Download - Prelim MathsInFocus EXT1
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TERMINOLOGY
1 Basic Arithmetic
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 3, 2, 1, 0, 1, 2, 3,f 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 2 2 2.3 # #= 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 ...
Scienti c 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 7.65 108# in scientifi c notation
ch1.indd 2 5/20/09 3:05:56 PM
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3Chapter 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 signi cant gures, scienti c 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
Irrationalnumbers
Unreal or imaginarynumbers
Integers
Rationalnumbers
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 form of a fraction
ba
where a and b are integers, .b 0! e.g. , . , . ,143 3 7 0 5 5
-
Irrational numbers cannot be written in the form of a fraction ba
(that is, they are not rational) e.g. ,2 r
EXAMPLE
Which of these numbers are rational and which are irrational?
, . , , , , .3 1 353 9
42 65
r-
Solution
34
and r are irrational as they cannot be written as fractions (r is irrational).
. , .1 3 131 9
13 2 65 2
2013and
= = - = - so they are all rational.
ch1.indd 3 5/20/09 3:06:00 PM
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4 Maths In Focus Mathematics Extension 1 Preliminary Course
Order of operations
1. Brackets: do calculations inside grouping symbols rst. (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?
Evaluate 1 219 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- +] g
press 40 3 ( 5 4 )13
#- + =
=
To evaluate 1.69 2.775.67 3.49
+
- correct to 1 decimal place
press ( ( 5.67 3.49 ) ( 1.69 2.77 ) )': - + =
0.7
correct to 1decimal place
=
ch1.indd 4 7/27/09 7:18:25 PM
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5Chapter 1 Basic Arithmetic
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
MEMORY KEYS
Use STO to store a number in memory. There are several memories that you can use at the same timeany 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 fi nd the reciprocal of x . For example, to evaluate
7.6 2.1
1#-
0.063= -
press ( ( ) 7.6 2.1 ) x 1#- =-
(correct to 3 decimal places)
Different calculators use different keys so check
the instructions for your calculator.
ch1.indd 5 7/24/09 8:09:51 PM
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6 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.
Solution
.
. .0 6825371
0 6825371 0 7 correct to 1 decimal place` =#
2. Round off 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.
ch1.indd 6 7/8/09 10:56:28 AM
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7Chapter 1 Basic Arithmetic
While using a xed 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.
Dont round off at each step of a series of
calculations.
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
FIX KEY
Use MODE or SET UP to fi x the number of decimal places (see the instructions for your calculator). This will cause all answers to have a fi xed number of decimal places until the calculator is turned off or switched back to normal.
1. State which numbers are rational and which are irrational.
(a) 169
0.546 (b)
(c) 17-
(d) 3r
(e) .0 34
(f) 218
(g) 2 2
(h) 271
17.4% (i)
(j) 5
1
ch1.indd 7 5/20/09 3:06:03 PM
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8 Maths In Focus Mathematics Extension 1 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#-
(f) 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
(f) 4.7 1.3
1+
(g) 4.51 3.28
1+
(h) 5.2 3.60.9 1.4
-
+
(i) 1.23 3.155.33 2.87
-
+
(j) 1.7 8.9 3.942 2 2+ -
4. Round off 1289 to the nearest hundred.
5. Write 947 to the nearest ten.
6. Round off 3200 to the nearest thousand.
7. A crowd of 10 739 spectators attended a tennis match. Write this gure 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.
10. A company made a pro t 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.
13. Round off 32.569148 to the nearest unit.
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, nd the cost of each pie.
18. Evaluate 7.48 correct to 2 decimal places.
19. Evaluate 8
6.4 2.3+ correct to
1 decimal place.
20. Find the length of each piece of material, to 1 decimal place, if 25 m of material is cut into 7 equal pieces.
ch1.indd 8 5/20/09 3:06:04 PM
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9Chapter 1 Basic Arithmetic
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- =
21. How much will 7.5 m2 of tiles cost, at $37.59 per m 2 ?
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?
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 .. ..
5 399 68 5 479 91
2
-- ] g
correct to 1 decimal place.
ch1.indd 9 7/8/09 10:56:29 AM
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10 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
AddSubtract-4 -3 -2 -1 0 1 2 3 4
Same signs
Different 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 left.
-4 -3 -2 -1 0 1 2 3 4
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. 32 - -
You can also do these on a calculator, or you may have a different way of working these out.
ch1.indd 10 5/20/09 3:06:04 PM
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11Chapter 1 Basic Arithmetic
EXAMPLES
Evaluate 1. 2 7#- Solution
Different 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 -- ] g
5. 28 7' -
6. . .4 9 3 7- +
7. . .2 14 5 37- -
8. . .4 8 7 4# -
9. . .1 7 4 87- -] g
10. 53 1
32
- -
11. 5 3 4#-
12. 2 7 3#- + -
13. 4 3 2#- -
14. 1 2- - -
15. 7 2+ -
16. 2 1- -] g
17. 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.
ch1.indd 11 7/9/09 1:58:45 AM
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12 Maths In Focus Mathematics Extension 1 Preliminary Course
Fractions, Decimals and Percentages
EXAMPLES
1. Write 0.45 as a fraction in its simplest form. Solution
.0 45
10045
55
209
'=
=
2. Convert 83 to a decimal.
Solution
..
.
8 3 0000 375
83 0 375So =
g
3. Change 35.5% to a fraction. Solution
. % .35 5
10035 5
22
20071
#=
=
4. Write 0.436 as a percentage. Solution
. . %
. %
0 436 0 436 100
43 6
#=
=
5. Write 20 g as a fraction of 1 kg in its simplest form. Solution
1 1000kg g=
1
201000
20
501
kg
gg
g=
=
Multiply by 100% to change a fraction or decimal to a percentage.
Conversions
You can do all these conversions on your calculator using the
acb
or S D+ key.
8
3 means 3 8.'
ch1.indd 12 5/20/09 3:06:06 PM
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13Chapter 1 Basic Arithmetic
Sometimes decimals repeat, or recur. Example
. 0.31 0 33333333 3
f= =
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
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 fraction. Solution
. ( )
. ( )
( ) ( ): .
.
n
n
n
n
1 3292929 1
100 132 9292929 2
2 1 99 131 6
99131 6
1010
9901316
1495163
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 of people who prefer to drink Lemon Fuzzy, if 24 out of every 30 people prefer it. Solution
% %3024
1100 80# =
CONTINUED
ch1.indd 13 7/8/09 10:56:31 AM
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14 Maths In Focus Mathematics Extension 1 Preliminary Course
1. Write each decimal as a fraction in its lowest terms.
0.64 (a) 0.051 (b) 5.05 (c) 11.8 (d)
2. Change each fraction into a decimal.
(a) 52
(b) 187
(c) 125
(d) 117
3. Convert each percentage to a fraction in its simplest form.
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 fraction as a percentage.
(a) 207
(b) 31
(c) 2154
(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 fraction.
52% (a) 7% (b) 16.8% (c) 109% (d) 43.4% (e)
(f) %1241
8. Write these fractions as recurring decimals.
(a) 65
(b) 799
(c) 9913
(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
9901316
1495163
Then
and
f
f
f
=
=
=
- =
=
=
This method avoids decimals in the fraction at the end.
ch1.indd 14 7/8/09 10:56:31 AM
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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 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.
(f) 335
(g) 71
(h) 1112
9. Express as fractions in lowest terms.
(a) .0 8
(b) .0 2
(c) .1 5
(d) .3 7
(e) .0 67
(f) .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) 19931
-
(e) 7 413 6
+
+
11. Evaluate and write as a fraction. (a) . . .7 5 4 1 7 9' +] g
(b) 4.5 1.315.7 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 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.
ch1.indd 15 5/20/09 3:06:07 PM
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16 Maths In Focus Mathematics Extension 1 Preliminary Course
DID YOU KNOW?
Some countries use a comma for the decimal pointfor example, 0,45 for 0.45. This is the reason that our large numbers now have spaces instead of commas between
digitsfor example, 15 000 rather than 15,000.
EXAMPLES
1. Evaluate 1 .52
43
- Solution
152
43
57
43
2028
2015
2013
- = -
= -
=
2. Evaluate 221 3' .
Solution
221 3
25
13
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 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.
ch1.indd 16 5/20/09 3:06:07 PM
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17Chapter 1 Basic Arithmetic
4. Evaluate . . .0 02 0 3# Solution
. . .0 02 0 3 0 006# =
5. Evaluate 10
8.753 . Solution
. .8 753 10 0 8753' =
6. The price of a $75 tennis racquet increased by %.521 Find the new
price. Solution
% $ . $
$ .
5 75 0 055 75
4 13
of` #=
=
% . % $ . $
$ .
521 0 055 105
21 75 1 055 75
79 13
21
or of #= =
=
So the price increases by $4.13 to $79.13.
7. The price of a book increased by 12%. If it now costs $18.00, what did it cost before the price rise? Solution
The new price is 112% (old price 100%, plus 12%)
1%$ .
100%$ .
$16.07
11218 00
11218 00
1100
`
#
=
=
=
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
(a) 53
41
+
(b) 352 2
107
-
(c) 43 1
52
#
(d) 73 4'
(e) 153 2
32
'
Multiply the numbers and count the number
of decimal places in the question.
Move the decimal point 1 place to
the left.
ch1.indd 17 5/20/09 3:06:07 PM
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18 Maths In Focus Mathematics Extension 1 Preliminary Course
4. Find 53 of $912.60.
5. Find 75 of 1 kg, in grams correct
to 1 decimal place.
6. Trinh spends 31 of her day
sleeping, 247 at work and
121
eating. What fraction of the day is left?
7. I get $150.00 a week for a casual
job. If I spend 101 on bus fares,
152 on lunches and
31 on outings,
how much money is left over for savings?
8. John grew by 20017 of his height
this year. If 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# (f) . .0 8 0 3# (g) . .0 02 0 009# (h) .5 72 1000#
(i) 1008.74
(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 profi t of $5 by selling a bike for $85, fi nd the profi t as a percentage of the selling price.
15. Increase 350 g by 15%.
16. Decrease 45 m by %.821
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 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. If 546 cars were stolen last month, how many were stolen this month?
20. Georges computer cost $3500. If it has depreciated by 17.2%, what is the computer worth now?
ch1.indd 18 7/9/09 1:58:53 AM
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19Chapter 1 Basic Arithmetic
Powers and Roots
A power (or index ) of a number shows how many times a number is multiplied by itself.
PROBLEM
If both the hour hand and minute hand start at the same position at 12 oclock, when is the rst time, correct to a fraction of 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 of a number is the inverse of 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 formulae use indices (powers and roots). For example the compound interest formula that you will study in Chapter 8 of the HSC
Course book is 1A P rn
= +^ h
Geometry uses formulae involving indices, such as 34
V r3r= . Do you know what this formula is for?
In Chapter 7, the formula for the distance between 2 points on a number plane is
d x x y y( ) ( )2 1
2
2 1
2= - + -
See if you can fi nd other formulae involving indices.
ch1.indd 19 5/20/09 3:06:11 PM
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20 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
( )( )( )
a aaa
a a aa a a m
na a a m n
a1
timestimes
times
m nn
m
m n
'
# # #
# # #
# # #
ff
f
=
=
=-
= -
Index laws
There are some general laws that simplify 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 for squares and cubes.
Use the xy or ^ key to fi nd powers of numbers.
Use the key for square roots.
Use the 3 key for cube roots. Use the x for other roots.
ch1.indd 20 5/20/09 3:06:11 PM
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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
ba
bann
n
=c m
Proof
( )
( )( )
ba
ba
ba
ba
ba n
b b b ba a a a n
n
ba
times
timestimes
n
n
n
# # # #
# # # #
# # # #
f
ff
=
=
=
c m
EXAMPLES
Simplify
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
ch1.indd 21 5/20/09 3:06:11 PM
-
22 Maths In Focus Mathematics Extension 1 Preliminary Course
1. Evaluate without using a calculator.
(a) 5 23 2# (b) 3 84 2+
(c) 41 3
c 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 #-
(f) 0.99 5.61
13 +
3. Simplify (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' (f) 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) yx
9
2 5
f 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. Simplify (a) x x5 9# (b) a a1 6#- -
(c) mm
3
7
(d) k k k13 6 9# ' (e) a a a5 4 7# #- -
(f) 5 5x x#2 3
(g) m nm n
4 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
# #=
=
=
=
- -
+ -
ch1.indd 22 5/20/09 3:06:12 PM
-
23Chapter 1 Basic Arithmetic
(h) 2 2
p
p p2
#1 1
(i) (3 )x11 2
(j) ( )
x
x3
4 6
5. Simplify (a) 5( )pq3
(b) ba 8
c m
(c) 4ba4
3
d n
(7 (d) a 5 b ) 2
(e) (2 )
m
m4
7 3
(f) ( )xy
xy xy3 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 a 3 b 2 when 2a = and
43b = .
7. If 32x = and
91,y = nd the value
of xy
x y5
3 2
.
8. If 21,
31a b= = and
41,c =
evaluate c
a b4
2 3
as a fraction .
9. (a) Simplify a ba b
8 7
11 8
.
Hence evaluate (b) a ba b
8 7
11 8
when
52a = and
85b = as a fraction .
10. (a) Simplify p q r
p q r4 6 2
5 8 4
.
(b) Hence evaluate p q r
p q r4 6 2
5 8 4
as a
fraction when 87,
32p q= = and
43r = .
11. Evaluate ( )a4 3 when 6.a
32
=
1
c m
12. Evaluate ba b
4
3 6
when a21
= and
b32
= .
13. Evaluate x y
x y5 5
4 7
when x31
= and
y92
= .
14. Evaluate kk
9
5
-
-
when .k31
=
15. Evaluate ( )a ba b3 2 2
4 6
when a43
= and
b91
= .
16. Evaluate a ba b
5 2
6 3
#
# as a fraction
when a91
= and b43
= .
17. Evaluate a ba b
3
2 7
as a fraction in
index form when a52 4
= c m and
b85 3
= c m .
18. Evaluate ( )
( )
a b c
a b c2 4 3
3 2 4
as a fraction
when ,a31
= b76
= and c97
= .
ch1.indd 23 5/20/09 3:06:12 PM
-
24 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
x x x
x
x xxx
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 simplify x x3 5' using (i) index laws and (ii) cancelling. (i) x x x3 5 2' = - by index laws
(ii) xx
x x x x xx x x
x1
5
3
2
# # # #
# #=
=
xx1So 2
2=-
Now simplify 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' (f) 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 =
ch1.indd 24 5/20/09 3:06:13 PM
-
25Chapter 1 Basic Arithmetic
1xx
nn=
-
Proof
x x x
x
x xxx
x
xx
1
1
n n
n
nn
n
nn
0 0
00
'
'
`
=
=
=
=
=
-
-
-
EXAMPLES
1. Simplify .abcab c
4
5 0
e o Solution
1abcab c
4
5 0
=e o
2. Evaluate .2 3- Solution
2
21
81
33
=
=
-
3. Write in index form.
(a) 1x2
(b) 3x5
(c) 51x
(d) x 1
1+
CONTINUED
ch1.indd 25 5/20/09 3:06:13 PM
-
26 Maths In Focus Mathematics Extension 1 Preliminary Course
1. Evaluate as a fraction or whole number.
(a) 3 3- (b) 4 1- (c) 7 3- (d) 10 4- (e) 2 8- 6 (f) 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) 65 2-
c m
(e) 3
2
x y
x y 0
-
+f p
(f) 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) 1x
x2
2= -
(b) x x
x
3 3 1
3
5 5
5
#=
= -
(c) x x
x
51
51 1
51 1
#=
= -
(d) ( )x xx
11
11
1
1
1
+=
+
= + -] g
4. Write a 3 without the negative index. Solution
aa13
3=-
ch1.indd 26 5/20/09 3:06:13 PM
-
27Chapter 1 Basic Arithmetic
(l) 98 0
c m
(m) 76 2-
c m
(n) 109 2-
c m
(o) 116 0
c m
(p) 41 2
--
c m
(q) 52 3
--
c m
(r) 372 1
--
c m
(s) 83 0
-c m
(t) 141 2
--
c m
3. Change into index form.
(a) 1m3
(b) 1x
(c) 1p7
(d) 1d9
(e) 1k5
(f) 1x2
(g) 2x4
(h) 3y2
(i) 21z6
(j) 53t8
(k) 72x
(l) 2
5m6
(m) 32y7
(n) (3 4)
1x 2+
(o) ( )
1a b 8+
(p) 2
1x -
(q) ( )p5 1
13+
(r) (4 9)
2t 5-
(s) ( )x4 1
111+
(t) 9( 3 )
5a b 7+
4. Write without negative indices.
(a) t 5-
(b) x 6-
(c) y 3-
(d) n 8-
(e) w 10-
(f) 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) 1x5-
b l
(p) y1 10-
c m
(q) 2p
1-
d n
(r) 1a b
2
+
-
c m
(s) x yx y 1
-
+ -e o
(t) 32x yw z 7
+
- -e o
ch1.indd 27 5/20/09 3:06:14 PM
-
28 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
n
n
a a
a a
a a
by index lawsn
n n
n`
=
=
=
1
1
` ^
^
j h
h
Fractional indices
Class Investigation
Explore fractional indices by looking at these questions. For example simplify (i) 2x
21` j and (ii) .x
2^ h
2( ) x x
xi by index laws
21=
=
1` ^j h
2
2
( ) x x
x x x
x x
ii
So
2
22
`
=
= =
=
1
1
^
` ^
h
j h
Now simplify these questions.
(a) 2x21
^ h
(b) x2
(c) 3x31
` j
(d) 3x31
^ h
(e) x33
^ h
(f) x33
(g) 4x41
` j
(h) 4x41
^ h
(i) x44
^ h
(j) x44
Use your results to complete:
nx =1
na an=1
ch1.indd 28 5/20/09 3:06:14 PM
-
29Chapter 1 Basic Arithmetic
EXAMPLES
1. Evaluate (a) 249
1
(b) 3271
Solution
(a) 249 497
=
=
1
(b) 3
27 273
3=
=
1
2. Write x3 2- in index form. Solution
2( )x x3 2 3 2- = -1
3. Write 7( )a b+1
without fractional indices. Solution
7( )a b a b7+ = +1
Proof
n n
n n
a a
a
aa
m
n m
m
mn
=
=
a =
=
m
m
1
1
`
^
^
j
h
h
Putting the fractional and negative indices together gives this rule.
- na
a1
n=
1
Here are some further rules.
n
( )a a
a
mn
n m
=
=
m
ch1.indd 29 5/20/09 3:06:14 PM
-
30 Maths In Focus Mathematics Extension 1 Preliminary Course
ba
abn n
=-
c bm l
EXAMPLES
1. Evaluate
(a) 384
(b) -
31251
(c) 32 3-
c m
Solution
(a) 3 ( ) ( )
8 8 8216
or3 4 434
=
=
=
4
(b) -
3
3125
125
1
1251
51
3
=
=
=
1
1
Proof
ba
ba
ba
ba
ab
ab
ab
1
1
1
1
n
n
n
n
n
n
n
n
n
n
n
'
#
=
=
=
=
=
=
-
c
c
b
m
m
l
ch1.indd 30 5/20/09 3:06:15 PM
-
31Chapter 1 Basic Arithmetic
(c) 32
23
827
383
3 3
=
=
=
-
c cm m
2. Write in index form. (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 fractional indices. Solution
-5
5
rr
r
1
135
=
=
3
3
DID YOU KNOW?
Nicole Oresme (132382) was the fi rst mathematician to use fractional indices. John Wallis (16161703) was the fi rst person to explain the signifi cance of zero, negative
and fractional indices. He also introduced the symbol 3 for infi nity. Do an Internet search on these mathematicians and fi nd out more about their work and
backgrounds. You could use keywords such as indices and infi nity as well as their names to fi nd this information.
ch1.indd 31 5/20/09 3:06:15 PM
-
32 Maths In Focus Mathematics Extension 1 Preliminary Course
1. Evaluate
(a) 2811
(b) 3271
(c) 2161
(d) 381
(e) 2491
(f) 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) 12.91
5
(e) . .. .
1 5 3 73 6 1 48
+
-
(f) . .
. .8 79 1 4
5 9 3 74 #-
3. Write without fractional indices.
(a) 3y1
(b) 3y2
(c) 2x-
1
(d) 2( )x2 5+1
(e) -
2( )x3 1-1
(f) 3( )q r6 +1
(g) -
5( )x 7+2
4. Write in index form.
(a) t
(b) y5
(c) x3
(d) 9 x3 - (e) s4 1+
(f) 2 3
1t +
(g) (5 )
1
x y 3-
(h) ( )x3 1 5+
(i) ( 2)
1
x 23 -
(j) 2 7
1y +
(k) 4
5x3 +
(l) y3 1
22 -
(m) 5 ( 2)
3
x2 34 +
5. Write in index form and simplify.
(a) x x
(b) xx
(c) xx
3
(d) x
x3
2
(e) x x4
1.7 Exercises
ch1.indd 32 5/20/09 3:06:15 PM
-
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) 1pp
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
76 3 8+
2
Scienti c notation (standard form)
Very large or very small numbers are usually written in scienti c notation to make them easier to read. What could be done to make the gures 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 scienti c notation is written as a number between 1 and 10 multiplied by a power of 10.
EXAMPLES
1. Write 320 000 000 in scienti c 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.
ch1.indd 33 8/1/09 2:15:28 PM
-
34 Maths In Focus Mathematics Extension 1 Preliminary Course
SIGNIFICANT FIGURES
The concept of signi cant gures is related to rounding off. When we look at very large (or very small) numbers, some of the smaller digits are not signi cant.
For example, in a football crowd of 49 976, the 6 people are not really signi cant in terms of a crowd of about 50 000! Even the 76 people are not signi cant.
When a company makes a pro t of $5 012 342.87, the amount of 87 cents is not exactly a signi cant sum! Nor is the sum of $342.87.
To round off to a certain number of signi cant gures, we count from the rst non-zero digit.
In any number, non-zero digits are always signi cant. Zeros are not signi cant, except between two non-zero digits or at the end of a decimal number.
Even though zeros may not be signi cant, they are still necessary. For example 31, 310, 3100, 31 000 and 310 000 all have 2 signi cant gures but are very different numbers!
Scienti c notation uses the signi cant gures in a number.
SCIENTIFIC NOTATION KEY
Use the EXP or 10x# key to put numbers in scientifi c 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 scientifi c notation, except the powers of 10 are always multiples of 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 signi cant gures, use the usual rules for rounding off.
ch1.indd 34 8/1/09 2:15:42 PM
-
35Chapter 1 Basic Arithmetic
EXAMPLES
1. Round off 4 592 170 to 3 signi cant gures. Solution
4 592 170 4 590 000= to 3 signi cant gures
2. Round off 0.248 391 to 2 signi cant gures. Solution
. .0 248 391 0 25= to 2 signi cant gures
3. Round off 1.396 794 to 3 signi cant gures. Solution
. .1 396 794 1 40= to 3 signi cant gures
1. Write in scienti c notation . 3 800 (a) 1 230 000 (b) 61 900 (c) 12 000 000 (d) 8 670 000 000 (e) 416 000 (f) 900 (g) 13 760 (h) 20 000 000 (i) 80 000 (j)
2. Write in scienti c notation. 0.057 (a) 0.000 055 (b) 0.004 (c) 0.000 62 (d) 0.000 002 (e) 0.000 000 08 (f) 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 78 107# (c) .9 25 103# (d) .6 33 106# (e) 4 105# (f) .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 signi cant gures.
235 980 (a) 9 234 605 (b) 10 742 (c) 0.364 258 (d) 1.293 542 (e) 8.973 498 011 (f) 15.694 (g) 322.78 (h) 2904.686 (i) 9.0741 (j)
1.8 Exercises
Remember to put the 0s in!
ch1.indd 35 5/20/09 3:06:16 PM
-
36 Maths In Focus Mathematics Extension 1 Preliminary Course
5. Evaluate correct to 3 signi cant gures.
(a) . .14 6 0 453# (b) .4 8 7' (c) 4. . .47 2 59 1 46#+
(d) . .3 47 2 7
1-
6. Evaluate . . ,4 5 10 2 9 104 5# # # giving your answer in scienti c notation.
7. Calculate ..
1 34 108 72 10
7
3
#
#-
and write
your answer in standard form correct to 3 signi cant gures.
Investigation
A logarithm is an index. It is a way of nding 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.
If a yx = then log y xa =
The expression log 1. 7 49 means the power of 7 that gives 49. The solution is 2 since .7 492 = The expression log 2. 2 16 means the power of 2 that gives 16. The solution is 4 since .2 164 =
Can you evaluate these logarithms? log 1. 3 27 log 2. 5 25 log 3. 10 10 000 log 4. 2 64 log 5. 4 4 log 6. 7 7 log 7. 3 1 log 8. 4 2
9. 31log3
10. 41log2
The a is called the base number and the x is the index or power.
ch1.indd 36 5/20/09 3:06:17 PM
-
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 of the side of 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 of our cars!
Absolute value defi nitions
We write the absolute value of x as x
xx x
x
0 when
when x 01
$=
-)
EXAMPLES
1. Evaluate .4 Solution
4 4 04 since $=
We can also defi ne x as the distance of x from 0 on the
number line. We will use this in Chapter 3.
CONTINUED
ch1.indd 37 7/9/09 1:59:20 AM
-
38 Maths In Focus Mathematics Extension 1 Preliminary Course
2. Evaluate .3- Solution
3 3 3 03
since 1- = - - -=
] g
The absolute value has some properties shown below.
Properties of absolute value
a 9= = =
| | | | | | | | | | | |
| | | |
| | | || | | | | | | |
| | | | | | | |
| | | | | | | | | | | | | | | | | |
ab a b
a
a aa a
a b b a
a b a b
2 3 2 3 6
3 3
5 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 3 2- - + - . 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
11
LHS = +
= - +
=
=
LHS means Left Hand Side.
ch1.indd 38 7/8/09 10:56:34 AM
-
39Chapter 1 Basic Arithmetic
a b
2 32 3
5
RHS = +
= - +
= +
=
a b a b
1 5Since 1
#+ +
3. Write expressions for 2 4x - without the absolute value signs. Solution
1
x x xx
x
x x xx x
x
2 4 2 4 2 4 02 4
2
2 4 2 4 2 4 02 4 2 4
2
wheni.e.
wheni.e.
1
1
$
$
$
- = - -
- = - - -
= - +
] g
Class Discussion
Are these statements true? If so, are there some values for which the expression is undefi ned (values of x or y that the expression cannot have)?
1. xx 1=
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. xx
3 23 2
1-
-=
9. x
x1
2=
10. x 0$
Discuss absolute value and its defi nition in relation to these statements.
RHS means Right Hand Side.
ch1.indd 39 7/8/09 10:56:35 AM
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40 Maths In Focus Mathematics Extension 1 Preliminary Course
1. Evaluate (a) 7 (b) 5- (c) 6- (d) 0 (e) 2 (f) 11- (g) 2 3- (h) 3 8- (i) 5 2- (j) 5 3-
2. Evaluate (a) 3 2+ - (b) 3 4- - (c) 5 3- + (d) 2 7#- (e) 3 1- + - (f) 5 2 6 2#- - (g) 2 5 1#- + - (h) 3 4- (i) 2 3 3 4- - - (j) 5 7 4 2- + -
3. Evaluate a b- if
(a) 5 2a band= = (b) 1 2a band= - = (c) 2 3a band= - = - (d) 4 7a band= = (e) .a b1 2and= - = -
4. Write an expression for
(a) a a 0when 2
(b) 0a awhen 1
(c) 0a awhen =
(d) 0a a3 when 2
(e) 0a a3 when 1
(f) 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 de nition of absolute value to write each expression without the absolute value signs
(a) x 5+ (b) 3b - (c) 4a + (d) 2 6y - (e) 3 9x + (f) 4 x- (g) k2 1+ (h) 5 2x - (i) a b+ (j) p q-
8. Find values of x for which .x 3=
9. Simplify nn
where .n 0!
10. Simplify 22
xx-
- and state which
value x cannot be.
1.9 Exercises
ch1.indd 40 5/20/09 3:06:19 PM
-
41Chapter 1 Basic Arithmetic
1. Convert 0.45 to a fraction (a) 14% to a decimal (b)
(c) 85 to a decimal
78.5% to a fraction (d) 0.012 to a percentage (e)
(f) 1511 to a percentage
2. Evaluate as a fraction.
(a) 7 2- (b) 5 1-
(c) 29-
1
3. Evaluate correct to 3 signi cant gures.
(a) . .4 5 7 62 2+
(b) 4.30.3
(c) 5.72
3
(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- + - (f) 12- - - (g) 24 6'- -
5. Simplify
(a) x x x5 7 3# ' (b) (5 )y3 2
(c) ( )
a b
a b9
5 4 7
(d) 3
2x6 3d n
(e) a bab
5 6
4 0
e o
6. Evaluate
(a) 153
87
-
(b) 76 3
32
#
(c) 943
'
(d) 52 2
101
+
(e) 1565
#
7. Evaluate (a) 4-
(b) 2361
(c) 5 2 32- - (d) 4 3- as fraction
(e) 382
(f) 2 1- -
(g) 249-
1
as a fraction
(h) 4161
(i) 3 0-] g (j) 4 7 2 32- - - -
8. Simplify (a) a a14 9' (b) x y5 3 6_ i (c) p p p6 5 2# '
(d) 2b9 4^ h
(e) (2 )
x y
x y10
7 3 2
9. Write in index form.
(a) n
(b) 1x5
(c) 1x y+
(d) x 14 +
Test Yourself 1
ch1.indd 41 5/20/09 3:06:19 PM
-
42 Maths In Focus Mathematics Extension 1 Preliminary Course
(e) a b7 +
(f) 2x
(g) 21x3
(h) x43
(i) (5 3)x 97 +
(j) 1
m34
10. Write without fractional or negative indices .
(a) a 5-
(b) 4n1
(c) 2( )x 1+1
(d) ( )x y 1- - (e) (4 7)t 4- -
(f) 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 a 2 b 4 when 259a = and 1
32b = .
13. If 31a
4
= c m and 43,b = evaluate ab3 as a
fraction.
14. Increase 650 mL by 6%.
15. Johan spends 31 of his 24-hour day
sleeping and 41 at work.
How many hours does Johan spend (a) at work?
What fraction of his day is spent at (b) work or sleeping?
If he spends 3 hours watching TV, (c) what fraction of the day is this?
What percentage of the day does he (d) spend sleeping?
16. The price of a car increased by 12%. If the car cost $34 500 previously, what is its new price?
17. Rachel scored 56 out of 80 for a maths test. What percentage did she score?
18. Evaluate ,2118 and write your answer in scienti c notation correct to 1 decimal place.
19. Write in index form. (a) x
(b) 1y
(c) 3x6 +
(d) (2 3)
1x 11-
(e) y73
20. Write in scienti c notation. 0.000 013 (a)
123 000 000 000 (b) 21. Convert to a fraction.
(a) .0 7
(b) .0 124
22. Write without the negative index. (a) x 3-
(b) ( )a2 5 1+ -
(c) ba 5-
c m
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 2a = - and 5.b = -
ch1.indd 42 5/20/09 3:06:20 PM
-
43Chapter 1 Basic Arithmetic
1. Simplify 843 3
32 4 1 .
52
87
'+ -c cm m
2. Simplify .53
125
180149
307
+ + -
3. Arrange in increasing order of size: 51%,
0.502, . ,0 5
.9951
4. Mark spends 31 of his day sleeping,
121
of the day eating and 201 of the day
watching TV. What percentage of the day is left?
5. Write -
3642
as a rational number.
6. Express . .3 2 0 01425' in scienti c notation correct to 3 signi cant gures.
7. Vinh scored 1721 out of 20 for a maths
test, 19 out of 23 for English and 5521
out of 70 for physics. Find his average score as a percentage, to the nearest whole percentage.
8. Write .1 3274
as a rational number.
9. The distance from 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 signi cant gures.
11. Show that ( ) ( ) .2 2 1 2 2 2 1k k k1 1- + = -+ +
12. Find the value of b ca3 2
in index form if
., a b c52
31
53and
4 3 2
= = - =c c cm m m
13. Which of the following are rational numbers: , . , , , . , ,3 0 34 2 3 1 5 0
73
r- ?
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 | |
x
x
1
12 -
+ for .x 1!!
16. Evaluate 2.4 3.314.3 2.9
3 2
1.36
+
- correct to
2 decimal places.
17. Write 15 g as a percentage of 2.5 kg.
18. Evaluate . .2 3 5 7 10.1 8 2#+ - correct to 3 signi cant gures.
19. Evaluate ( . )
. .6 9 10
3 4 10 1 7 105 3
3 2
#
# #- +- - and
express your answer in scienti c notation correct to 3 signi cant gures.
20. Prove | | | | | |a b a b#+ + for all real a , b .
Challenge Exercise 1
ch1.indd 43 5/20/09 3:06:21 PM
-
TERMINOLOGY
2 Algebra and Surds
Binomial: A mathematical expression consisting of two terms such as 3x + or x3 1-
Binomial product: The product of two binomial expressions such as ( 3) (2 4)x x+ -
Expression: A mathematical statement involving numbers, pronumerals and symbols e.g. x2 3-
Factorise: The process of writing an expression as a product of its factors. It is the reverse operation of expanding brackets i.e. take out the highest common factor in an expression and place the rest in brackets e.g. 2 8 2( 4)y y= --
Pronumeral: A letter or symbol that stands for a number
Rationalising the denominator: A process for replacing a surd in the denominator by a rational number without altering its value
Surd: From absurd. The root of a number that has an irrational value e.g. 3 . It cannot be expressed as a rational number
Term: An element of an expression containing pronumerals and/or numbers separated by an operation such as , , or# '+ - e.g. 2 , 3x -
Trinomial: An expression with three terms such as x x3 2 12 - +
ch2.indd 44 7/8/09 1:40:51 PM
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45Chapter 2 Algebra and Surds
INTRODUCTION
THIS CHAPTER REVIEWS ALGEBRA skills, including simplifying expressions, removing grouping symbols, factorising, completing the square and simplifying algebraic fractions . Operations with surds , including rationalising the denominator , are also studied in this chapter .
DID YOU KNOW?
One of the earliest mathematicians to use algebra was Diophantus of Alexandria . It is not known when he lived, but it is thought this may have been around 250 AD.
In Baghdad around 700800 AD a mathematician named Mohammed Un-Musa Al-Khowarezmi wrote books on algebra and Hindu numerals. One of his books was named Al-Jabr wal Migabaloh , and the word algebra comes from the fi rst word in this title.
Simplifying Expressions
Addition and subtraction
EXAMPLES
Simplify
1. x x7 - Solution
7 7 1
6
x x x x
x
- = -
=
2. x x x4 3 62 2 2- + Solution
4 3 6 6
7
x x x x x
x
2 2 2 2 2
2
- + = +
=
Here x is called a pronumeral.
CONTINUED
ch2.indd 45 7/8/09 1:41:05 PM
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46 Maths In Focus Mathematics Extension 1 Preliminary Course
3. x x x3 5 43 - - + Solution
x x x x x3 5 4 8 43 3- - + = - +
4. a b a b3 4 5- - - Solution
3 4 5 3 5 4
2 5
a b a b a a b b
a b
- - - = - - -
= - -
Only add or subtract like terms. These have the same pronumeral (for example, 3 x and 5 x ).
1. 2 5x x+
2. 9 6a a-
3. 5 4z z-
4. 5a a+
5. b b4 -
6. r r2 5-
7. y y4 3- +
8. x x2 3- -
9. 2 2a a-
10. k k4 7- +
11. 3 4 2t t t+ +
12. w w w8 3- +
13. m m m4 3 2- -
14. 3 5x x x+ -
15. 8 7h h h- -
16. b b b7 3+ -
17. 3 5 4 9b b b b- + +
18. x x x x5 3 7- + - -
19. x y y6 5- -
20. a b b a8 4 7+ - -
21. 2 3xy y xy+ +
22. 2 5 3ab ab ab2 2 2- -
23. m m m5 122 - - +
24. 7 5 6p p p2 - + -
25. 3 7 5 4x y x y+ + -
26. 2 3 8ab b ab b+ - +
27. ab bc ab ac bc+ - - +
28. a x a x7 2 15 3 5 3- + - +
29. 3 4 2x xy x y x y xy y3 2 2 2 2 3- + - + +
30. 3 4 3 5 4 6x x x x x3 2 2- - + - -
2.1 Exercises
Simplify
ch2.indd 46 7/8/09 1:41:09 PM
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47Chapter 2 Algebra and Surds
Multiplication
EXAMPLES
Simplify
1. x y x5 3 2# #- Solution
5 3 2 30
30
x y x xyx
x y2# #- = -
= -
2. 3 4x y xy3 2 5#- - Solution
x y xy x y3 4 123 2 5 4 7#- - =
Use index laws to simplify this
question.
1. b5 2#
2. x y2 4#
3. p p5 2#
4. z w3 2#-
5. a b5 3#- -
6. x y z2 7# #
7. ab c8 6#
8. d d4 3#
9. a a a3 4# #
10. y3 3-^ h
11. 2x2 5^ h
12. ab a2 33 #
13. a b ab5 22 # -
14. pq p q7 32 2 2#
15. ab a b5 2 2#
16. h h4 23 7# -
17. k p p3 2#
18. t3 3 4-^ h
19. m m7 26 5# -
20. x x y xy2 3 42 3 2# #- -
2.2 Exercises
Simplify
ch2.indd 47 7/8/09 1:41:11 PM
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48 Maths In Focus Mathematics Extension 1 Preliminary Course
Division
Use cancelling or index laws to simplify divisions.
EXAMPLES
Simplify
1. v y vy6 22 ' Solution
By cancelling,
v y vyvy
v y
v y
v v y
v
6 22
6
2
6
3
22
1 1
3 1 1
'
# #
# # #
=
=
=
Using index laws,
v y vy v y
v yv
6 2 3
33
2 2 1 1 1
1 0
' =
=
=
- -
2. 155
aba b
2
3
Solution
3
3
aba b a b
a b
ba
155
3
2
33 1 1 2
2 1
2
=
=
=
- -
-
1
1
1. x30 5'
2. y y2 '
3. 2
8a2
4. 8aa2
5. aa
28 2
6. x
xy
2
7. p p12 43 2'
8. 6
3ab
a b2 2
9. 1520
xyx
10. xx
39
4
7-
2.3 Exercises
Simplify
ch2.indd 48 7/9/09 2:18:39 AM
-
49Chapter 2 Algebra and Surds
11. ab b15 5'- -
12. 62a bab2 3
13. pqs
p4
8-
14. cd c d14 212 3 3'
15. 4
2
x y z
xy z3 2
2 3
16. pq
p q
7
423
5 4
17. a b c a b c5 209 4 2 5 3 1'- - -
18. a b
a b
4
29 2 1
5 2 4
- -
-
^
^
h
h
19. x y z xy z5 154 7 8 2'- -
20. a b a b9 184 1 3 1 3'- -- -^ h
Removing grouping symbols
The distributive law of numbers is given by
a b c ab ac+ = +] g
EXAMPLE
( )7 9 11 7 20
140
# #+ =
=
Using the distributive law,
( )7 9 11 7 9 7 11
63 77
140
# # #+ = +
= +
=
EXAMPLES
Expand and simplify. 1. a2 3+] g Solution
2( 3) 2 2 3
2 6
a a
a
# #+ = +
= +
This rule is used in algebra to help remove grouping symbols.
CONTINUED
ch2.indd 49 7/8/09 1:41:13 PM
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50 Maths In Focus Mathematics Extension 1 Preliminary Course
2. x2 5- -] g Solution
( ) ( )x x
x
x
2 5 1 2 5
1 2 1 5
2 5
# #
- - = - -
= - - -
= - +
3. a ab c5 4 32 + -] g Solution
( )a ab c a a ab a c
a a b a c
5 4 3 5 4 5 3 5
20 15 5
2 2 2 2
2 3 2
# # #+ - = + -
= + -
4. y5 2 3- +^ h Solution
( )y y
y
y
5 2 3 5 2 2 3
5 2 6
2 1
# #- + = - -
= - -
= - -
5. b b2 5 1- - +] ]g g Solution
( ) ( )b b b b
b b
b
2 5 1 2 2 5 1 1 1
2 10 1
11
# # # #- - + = + - - -
= - - -
= -
1. x2 4-] g
2. h3 2 3+] g
3. a5 2- -] g
4. x y2 3+^ h
5. x x 2-] g
6. a a b2 3 8-] g
7. ab a b2 +] g
8. n n5 4-] g
9. x y xy y3 22 2+_ i
10. k3 4 1+ +] g
11. t2 7 3- -] g
12. y y y4 3 8+ +^ h
2.4 Exercises Expand and simplify
ch2.indd 50 7/8/09 1:41:13 PM
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51Chapter 2 Algebra and Surds
13. b9 5 3- +] g
14. x3 2 5- -] g
15. m m5 3 2 7 2- + -] ]g g
16. h h2 4 3 2 9+ + -] ]g g
17. d d3 2 3 5 3- - -] ]g g
18. a a a a2 1 3 42+ - + -] ^g h
19. x x x3 4 5 1- - +] ]g g
20. ab a b a2 3 4 1- - -] ]g g
21. x x5 2 3- - -] g
22. y y8 4 2 1- + +^ h
23. a b a b+ --] ]g g
24. t t2 3 4 1 3- - + +] ]g g
25. a a4 3 5 7+ + --] ]g g
Binomial Products
A binomial expression consists of two numbers , for example 3.x + A set of two binomial expressions multiplied together is called a binomial
product. Example: x x3 2+ -] ]g g . Each term in the fi rst bracket is multiplied by each term in the second
bracket.
a b x y ax ay bx by+ + = + + +] ^g h
Proof a b c d a c d b c d
ac ad bc bd+ + = + + +
= + + +
] ] ] ]g g g g
EXAMPLES
Expand and simplify 1. 3 4p q+ -^ ^h h
Solution
p q pq p q3 4 4 3 12+ - = - + -^ ^h h
2. 5a 2+] g
Solution
( 5)( 5)
5 5 25
10 25
a a a
a a a
a a
5 2
2
2
+ = + +
= + + +
= + +
] g
Can you see a quick way of doing this?
ch2.indd 51 7/8/09 1:41:14 PM
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52 Maths In Focus Mathematics Extension 1 Preliminary Course
The rule below is not a binomial product (one expression is a trinomial), but it works the same way.
a b x y z ax ay az bx by bz+ + + = + + + + +] ^g h
EXAMPLE
Expand and simplify .x x y4 2 3 1+ - -] ^g h
Solution
( ) ( )x x y x xy x x y
x xy x y
4 2 3 1 2 3 8 12 4
2 3 7 12 4
2
2
+ - - = - - + - -
= - + - -
1. 5 2a a+ +] ]g g
2. x x3 1+ -] ]g g
3. 2 3 5y y- +^ ^h h
4. 4 2m m- -] ]g g
5. 4 3x x+ +] ]g g
6. 2 5y y+ -^ ^h h
7. 2 3 2x x- +] ]g g
8. 7 3h h- -] ]g g
9. 5 5x x+ -] ]g g
10. a a5 4 3 1- -] ]g g
11. 2 3 4 3y y+ -^ ^h h
12. 4 7x y- +] g^ h
13. 3 2x x2 + -^ ]h g
14. 2 2n n+ -] ]g g
15. 2 3 2 3x x+ -] ]g g
16. 4 7 4 7y y- +^ ^h h
17. 2 2a b a b+ -] ]g g
18. 3 4 3 4x y x y- +^ ^h h
19. 3 3x x+ -] ]g g
20. 6 6y y- +^ ^h h
21. a a3 1 3 1+ -] ]g g
22. 2 7 2 7z z- +] ]g g
23. 9 2 2x x y+ - +] g^ h
24. b a b3 2 2 1- + -] ]g g
25. 2 2 4x x x2+ - +] g^ h
26. 3 3 9a a a2- + +] g^ h
27. 9a 2+] g
28. 4k 2-] g
29. 2x 2+] g
30. 7y 2-^ h
31. 2 3x 2+] g
32. 2 1t 2-] g
2.5 Exercises
Expand and simplify
ch2.indd 52 7/8/09 1:41:15 PM
-
53Chapter 2 Algebra and Surds
33. 3 4a b 2+] g
34. 5x y 2-^ h
35. 2a b 2+] g
36. a b a b- +] ]g g
37. a b 2+] g
38. a b 2-] g
39. a b a ab b2 2+ - +] ^g h
40. a b a ab b2 2- + +] ^g h
Some binomial products have special results and can be simplifi ed quickly using their special properties. Binomial products involving perfect squares and the difference of two squares occur in many topics in mathematics. Their expansions are given below.
Difference of 2 squares
a b a b a b2 2+ - = -] ]g g
Proof
( ) ( )a b a b a ab ab b
a b
2 2
2 2
+ - = - + -
= -
a b a ab b22 2 2+ = + +] g
Perfect squares
Proof
( ) ( )
2
a b a b a b
a ab ab b
a ab b
2
2 2
2 2
+ = + +
= + + +
= + +
] g
2a b a ab b2 2 2- = - +] g
Proof
( ) ( )
2
a b a b a b
a ab ab b
a ab b
2
2 2
2 2
- = - -
= - - +
= - +
] g
ch2.indd 53 7/8/09 1:41:16 PM
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54 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
Expand and simplify 1. 2 3x 2-] g
Solution
( )x x x
x x
2 3 2 2 2 3 3
4 12 9
2 2 2
2
- = - +
= - +
] ]g g
2. 3 4 3 4y y- +^ ^h h
Solution
(3 4)(3 4) 4
9 16
y y y
y
3 2 2
2
- + = -
= -
^ h
1. 4t 2+] g
2. 6z 2-] g
3. x 1 2-] g
4. 8y 2+^ h
5. 3q 2+^ h
6. 7k 2-] g
7. n 1 2+] g
8. 2 5b 2+] g
9. 3 x 2-] g
10. y3 1 2-^ h
11. x y 2+^ h
12. a b3 2-] g
13. 4 5d e 2+] g
14. 4 4t t+ -] ]g g
15. x x3 3- +] ]g g
16. p p1 1+ -^ ^h h
17. 6 6r r+ -] ]g g
18. x x10 10- +] ]g g
19. 2 3 2 3a a+ -] ]g g
20. 5 5x y x y- +^ ^h h
21. a a4 1 4 1+ -] ]g g
22. 7 3 7 3x x- +] ]g g
23. 2 2x x2 2+ -^ ^h h
24. 5x2 2+^ h
25. 3 4 3 4ab c ab c- +] ]g g
26. 2x x2
+b l
27. 1 1a a a a- +b bl l
28. x y x y2 2+ - - -_ _i i6 6@ @
29. a b c 2+ +] g6 @
2.6 Exercises Expand and simplify
ch2.indd 54 7/8/09 1:41:17 PM
-
55Chapter 2 Algebra and Surds
30. x y1 2+ -] g7 A
31. a a3 32 2+ - -] ]g g
32. 16 4 4z z- - +] ]g g
33. 2 3 1 4x x 2+ + -] g
34. 2x y x y2+ - -^ ^h h
35. n n n4 3 4 3 2 52- + - +] ]g g
36. x 4 3-] g
37. x x x1 1 2
2 2
- - +b bl l
38. x y x y42 2 2 2 2+ -_ i
39. 2 5a 3+] g
40. x x x2 1 2 1 2 2- + +] ] ]g g g
Expand (x 4) (x 4) .- - 2
PROBLEM
Find values of all pronumerals that make this true.
i i c c b
a b c
d e
f e b
i i i h g
#
Try c 9.=
Factorisation
Simple factors
Factors are numbers that exactly divide or go into an equal or larger number, without leaving a remainder.
EXAMPLES
The numbers 1, 2, 3, 4, 6, 8, 12 and 24 are all the factors of 24. Factors of 5 x are 1, 5, x and 5 x .
To factorise an expression, we use the distributive law.
a bax bx x ++ = ] g
ch2.indd 55 7/8/09 1:41:18 PM
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56 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
Factorise
1. 3 12x + Solution
The highest common factor is 3. x x3 12 3 4+ = +] g
2. 2y y2 - Solution
The highest common factor is y. y y y y2 22 - = -^ h
3. 2x x3 2- Solution
x and x 2 are both common factors. We take out the highest common factor which is x 2 . x x x x2 23 2 2- = -] g
4. x xy5 3 32+ ++] ]g g Solution
The highest common factor is 3x + . x x x yy5 3 3 3 5 22+ + + ++ =] ] ] ^g g g h
5. 8 2a b ab3 2 3- Solution
There are several common factors here. The highest common factor is 2 ab 2 . 8 2 2 4a b ab ab a b3 2 3 2 2- = -^ h
Check answers by expanding brackets.
Divide each term by 3 to fi nd the terms inside the brackets.
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57Chapter 2 Algebra and Surds
1. 2 6y +
2. x5 10-
3. 3 9m -
4. 8 2x +
5. y24 18-
6. 2x x2 +
7. 3m m2 -
8. 2 4y y2 +
9. 15 3a a2-
10. ab ab2 +
11. 4 2x y xy2 -
12. 3 9mn mn3 +
13. 8 2x z xz2 2-
14. 6 3 2ab a a2+ -
15. 5 2x x xy2 - +
16. 3 2q q5 2-
17. 5 15b b3 2+
18. 6 3a b a b2 3 3 2-
19. x m m5 7 5+ + +] ]g g
20. y y y2 1 1- - -^ ^h h
21. 4 7 3 7y x y+ - +^ ^h h
22. 6 2 5 2x a a- + -] ]g g
23. x t y t2 1 2 1+ - +] ]g g
24. a x b x3 2 2 3 2- + -] ]g g c x3 3 2- -] g
25. 6 9x x3 2+
26. 3 6pq q5 3-
27. 15 3a b ab4 3 +
28. 4 24x x3 2-
29. 35 25m n m n3 4 2-
30. 24 16a b ab2 5 2+
31. r rh2 22r r+
32. 3 5 3x x2- + -] ]g g
33. 4 2 4y x x2 + + +] ]g g
34. a a a1 1 2+ - +] ]g g
35. ab a a4 1 3 12 2+ - +^ ^h h
2.7 Exercises
Factorise
Grouping in pairs
If an expression has 4 terms, it may be factorised in pairs.
( ) ( )
( ) ( )
ax bx ay by x a b y a b
a b x y
+ + + = + + +
= + +
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58 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
Factorise
1. 2 3 6x x x2 - + - Solution
2 3 6 ( 2) 3( 2)
( 2)( 3)x x x x x x
x x
2 - + - = - + -
= - +
2. 2 4 6 3x y xy- + - Solution
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
x y xy x y x
x y x
x y
x y xy x y x
x y x
x y
2 4 6 3 2 2 3 2
2 2 3 2
2 2 3
2 4 6 3 2 2 3 2
2 2 3 2
2 2 3
or
- + - = - + -
= - - -
= - -
- + - = - - - +
= - - -
= - -
1. 2 8 4x bx b+ + +
2. 3 3ay a by b- + -
3. x x x5 2 102 + + +
4. 2 3 6m m m2 - + -
5. ad ac bd bc- + -
6. 3 3x x x3 2+ + +
7. ab b a5 3 10 6- + -
8. 2 2xy x y xy2 2- + -
9. ay a y 1+ + +
10. 5 5x x x2 + - -
11. 3 3y ay a+ + +
12. 2 4 2m y my- + -
13. x xy xy y2 10 3 152 2+ - -
14. 4 4a b ab a b2 3 2+ - -
15. x x x5 3 152- - +
16. 7 4 28x x x4 3+ - -
17. 7 21 3x xy y- - +
18. 4 12 3d de e+ - -
19. x xy y3 12 4+- -
20. a ab b2 6 3+ - -
21. x x x3 6 183 2 +- -
22. pq p q q3 32+- -
2.8 Exercises
Factorise
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59Chapter 2 Algebra and Surds
23. x x x3 6 5 103 2- - +
24. 4 12 3a b ac bc- + -
25. 7 4 28xy x y+ - -
26. x x x4 5 204 3- - +
27. x x x4 6 8 123 2- + -
28. 3 9 6 18a a ab b2 + + +
29. y xy x5 15 10 30+- -
30. r r r2 3 62r r+ - -
Trinomials
A trinomial is an expression with three terms, for example 4 3.x x2 - + Factorising a trinomial usually gives a binomial product.
x a b x a x bx ab2 + + ++ + =] ] ]g g g
Proof
( )
( ) ( )
( ) ( )
x a b x ab x ax bx abx x a b x a
x a x b
2 2+ + + = + + +
= + + +
= + +
EXAMPLES
Factorise
1. 5 6m m2 - + Solution
a b 5+ = - and 6ab = +
6235
+-
-
-
'
Numbers with sum 5- and product 6+ are 2- and 3.-
[ ] [ ]m m m mm m
5 6 2 32 3
2` - + = + - + -
= - -
] ]
] ]
g g
g g
2. 2y y2 + - Solution
1a b+ = + and 2ab = -
2211
-+
-+
'
Two numbers with sum 1+ and product 2- are 2+ and 1- . y y y y2 2 12` + - = + -^ ^h h
Guess and check by trying 2- and 3-
or 1- and .6-
Guess and check by trying 2 and 1- or
2- and 1.
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60 Maths In Focus Mathematics Extension 1 Preliminary Course
The result x a b x a x bx ab2 + + + ++ =] ] ]g g g only works when the coeffi cient of x2 (the number in front of x2 ) is 1. When the coeffi cient of x2 is not 1, for example in the expression 5 2 4,x x2 - + we need to use a different method to factorise the trinomial.
There are different ways of factorising these trinomials. One method is the cross method . Another is called the PSF method . Or you can simply guess and check.
1. 4 3x x2 + +
2. y y7 122 + +
3. m m2 12 + +
4. t t8 162 + +
5. 6z z2 + -
6. 5 6x x2 - -
7. v v8 152 - +
8. 6 9t t2 - +
9. x x9 102 + -
10. 10 21y y2 - +
11. m m9 182 - +
12. y y9 362 + -
13. 5 24x x2 - -
14. 4 4a a2 - +
15. x x14 322 + -
16. 5 36y y2 - -
17. n n10 242 +-
18. x x10 252 +-
19. p p8 92 + -
20. k k7 102 +-
21. x x 122 + -
22. m m6 72 - -
23. 12 20q q2 + +
24. d d4 52 - -
25. l l11 182 +-
2.9 Exercises
Factorise
EXAMPLES
Factorise
1. 5 13 6y y2 - +
Solutionguess and check
For 5 y 2 , one bracket will have 5 y and the other y : .y y5^ ^h h Now look at the constant (term without y in it): .6+
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61Chapter 2 Algebra and Surds
The two numbers inside the brackets must multiply to give 6.+ To get a positive answer, they must both have the same signs. But there is a negative sign in front of 13 y so the numbers cannot be both positive. They must both be negative. y y5 - -^ ^h h To get a product of 6, the numbers must be 2 and 3 or 1 and 6. Guess 2 and 3 and check:
3 5 15 2 6
5 17 6
y y y y y
y y
5 2 2
2
- = - - +
= - +
-^ ^h h
This is not correct. Notice that we are mainly interested in checking the middle two terms, .y y15 2and- - Try 2 and 3 the other way around: .y y5 3 2- -^ ^h h Checking the middle terms: y y y10 3 13- - = - This is correct, so the answer is .y y5 3 2- -^ ^h h Note: If this did not check out, do the same with 1 and 6.
Solution cross method
Factors of 5y2 are 5 y and y. Factors of 6 are 1- and 6- or 2- and .3- Possible combinations that give a middle term of y13- are
By guessing and checking, we choose the correct combination.
y13-
y y
y y
5 2 10
3 3
#
#
- = -
- = -
y y y y5 13 6 5 3 22` - + = - -^ ^h h
Solution PSF method
P: Product of fi rst and last terms 30y2 S: Sum or middle term y13- F: Factors of P that give S ,y y3 10- -
y
yyy
3031013
2 -
-
-
)
y y y y y
y y y
y y
5 13 6 5 3 10 65 3 2 5 3
5 3 2
2 2` - + = - - +
= - - -
= - -
^ ^
^ ^
h h
h h
5y
y 3-
2- 5y
y 2-
3- 5y
y 6-
1- 5y
y 1-
6-
5y
y 2-
3-
CONTINUED
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62 Maths In Focus Mathematics Extension 1 Preliminary Course
2. 4 4 3y y2 + -
Solutionguess and check
For 4 y 2 , both brackets will have 2 y or one bracket will have 4 y and the other y . Try 2 y in each bracket: .y y2 2^ ^h h Now look at the constant: .3- The two numbers inside the brackets must multiply to give .3- To get a negative answer, they must have different signs. y y2 2 +-^ ^h h To get a product of 3, the numbers must be 1 and 3. Guess and check: y y2 3 2 1+-^ ^h h Checking the middle terms: y y y2 6 4- = - This is almost correct, as the sign is wrong but the coeffi cient is right (the number in front of y ). Swap the signs around:
4 6 2 3