Preliminary Mathematics textbook

TERMINOLOGYTERMINOLOGY 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

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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 signifi cant fi gures, scientifi 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.

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4 Maths In Focus Mathematics 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 )

31#

=

- + =

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 8/7/09 11:30:59 AM


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.

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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.

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While using a fi 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

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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 fi 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 profi 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, fi 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.

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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?


(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.

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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.

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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.

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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.'

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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

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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 7/16/09 1:12:25 PM


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.

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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 7/16/09 1:12:27 PM


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.

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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 7/16/09 1:12:28 PM


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 fi 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 7/16/09 1:12:32 PM


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 7/16/09 1:12:33 PM


( )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

ban

n

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 7/16/09 1:12:33 PM


1. Evaluate without using a calculator.

(a) 5 23 2# (b) 3 84 2+

(c) 41 3c m

(d) 273 (e) 164

2. Evaluate correct to 1 decimal place.

(a) 3.72 (b) 1.061.5 (c) 2.3 0.2- (d) 193 (e) . . .34 8 1 2 43 13 #-

(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 5f p (m)

ww w

3

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 7/16/09 1:12:34 PM


(h) 2 2

p

p p2

#1 1

(i) (3 )x11 2

(j) ( )

x

x3

4 6

5. Simplify (a) 5( )pq3

(b) ba 8c m

(c) 4ba4

3d n (7 (d) a 5 b ) 2

(e) (2 )

m

m4

7 3

(f) ( )

xyxy xy3 2 4#

(g) 3

4

( )

( )

k

k

6

23

8

(h) yy

28

5 712

#_ i (i)

aa a

11

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 = fi 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 b

a 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 b

85 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 7/16/09 1:12:35 PM


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 7/16/09 1:12:36 PM


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 0e 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 7/16/09 1:12:37 PM


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 7/16/09 1:12:38 PM


(l) 98 0c m

(m) 76 2-c m

(n) 109 2-c m

(o) 116 0c m

(p) 41 2

--c m

(q) 52 3

--c m

(r) 372 1

--c m

(s) 83 0

-c m (t) 1

41 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) 1x

5-b l (p) y

1 10-c m (q) 2p

1-d n (r) 1a b

2

+

-c m (s) x y

x y 1

-

+ -e o (t)

32

x yw z 7

+

- -e o

ch1.indd 27 7/16/09 1:12:39 PM


Proof

n

n

a a

a a

a a

by index lawsn

n n

n`

=

=

=

1

1

` ^^

j hh

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 xx

i by index laws2

1=

=

1` ^j h

2

2

( ) x x

x x x

x x

ii

So

2

22

`

=

= =

=

1

1

^` ^

hj h

Now 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 7/16/09 1:12:40 PM


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

`^^

jhh

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 7/16/09 1:12:41 PM


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 7/16/09 1:12:42 PM


(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 7/16/09 1:12:42 PM


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


(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


(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) x

x3

(d) x

x3

2

(e) x x4

1.7 Exercises

ch1.indd 32 7/16/09 1:12:43 PM


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 7/31/09 3:40:42 PM


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 7/31/09 3:40:43 PM


EXAMPLES

1. Round off 4 592 170 to 3 signifi cant fi gures. Solution

4 592 170 4 590 000= to 3 signifi cant fi gures

2. Round off 0.248 391 to 2 signifi cant fi gures. Solution

. .0 248 391 0 25= to 2 signifi cant fi gures

3. Round off 1.396 794 to 3 signifi cant fi gures. Solution

. .1 396 794 1 40= to 3 signifi cant fi gures

1. Write in scientifi 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 scientifi 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 signifi cant fi 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 7/16/09 1:12:46 PM


5. Evaluate correct to 3 signifi cant fi 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 scientifi c notation.

7. Calculate ..

1 34 108 72 10

7

3

#

#-

and write

your answer in standard form correct to 3 signifi cant fi gures.

Investigation

A logarithm is an index. It is a way of fi 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 7/16/09 1:12:47 PM


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 7/16/09 1:12:47 PM


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 7/16/09 1:12:49 PM


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 7/16/09 1:12:50 PM


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 defi 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 7/16/09 1:12:51 PM


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 signifi cant fi 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 0e 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


(a) n

(b) 1x5

(c) 1x y+

(d) x 14 +

Test Yourself 1

ch1.indd 41 7/16/09 1:12:52 PM


(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 scientifi 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 scientifi 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 7/16/09 1:12:54 PM


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 scientifi c notation correct to 3 signifi cant fi 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 signifi cant fi 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 5 073

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 signifi cant fi gures.

19. Evaluate ( . )

. .6 9 10

3 4 10 1 7 105 3

3 2

#

# #- +- - and

express your answer in scientifi c notation correct to 3 signifi cant fi gures.

20. Prove | | | | | |a b a b#+ + for all real a , b .

Challenge Exercise 1

ch1.indd 43 7/16/09 1:12:56 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 7/17/09 11:54:59 AM

45Chapter 2 Algebra and Surds

DID YOU KNOW?

Box text...

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 7/17/09 11:55:13 AM


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 7/17/09 11:55:25 AM


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 7/17/09 11:55:28 AM


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. 155aba 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

3aba b2 2

9. 1520xyx

10. xx

39

4

7-

2.3 Exercises

Simplify

ch2.indd 48 7/17/09 11:55:29 AM


11. ab b15 5'- -

12. 62a bab2 3

13. pqsp

48-

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

- -

-

^^

hh

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 7/17/09 11:55:30 AM


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 7/17/09 11:55:31 AM


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 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 7/31/09 3:43:28 PM


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 7/31/09 3:43:29 PM


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 7/17/09 11:55:35 AM


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 x

2

+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 7/17/09 11:55:36 AM


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 x

1 1 22 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 7/17/09 11:55:38 AM


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.

ch2.indd 56 7/17/09 11:55:39 AM


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

+ + + = + + +

= + +

ch2.indd 57 7/17/09 11:55:40 AM


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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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 gg 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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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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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 hh 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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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 coef cient is right (the number in front of y ). Swap the signs around:

4 6 2 3

4 4 3

y y y y y

y y

2 1 2 3 2

2

+ = +

= +

- - -

-

^ ^h h

This is correct, so the answer is .y y2 1 2 3- +^ ^h h

Solution cross method

Factors of 4y2 are 4 y and y or 2 y and 2 y . Factors of 3 are 1- and 3 or 3- and 1. Trying combinations of these factors gives

2y# 3

2 1 2y y

y4

#- = -

= 6y

y y y y4 4 3 2 3 2 12` + - = + -^ ^h h

Solution PSF method

P: Product of rst and last terms y12 2- S: Sum or middle term 4 y F: Factors of P that give S ,y y6 2+ -

y

yyy

12624

2-+

-

+

)

y y y y y

y y y

y y

4 4 3 4 6 2 32 2 3 1 2 3

2 3 2 1

2 2` + - = + - -

= + - +

= + -

^ ^

^ ^

h h

h h

2y

2y 1-

3

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Perfect squares

You have looked at some special binomial products, including 2a

Documents

Preliminary Mathematics textbook