.15
•
(h) 0.216
• •
(i) 0.2 19
• •
(j) 1.074
10. Evaluate and express as a decimal.
5
(a)
3+6
(b) 8 - 3 ' 5
4+7
(c)
12 + 3
11. Evaluate and write as a fraction.
(a) 7.5 ' ] 4.1 + 7.9 g
15.7 - 8.9
(b)
4.5 - 1.3
6.3 + 1.7
(c)
12.3 - 8.9 + 7.6
4 .3
(d)
11.5 - 9.7
64
(e)
8100
12. Angel scored 17 out of 23 in a
class test. What was her score as a
percentage, to the nearest unit?
13. A survey showed that 31 out of
40 people watched the news on
Monday night. What percentage
of people watched the news?
14. What percentage of 2 kg is 350 g?
15. Write 25 minutes as a percentage
of an hour.
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 if a fraction has 3 in
the denominator? What about 9 in the denominator? What about 11?
Can you predict what fraction certain recurring decimals will be? What
denominator would 1 digit recurring give? What denominator would you
have for 2 digits recurring?
Operations with fractions, decimals and percentages
You will need to know how to work with fractions without using a calculator,
as they occur in other areas such as algebra, trigonometry and surds.
15
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Maths In Focus Mathematics Extension 1 Preliminary Course
The examples on fractions show how to add, subtract, multiply or divide
fractions both with and without the calculator. The decimal examples will
help with some simple multiplying and the percentage examples will be useful
in Chapter 8 of the HSC Course book when doing compound interest.
Most students use their calculators for decimal calculations. However, it
is important for you to know how to operate with decimals. Sometimes the
calculator can give a wrong answer if the wrong key is pressed. If you can
estimate the size of the answer, you can work out if it makes sense or not. You
can also save time by doing simple calculations in your head.
DID YOU KNOW?
Some countries use a comma for the decimal point—for example, 0,45 for 0.45.
This is the reason that our large numbers now have spaces instead of commas between
digits—for example, 15 000 rather than 15,000.
EXAMPLES
1. Evaluate 1
3
2
- .
5
4
Solution
1
3
3
2
7
- = 5
4
5
4
28
15
=
20
20
13
=
20
2. Evaluate 2
1
' 3.
2
Solution
2
3
5
1
'3 = '
2
2
1
5 1
= #
2 3
5
=
6
3. Evaluate 0.056 # 100.
Move the decimal point
2 places to the right.
Solution
0.056 #100 = 5.6
Chapter 1 Basic Arithmetic
17
4. Evaluate 0.02 # 0.3.
Multiply the numbers
and count the number
of decimal places in
the question.
Solution
0.02 # 0.3 = 0.006
5. Evaluate
8.753
.
10
Solution
Move the decimal
point 1 place to
the left.
8.753 ' 10 = 0.8753
1
6. The price of a $75 tennis racquet increased by 5 %. Find the new
2
price.
Solution
1
5 % = 0.055
2
1
` 5 % of $75 = 0.055#$75
2
= $4.13
1
or 105 % of $75 = 1.055#$75
2
= $79.13
So the price increases by $4.13 to $79.13.
7. The price of a book increased by 12%. If it now costs $18.00, what did
it cost before the price rise?
Solution
The new price is 112% (old price 100%, plus 12%)
$18.00
` 1% =
112
$18.00 100
100% =
#
112
1
= $16.07
So the old price was $16.07.
1.4 Exercises
1.
Write 18 minutes as a fraction of
2 hours in its lowest terms.
2.
Write 350 mL as a fraction of
1 litre in its simplest form.
3.
Evaluate
3
1
(a)
+
5
4
2
7
-2
5
10
3
2
(c) #1
5
4
3
(d) ' 4
7
3
2
(e) 1 ' 2
5
3
(b) 3
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Maths In Focus Mathematics Extension 1 Preliminary Course
3
of $912.60.
5
4.
Find
5.
5
Find of 1 kg, in grams correct
7
to 1 decimal place.
6.
Trinh spends
sleeping,
1
of her day
3
7
1
at work and
24
12
eating. What fraction of the day
is left?
7.
I get $150.00 a week for a casual
1
job. If I spend
on bus fares,
10
2
1
on lunches and on outings,
15
3
how much money is left over for
savings?
8.
John grew by
9.
17
of his height
200
this year. If he was 165 cm tall
last year, what is his height now,
to the nearest cm?
Evaluate
(a) 8.9 + 3
(b) 9 - 3.7
(c) 1.9 #10
(d) 0.032 #100
(e) 0.7 # 5
(f) 0.8 # 0.3
(g) 0.02 # 0.009
(h) 5.72 #1000
8.74
(i)
100
(j) 3.76 # 0.1
10. Find 7% of $750.
11. Find 6.5% of 845 mL.
12. What is 12.5% of 9217 g?
13. Find 3.7% of $289.45.
14. If Kaye makes a profit of $5 by
selling a bike for $85, find the
profit as a percentage of the
selling price.
15. Increase 350 g by 15%.
1
16. Decrease 45 m by 8 %.
2
17. The cost of a calculator is now
$32. If 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
year’s height. How high was the
tree then, to 1 decimal place?
19. This month there has been a
4.9% increase in stolen cars. If
546 cars were stolen last month,
how many were stolen this
month?
20. George’s computer cost $3500. If
it has depreciated by 17.2%, what
is the computer worth now?
Chapter 1 Basic Arithmetic
19
PROBLEM
If both the hour hand and minute hand start at the same position at
12 o’clock, when is the first time, correct to a fraction of a minute, that
the two hands will be together again?
Powers and Roots
A power (or index) of a number shows how many times a number is
multiplied by itself.
EXAMPLES
1. 4 3 = 4 # 4 # 4 = 64
2. 2 5 = 2 # 2 # 2 # 2 # 2 = 32
A root of a number is the inverse of the power.
EXAMPLES
1.
36 = 6 since 6 2 = 36
2.
3
8 = 2 since 2 3 = 8
3.
6
64 = 2 since 2 6 = 64
DID YOU KNOW?
Many formulae use indices (powers and roots).
For example the compound interest formula that you will study in Chapter 8 of the HSC
n
Course book is A = P ^ 1 + r h
4
Geometry uses formulae involving indices, such as V = rr 3. Do you know what this
3
formula is for?
In Chapter 7, the formula for the distance between 2 points on a number plane is
d=
2
(x 2 - x 1) + (y 2 - y 1)
2
See if you can find other formulae involving indices.
In 4 3 the 4 is called the base
number and the 3 is called
the index or power.
20
Maths In Focus Mathematics Extension 1 Preliminary Course
POWER AND ROOT KEYS
Use the x 2 and x 3 keys for squares and cubes.
y
Use the x or ^ key to find powers of numbers.
key for square roots.
Use the
These laws work for any m
and n, including fractions and
negative numbers.
Use the
3
key for cube roots.
Use the
x
for other roots.
Index laws
There are some general laws that simplify calculations with indices.
am # an = am + n
Proof
a m # a n = (a # a #f# a) # (a # a #f# a)
14444244443 14444244443
m times
n times
=a
#
#
f
#
a
a
14444244443
m + n times
= am + n
am ' an = am - n
Proof
am
an
a # a #f# a (m times)
=
a # a #f# a (n times)
a # a #f# a (m - n times)
=
1
= am - n
am ' an =
(a m)n = a mn
Proof
(a m) n = a m # a m # a m #f# a m
= am + m + m + f + m
= a mn
(n times)
(n times)
Chapter 1 Basic Arithmetic
(ab) n = a n b n
Proof
(ab) n = ab # ab # ab #f# ab (n times)
= (a # a #f# a) # (b # b #f# b)
14444244443 14444244443
n times
n times
= an bn
a n an
c m = n
b
b
Proof
a n a a a
a
c m = # # #f#
b
b b b
b
a # a # a #f # a
=
b # b # b #f # b
an
= n
b
(n times)
(n times)
(n times)
EXAMPLES
Simplify
1. m 9 # m 7 ' m 2
Solution
m9 #m7 ' m2 = m9 + 7 - 2
= m 14
2. (2y 4)3
Solution
(2y 4) 3 = 2 3 (y 4) 3
= 23 y4 # 3
= 8y 12
CONTINUED
21
22
Maths In Focus Mathematics Extension 1 Preliminary Course
3.
(y 6) 3 # y - 4
y5
Solution
(y 6) 3 # y - 4
y5
=
=
=
y 18 # y - 4
y5
y 18 + (- 4)
y5
y
14
y5
= y9
1.5 Exercises
1.
Evaluate without using a
calculator.
(a) 5 3 # 2 2
(b) 3 4 + 8 2
1 3
(c) c m
4
(d)
(e)
2.
3.
3
4
(h)
(i)
(j)
(k)
5
x2
p
y9
w6 # w7
(m)
w3
2
p #(p 3) 4
(n)
p9
6
x ' x7
(o)
x2
2
a # ( b 2) 6
(p)
a4 # b9
(x 2) - 3 #(y 3) 2
(q)
x -1 # y 4
(l) f
27
16
Evaluate correct to 1 decimal
place.
(a) 3.7 2
(b) 1.06 1.5
(c) 2.3 - 0.2
(d) 3 19
(e) 3 34.8 - 1.2 # 43.1
1
(f) 3
0.99 + 5.61
Simplify
(a) a 6 # a 9 # a 2
(b) y 3 # y - 8 # y 5
(c) a -1 # a -3
1
1
(d) w 2 # w 2
(e) x 6 ' x
(f) p 3 ' p - 7
y 11
(g) 5
y
(x 7) 3
(2x 5) 2
(3y - 2) 4
a3 #a5 ' a7
4.
Simplify
(a) x 5 # x 9
(b) a -1 # a - 6
m7
(c)
m3
(d) k 13 # k 6 ' k 9
(e) a - 5 # a 4 # a - 7
2
3
(f) x 5 # x 5
m5 # n4
(g) 4
m # n2
Chapter 1 Basic Arithmetic
1
1
p2 # p2
(h)
10. (a) Simplify
p
(i) (3x 11) 2
(x 4) 6
(j)
x3
5.
2
1
2 6
11. Evaluate (a ) when a = c m .
3
12. Evaluate
b=
(2m 7) 3
m4
xy 3 #(xy 2) 4
(f)
xy
8 4
(2k )
(g)
(6k 3) 3
y 12
7
(h) _ 2y 5 i #
8
y=
Evaluate a3b2 when a = 2 and
3
b= .
4
7.
If x =
of
8.
9.
2
1
and y = , find the value
3
9
x3 y2
xy 5
.
1
1
1
, b = and c = ,
4
2
3
a2 b3
evaluate 4 as a fraction.
c
If a =
a b
.
a8 b7
11
(a) Simplify
8
(b) Hence evaluate
a=
a 11 b 8
when
a8 b7
5
2
and b = as a fraction.
5
8
x5 y5
when x =
1
and
3
14. Evaluate
k-5
1
when k = .
3
k-9
15. Evaluate
a4 b6
3
when a = and
3
2 2
4
a (b )
b=
6.
x4 y7
2
.
9
-3
p
a3 b6
1
when a = and
2
b4
2
.
3
13. Evaluate
a6 # a4
o
a 11
3
5xy 9
x8 # y3
p5 q8 r4
4 3
(d) (7a5b)2
(j) f
.
as a
p4 q6 r2
7
2
fraction when p = , q = and
8
3
3
r= .
4
a 8
(b) c m
b
4a 3
(c) d 4 n
b
(i) e
p4 q6 r2
(b) Hence evaluate
Simplify
(a) (pq 3) 5
(e)
p5 q8 r4
1
.
9
a6 # b3
as a fraction
a5 # b2
3
1
when a = and b = .
4
9
16. Evaluate
a2 b7
as a fraction in
a3 b
2 4
index form when a = c m and
5
5 3
b=c m.
8
17. Evaluate
18. Evaluate
(a 3) 2 b 4 c
as a fraction
a (b 2) 4 c 3
6
1
7
when a = , b = and c = .
7
3
9
23
24
Maths In Focus Mathematics Extension 1 Preliminary Course
Negative and zero indices
Class Investigation
Explore zero and negative indices by looking at these questions.
For example simplify x 3 ' x 5 using (i) index laws and (ii) cancelling.
(i) x 3 ' x 5 = x - 2 by index laws
3
x# x# x
(ii) x =
5
x
x# x# x# x
#
x
1
= 2
x
1
So x - 2 = 2
x
Now simplify these questions by (i) index laws and (ii) cancelling.
(a) x 2 ' x 3
(b) x 2 ' x 4
(c) x 2 ' x 5
(d) x 3 ' x 6
(e) x 3 ' x 3
(f) x 2 ' x 2
(g) x ' x 2
(h) x 5 ' x 6
(i) x 4 ' x 7
(j) x ' x 3
Use your results to complete:
x0 =
x-n =
x0 = 1
Proof
xn ' xn = xn - n
= x0
xn
xn ' xn = n
x
=1
`
x0 = 1
Chapter 1 Basic Arithmetic
x-n =
1
xn
Proof
x0 ' xn = x0 - n
= x-n
x0
x0 ' xn = n
x
1
= n
x
1
` x-n = n
x
EXAMPLES
0
1. Simplify e
Solution
ab 5 c
o .
abc 4
0
e
ab 5 c
o =1
abc 4
2. Evaluate 2 - 3 .
Solution
1
23
1
=
8
2-3 =
3. Write in index form.
1
x2
3
(b) 5
x
1
(c)
5x
1
(d)
x +1
(a)
CONTINUED
25
26
Maths In Focus Mathematics Extension 1 Preliminary Course
Solution
1
= x-2
x2
3
(b) 5 = 3# 15
x
x
-5
= 3x
1
1 1
= #x
(c)
5x
5
1 -1
= x
5
1
1
=
(d)
x +1
(x + 1) 1
= ] x + 1 g-1
(a)
4. Write a−3 without the negative index.
Solution
a-3 =
1
a3
1.6 Exercises
1.
Evaluate as a fraction or whole
number.
(a) 3 - 3
(b) 4 - 1
(c) 7 - 3
(d) 10 - 4
(e) 2 - 8
(f) 60
(g) 2 - 5
(h) 3 - 4
(i) 7 - 1
(j) 9 - 2
(k) 2 - 6
(l) 3 - 2
(m) 40
(n) 6 - 2
(o) 5 - 3
(p) 10 - 5
(q) 2 - 7
(r) 2 0
(s) 8 - 2
(t) 4 - 3
2.
Evaluate
(a) 2 0
1 -4
(b) c m
2
2 -1
(c) c m
3
5 -2
(d) c m
6
x + 2y 0
p
(e) f
3x - y
1 -3
(f) c m
5
3 -1
(g) c m
4
1 -2
(h) c m
7
2 -3
(i) c m
3
1 -5
(j) c m
2
3 -1
(k) c m
7
Chapter 1 Basic Arithmetic
8 0
(l) c m
9
6 -2
(m)c m
7
9 -2
(n) c m
10
6 0
(o) c m
11
1 -2
(p) c - m
4
2 -3
(q) c - m
5
2 -1
(r) c - 3 m
7
3 0
(s) c - m
8
1 -2
(t) c - 1 m
4
3.
Change into index form.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
1
m3
1
x
1
p7
1
d9
1
k5
1
x2
2
x4
3
y2
1
2z 6
3
5t 8
2
7x
5
2m 6
2
(m) 7
3y
(l)
1
(3x + 4) 2
1
(o)
( a + b) 8
1
(p)
x-2
(n)
1
(5p + 1) 3
2
(r)
(4t - 9) 5
1
(s)
4 (x + 1) 11
5
(t)
9 ( a + 3 b) 7
(q)
4.
Write without negative indices.
(a) t - 5
(b) x - 6
(c) y - 3
(d) n - 8
(e) w - 10
(f) 2x -1
(g) 3m - 4
(h) 5x - 7
(i) ]2xg- 3
(j) ] 4n g-1
(k) ] x + 1 g- 6
(l) ^ 8y + z h-1
(m) ]k - 3g- 2
(n) ^ 3x + 2y h- 9
1 -5
(o) b x l
1 -10
(p) c y m
2 -1
(q) d n
p
1 -2
m
a+b
x + y -1
(s) e x - y o
(r) c
(t) e
2w - z - 7
o
3x + y
27
28
Maths In Focus Mathematics Extension 1 Preliminary Course
Fractional indices
Class Investigation
Explore fractional indices by looking at these questions.
1 2
For example simplify (i) ` x 2 j and (ii) ^ x h .
1 2
(i) ` x 2 j = x 1
=x
2
(ii) ^ x h = x
2
^ by index laws h
1 2
So ` x 2 j = ^ x h = x
2
1
`
x2 =
x
Now simplify these questions.
1
(a) ^ x 2 h 2
x2
(b)
1 3
(c) ` x 3 j
1
(d) ^ x 3 h 3
3
(e) ^ 3 x h
(f)
3
x3
1 4
4
(g) ` x j
1
(h) ^ x 4 h 4
4
(i) ^ 4 x h
(j)
4
x4
Use your results to complete:
1
xn =
1
n
a =n a
Proof
1 n
`an j = a
^ n a hn = a
1
n
` a =n a
^ by index laws h
Chapter 1 Basic Arithmetic
EXAMPLES
1. Evaluate
(a) 49
1
2
1
(b) 27 3
Solution
1
2
(a) 49 = 49
=7
1
3
(b) 27 = 3 27
=3
2. Write
3x - 2 in index form.
Solution
1
3x - 2 = (3x - 2) 2
1
3. Write (a + b) 7 without fractional indices.
Solution
1
( a + b) 7 = 7 a + b
Putting the fractional and negative indices together gives this rule.
a
1
-n
=
n
1
a
Here are some further rules.
m
n
a = n am
= (n a ) m
Proof
m
1 m
m
n
1
n
n
n
a = `a j
m
= ^n a h
a = ^ am h
= n am
29
30
Maths In Focus Mathematics Extension 1 Preliminary Course
a -n
b n
c m = bal
b
Proof
a -n
1
c m =
b
a n
c m
b
1
= n
a
bn
an
bn
bn
=1# n
a
bn
= n
a
b n
= bal
=1'
EXAMPLES
1. Evaluate
4
(a) 8 3
(b) 125
-
1
3
2 -3
(c) c m
3
Solution
4
(a) 8 3 = (3 8 ) 4 (or 3 8 4 )
= 24
= 16
(b) 125
-
1
3
=
1
1
125 3
1
=3
125
1
=
5
Chapter 1 Basic Arithmetic
-3
(c) c 2 m
3
3 3
=c m
2
27
=
8
3
=3
8
2. Write in index form.
x5
(a)
(b)
1
(4x - 1) 2
2
3
Solution
5
x5 = x 2
1
(a)
(b)
(4x - 1)
2
3
2
=
1
2
(4x 2 - 1) 3
-
= (4x 2 - 1)
3. Write r
-
3
5
2
3
without the negative and fractional indices.
Solution
r
-
3
5
=
=
1
3
r5
1
5
r3
DID YOU KNOW?
Nicole Oresme (1323–82) was the first mathematician to use fractional indices.
John Wallis (1616–1703) was the first person to explain the significance of zero, negative
and fractional indices. He also introduced the symbol 3 for infinity.
Do an Internet search on these mathematicians and find out more about their work and
backgrounds. You could use keywords such as indices and infinity as well as their names to find
this information.
31
32
Maths In Focus Mathematics Extension 1 Preliminary Course
1.7 Exercises
1.
3.
Evaluate
(a) 81
1
2
Write without fractional indices.
1
(a) y 3
1
2
(b) y 3
(b) 27 3
1
(c) x
(c) 16 2
1
-
1
2
1
(d) (2x + 5) 2
(d) 8 3
1
(e) (3x - 1)
(e) 49 2
1
-
1
2
1
(f) (6q + r) 3
(f) 1000 3
1
(g) (x + 7)
(g) 16 4
-
2
5
1
(h) 64 2
(i) 64
(j) 1
4.
1
3
(l) 32
t
(a)
1
7
(k) 81
Write in index form.
(b)
5
x3
(c)
1
4
(d)
(e)
1
5
3
1
(m) 0 8
(f)
(n) 125
1
3
(g)
1
1
1
(r) 9
(s) 8
(i)
(t) 64
2.
(x - 2) 2
1
(j)
2 y+7
5
(k) 3
x+4
2
(l)
3 y2 - 1
3
(m)
5 4 (x 2 + 2) 3
3
2
-
1
3
-
2
3
Evaluate correct to 2 decimal
places.
1
(a) 23 4
(b) 4 45.8
(c)
(d)
(e)
7
5
8
5 .9 # 3 .7
8.79 - 1.4
4
(f)
1.24 + 4.3 2
1
12.9
3 .6 - 1 .4
1 .5 + 3 .7
(3x + 1) 5
1
(h)
(q) 256 4
9-x
4s + 1
1
2t + 3
1
(5x - y) 3
(o) 343 3
(p) 128 7
y
5.
3
Write in index form and simplify.
(a) x x
x
(b) x
x
(c) 3
x
x2
(d) 3
x
(e) x 4 x
Chapter 1 Basic Arithmetic
6.
Expand and simplify, and write in
index form.
7.
(a) ( x + x) 2
(b) (3 a + 3 b ) (3 a - 3 b )
1 2
(c) f p +
p
p
1 2
)
x
x ( x 2 - 3x + 1 )
(d) ( x +
(e)
x3
33
Write without fractional or
negative indices.
(a) (a - 2b)
(b) (y - 3)
-
-
1
3
2
3
-
4
7
-
2
9
(c) 4 (6a + 1)
( x + y)
(d)
3
-
5
4
6 (3 x + 8 )
(e)
7
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 of Fundy, Canada, has the largest tidal changes in the world. About 100 000 000 000
tons of water are moved with each tide change.
The dinosaurs dwelt on Earth for 185 000 000 years until they died out 65 000 000 years ago.
The width of one plant cell is about 0.000 06 m.
In 2005, the total storage capacity of dams in Australia was 83 853 000 000 000 litres and
households in Australia used 2 108 000 000 000 litres of 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 #10 8
Write the number
between 1 and 10
and count the decimal
places moved.
2. Write 7.1#10 -5 as a decimal number.
Solution
7.1#10
-5
= 7.1 ' 10
= 0.000 071
5
Count 5 places to
the left.
34
Maths In Focus Mathematics Extension 1 Preliminary Course
SCIENTIFIC NOTATION KEY
Use the EXP or #10 x key to put numbers in scientific notation.
For example, to evaluate 3.1#10 4 ' 2.5 #10 - 2,
press 3.1 EXP 4 ' 2.5 EXP (-) 2 =
= 1 240 000
DID YOU KNOW?
Engineering notation is similar to scientific notation, except the powers of 10 are always
multiples of 3. For example,
3.5 # 10
3
15.4 # 10
-6
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 of
87 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.
EXAMPLES
12 000 = 1.2 #10 4
0.000 043 5 = 4.35#10 - 5
0.020 7 = 2.07 #10 - 2
(2 significant figures)
(3 significant figures)
(3 significant figures)
When rounding off to significant figures, use the usual rules for rounding off.
Chapter 1 Basic Arithmetic
35
EXAMPLES
1. Round off 4 592 170 to 3 significant figures.
Solution
4 592 170 = 4 590 000 to 3 significant figures
2. Round off 0.248 391 to 2 significant figures.
Solution
0.248 391 = 0.25 to 2 significant figures
3. Round off 1.396 794 to 3 significant figures.
Solution
1.396 794 = 1.40 to 3 significant figures
1.8 Exercises
1.
Write in scientific notation.
(a) 3 800
(b) 1 230 000
(c) 61 900
(d) 12 000 000
(e) 8 670 000 000
(f) 416 000
(g) 900
(h) 13 760
(i) 20 000 000
(j) 80 000
3.
Write as a decimal number.
(a) 3.6 #10 4
(b) 2.78 #10 7
(c) 9.25#10 3
(d) 6.33#10 6
(e) 4 #10 5
(f) 7.23#10 - 2
(g) 9.7 #10 - 5
(h) 3.8 # 10 - 8
(i) 7 #10 - 6
(j) 5#10 - 4
2.
Write in scientific notation.
(a) 0.057
(b) 0.000 055
(c) 0.004
(d) 0.000 62
(e) 0.000 002
(f) 0.000 000 08
(g) 0.000 007 6
(h) 0.23
(i) 0.008 5
(j) 0.000 000 000 07
4.
Round these numbers to
2 significant figures.
(a) 235 980
(b) 9 234 605
(c) 10 742
(d) 0.364 258
(e) 1.293 542
(f) 8.973 498 011
(g) 15.694
(h) 322.78
(i) 2904.686
(j) 9.0741
Remember to put
the 0’s in!
36
Maths In Focus Mathematics Extension 1 Preliminary Course
5.
Evaluate correct to 3 significant
figures.
(a) 14.6 # 0.453
(b) 4.8 ' 7
(c) 4.47 + 2.59 #1.46
1
(d)
3.47 - 2.7
6.
Evaluate 4.5#10 4 # 2.9 #10 5,
giving your answer in scientific
notation.
7.
Calculate
8.72 #10 - 3
and write
1.34 #10 7
your answer in standard form
correct to 3 significant figures.
Investigation
A logarithm is an index. It is a way of finding the power (or index) to
which a base number is raised. For example, when solving 3 x = 9, 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.
The a is called the base
number and the x is the
index or power.
If a x = y then log a y = x
1. The expression log7 49 means the power of 7 that gives 49.
The solution is 2 since 7 2 = 49.
2. The expression log2 16 means the power of 2 that gives 16.
The solution is 4 since 2 4 = 16.
Can you evaluate these logarithms?
1. log3 27
2. log5 25
3. log10 10 000
4. log2 64
5. log4 4
6. log7 7
7. log3 1
8. log4 2
1
9. log 3
3
1
10. log 2
4
Chapter 1 Basic Arithmetic
37
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 c 2 = 9 gives two solutions, c = !3.
However when solving c 2 = 9, using Pythagoras’ theorem, we only use
the positive answer, c = 3, as this gives the length of the side of a triangle. The
negative answer doesn’t make sense.
We don’t 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
of our cars!
Absolute value definitions
We write the absolute value of x as x
x =)
We can also define
x as the distance
of x from 0 on the
number line. We will
use this in Chapter 3.
x when x $ 0
- x when x 1 0
EXAMPLES
1. Evaluate 4 .
Solution
4 = 4 since 4 $ 0
CONTINUED
38
Maths In Focus Mathematics Extension 1 Preliminary Course
2. Evaluate - 3 .
Solution
-3 = - ] - 3 g since - 3 1 0
=3
The absolute value has some properties shown below.
Properties of absolute value
| ab | = | a |#| b |
e.g. | 2 # - 3 | = | 2 |#| - 3 | = 6
|a | = a
e.g. | - 3 | 2 = ] - 3 g2 = 9
2
2
a2 = | a |
|- a | = | a |
|a - b | = | b - a |
| a + b |#| a | + | b |
e.g. 5 2 = | 5 | = 5
e.g. | -7 | = | 7 | = 7
e.g. | 2 - 3 | = | 3 - 2 | = 1
e.g. | 2 + 3 | = | 2 | + | 3 | but | - 3 + 4 | 1 | - 3 | + | 4 |
EXAMPLES
1. Evaluate 2 - -1 + - 3 2.
Solution
2 - -1 + - 3 2 = 2 - 1 + 3 2
=2 -1 + 9
= 10
2. Show that a + b # a + b when a = - 2 and b = 3.
Solution
LHS means Left Hand Side.
LHS = a + b
= -2 + 3
= 1
=1
Chapter 1 Basic Arithmetic
RHS means Right Hand Side.
RHS = a + b
= -2 + 3
= 2+3
=5
Since 11 5
a+b # a + b
3. Write expressions for 2x - 4 without the absolute value signs.
Solution
2x - 4 = 2x - 4 when 2x - 4 $ 0
i.e.
2x $ 4
x$2
2x - 4 = - ] 2x - 4 g when 2x - 4 1 0
= - 2x + 4 i.e.
2x 1 4
x12
Class Discussion
Are these statements true? If so, are there some values for which the
expression is undefined (values of x or y that the expression cannot
have)?
2.
x
=1
x
2x = 2x
3.
2x = 2 x
4.
x + y = x+y
5.
2
x = x2
6.
7.
3
x = x3
x +1 = x +1
1.
3x - 2
=1
3x - 2
x
9.
=1
x2
10. x $ 0
8.
Discuss absolute value and its definition in relation to these statements.
39
40
Maths In Focus Mathematics Extension 1 Preliminary Course
1.9 Exercises
1.
2.
3.
Evaluate
(a) 7
(b) - 5
(c) - 6
(d) 0
(e) 2
(f) -11
(g) - 2 3
(h) 3 - 8
2
(i) - 5
(j) - 5 3
Evaluate
(a) 3 + - 2
(b) - 3 - 4
(c) - 5 + 3
(d) 2 #-7
(e) - 3 + -1
2
(f) 5 - - 2 # 6
(g) - 2 + 5# -1
(h) 3 - 4
(i) 2 - 3 - 3 - 4
(j) 5 - 7 + 4 - 2
(i)
(j)
Show that a + b # a + b
when
(a) a = 2 and b = 4
(b) a = -1 and b = - 2
(c) a = - 2 and b = 3
(d) a = - 4 and b = 5
(e) a = -7 and b = - 3.
6.
Show that x 2 = x when
(a) x = 5
(b) x = - 2
(c) x = - 3
(d) x = 4
(e) x = - 9.
7.
Use the definition of absolute
value to write each expression
without the absolute value signs
(a) x + 5
(b) b - 3
(c) a + 4
(d) 2y - 6
(e) 3x + 9
(f) 4 - x
(g) 2k + 1
(h) 5x - 2
(i) a + b
(j) p - q
8.
Find values of x for which x = 3.
9.
n
Simplify n where n ! 0.
a = 5 and b = 2
a = -1 and b = 2
a = - 2 and b = - 3
a = 4 and b = 7
a = -1 and b = - 2.
Write an expression for
(a) a when a 2 0
(b)
(c)
(d)
(e)
(f)
(g)
a when a 1 0
a when a = 0
3a when a 2 0
3a when a 1 0
3a when a = 0
a + 1 when a 2 -1
x - 2 when x 2 2
x - 2 when x 1 2.
5.
Evaluate a - b if
(a)
(b)
(c)
(d)
(e)
4.
(h) a + 1 when a 1 -1
x-2
and state which
x-2
value x cannot be.
10. Simplify
Chapter 1 Basic Arithmetic
Test Yourself 1
1.
2.
Convert
(a) 0.45 to a fraction
(b) 14% to a decimal
5
(c)
to a decimal
8
(d) 78.5% to a fraction
(e) 0.012 to a percentage
11
(f)
to a percentage
15
(c) 9
-
(b)
(c)
(d)
(e)
7.
1
2
4.5 2 + 7.6 2
(e) 6
4.
5.
2
(e) 8 3
(f) - 2 - 1
1.3#10 9
3.8 #10 6
-
(g) 49
2
3
-
1
2
as a fraction
1
4
Evaluate
(a) |-3 | -| 2 |
(b) | 4 - 5 |
(c) 7 + 4 # 8
(d) [(3 + 2)#(5 - 1) - 4] ' 8
(e) - 4 + 3 - 9
(f) - 2 - -1
(g) - 24 ' - 6
(h) 16
(i) ] -3 g0
(j) 4 - 7 2 - -2 - 3
8.
(a) x 5 # x 7 ' x 3
(b) (5y 3) 2
(a 5) 4 b 7
(c)
a9 b
3
2x 6 n
(d) d
3
0
ab 4
o
a5 b6
Simplify
(a) a 14 ' a 9
6
(b) _ x 5 y 3 i
(c) p 6 # p 5 ' p 2
4
(d) ^ 2b 9h
(2x 7) 3 y 2
(e)
x 10 y
Simplify
(e) e
Evaluate
(a) - 4
(b) 36 2
(c) - 5 2 - 2 3
(d) 4 - 3 as fraction
(b) 4.3 0.3
2
(c) 3
5.7
(d)
3
7
5
8
6
2
#3
7
3
3
9'
4
2
1
+2
5
10
5
15#
6
1
Evaluate correct to 3 significant figures.
(a)
Evaluate
(a) 1
Evaluate as a fraction.
(a) 7 - 2
(b) 5 -1
3.
6.
9.
Write in index form.
n
1
(b) 5
x
1
(c)
x+y
(a)
(d)
4
x +1
41
42
Maths In Focus Mathematics Extension 1 Preliminary Course
(e)
7
(c) If he spends 3 hours watching TV,
what fraction of the day is this?
(d) What percentage of the day does he
spend sleeping?
a+b
2
(f) x
1
(g)
2x 3
(h)
3
x4
(i)
7
(5x + 3) 9
1
4
m3
(j)
10. Write without fractional or negative
indices.
(a) a - 5
1
(b) n 4
1
(c) (x + 1) 2
(d) (x - y) -1
(e) (4t - 7) - 4
1
(f) (a + b) 5
(g) x
3
(h) b 4
(j) x
-
17. Rachel scored 56 out of 80 for a maths
test. What percentage did she score?
18. Evaluate 2118, and write your answer in
scientific notation correct to 1 decimal
place.
19. Write in index form.
(a) x
1
(b) y
x+3
1
(d)
(2x - 3) 11
1
3
(i) (2x + 3)
16. The price of a car increased by 12%. If
the car cost $34 500 previously, what is
its new price?
4
3
3
2
11. Show that a + b # a + b when a = 5
and b = - 3.
9
2
12. Evaluate a b when a =
and b = 1 .
25
3
2 4
3
1 4
13. If a = c m and b = , evaluate ab 3 as a
4
3
fraction.
14. Increase 650 mL by 6%.
1
of his 24-hour day
3
1
sleeping and at work.
4
(a) How many hours does Johan spend
at work?
(b) What fraction of his day is spent at
work or sleeping?
15. Johan spends
(c)
6
(e)
3
y7
20. Write in scientific notation.
(a) 0.000 013
(b) 123 000 000 000
21. Convert to a fraction.
•
(a) 0. 7
• •
(b) 0.124
22. Write without the negative index.
(a) x - 3
(b) (2a + 5)- 1
a -5
(c) c m
b
23. The number of people attending a
football match increased by 4% from last
week. If there were 15 080 people at the
match this week, how many attended
last week?
24. Show that | a + b | # a + b when
a = - 2 and b = - 5.
Chapter 1 Basic Arithmetic
Challenge Exercise 1
3
2
2
7
+ 3 m ' c4 - 1 m.
4
5
3
8
1.
Simplify c 8
2.
3
5
149
7
Simplify +
+
.
5
12
180
30
3.
Arrange in increasing order of size: 51%,
• 51
0.502, 0. 5,
.
99
4.
1
1
of his day sleeping,
3
12
1
of the day eating and
of the day
20
watching TV. What percentage of the day
is left?
5.
Write 64
6.
Express 3.2 ' 0.014 in scientific
notation correct to 3 significant figures.
7.
Mark spends
-
2
3
as a rational number.
11. Show that 2 (2 k - 1) + 2 k + 1 = 2 (2 k + 1 - 1) .
12. Find the value of
3 2
2 4
1 3
a = c m , b = c - m and c = c m .
5
5
3
13. Which of the following are rational
•
3
numbers: 3 , - 0.34, 2, 3r, 1. 5, 0, ?
7
14. The percentage of salt in 1 L of water is
10%. If 500 mL of water is added to this
mixture, what percentage of salt is there
now?
15. Simplify
25
1
out of 20 for a maths
2
1
test, 19 out of 23 for English and 55
2
out of 70 for physics. Find his average
score as a percentage, to the nearest
whole percentage.
Vinh scored 17
• • •
8.
Write 1.3274 as a rational number.
9.
The distance from the Earth to the moon
is 3.84 #10 5 km. How long would it take
a rocket travelling at 2.13#10 4 km h to
reach the moon, to the nearest hour?
8.3# 4.1
correct to
0.2 + 5.4 ' 1.3
3 significant figures.
10. Evaluate 3
a
in index form if
b3 c2
|x + 1 |
x2 - 1
for x ! !1.
4.3 1.3 - 2.9
correct to
2.4 3 + 3.31 2
2 decimal places.
16. Evaluate 6
17. Write 15 g as a percentage of 2.5 kg.
18. Evaluate 2.3 1.8 + 5.7 #10 - 2 correct to
3 significant figures.
- 3.4 #10 - 3 + 1.7 #10 - 2
and
(6.9 #10 5) 3
express your answer in scientific notation
correct to 3 significant figures.
19. Evaluate
20. Prove | a + b | # | a | + | b | for all real a, b.
43