maths in focus - margaret grove - ch1

7/30/2019 Maths in Focus - Margaret Grove - ch1

1/42

TERMINOLOGYTERMINOLOGTERMINOLOGY

1Basic Arithmetic

Absolute value: The distance o a number rom zero onthe number line. Hence it is the magnitude or value o anumber without the sign

Directed numbers: The set o integers or wholenumbers 3, 2, 1, 0, 1, 2, 3,f f- - -

Exponent: Power or index o a number. For example 23has a base number o 2 and an exponent o 3

Index: The power o a base number showing howmany times this number is multiplied by itsel

e.g. 2 2 2 2.3 # #= The index is 3

Indices: More than one index (plural)

Recurring decimal: A repeating decimal that does notterminate e.g. 0.777777 is a recurring decimal that canbe written as a raction. More than one digit can recure.g. 0.14141414 ...

Scientific notation: Sometimes called standard notation.A standard orm to write very large or very small numbersas a product o a number between 1 and 10 and a powero 10 e.g. 765 000 000 is 7.65 108# in scientifc notation


2/42

3Chapter 1 Basic Arithmetic

INTRODUCTION

THIS CHAPTER GIVES A review o basic arithmetic skills, including knowing the

correct order o operations, rounding o, and working with ractions, decimals

and percentages. Work on signifcant fgures, scientifc notation and indices isalso included, as are the concepts o absolute values. Basic calculator skills are

also covered in this chapter.

Real Numbers

Types of numbers

Irrational

numbers

Unreal or imaginary

numbers

Integers

Rational

numbers

Real numbers

Integers are whole numbers that may be positive, negative or zero.

e.g. , , ,4 7 0 11- -

Rational numbers can be written in the orm o a ractionb

a

where a and b are integers, .b 0! e.g. , . , . ,143

3 7 0 5 5

-

Irrational numbers cannot be written in the orm o a ractionb

a(that

is, they are not rational) e.g. ,2 r

EXAMPLE

Which o these numbers are rational and which are irrational?

, . , , , , .3 1 353

94

2 65 r

-

Solution

34

andr

are irrational as they cannot be written as ractions (r is irrational).

. , .1 3 131

91

32 65 2

20

13and

= = - = - so they are all rational.


3/42

4 Maths In Focus Mathematics Preliminary Course

Order of operations

1. Brackets: do calculations inside grouping symbols first. (For example,

a fraction line, square root sign or absolute value sign can act as a

grouping symbol.)2. Multiply or divide from left to right.

3. Add or subtract from left to right.

EXAMPLE

Evaluate .40 3 5 4- +] g

Solution

40 3(5 4) 40 3 9

40 27

13

#- + = -

= -

=

PROBLEM

What is wrong with this calculation?

Evaluate1 2

19 4

+

-

- +Press19 4 1 2 19 4 1 2'+- ='17

What is the correct answer?

BRACKETS KEYS

Use ( and ) to open and close brackets. Always use them in pairs.

For example, to evaluate 40 5 43- +] gpress 40 3 ( 5 4 )

31

#

=

- + =

To evaluate1.69 2.77

5.67 3.49

+

-correct to 1 decimal place

press ( ( 5.67 3.49 ) ( 1.69 2.77 ) )': - + =

0.7

correct to 1decimal place

=


4/42


Rounding off

Rounding o is oten done in everyday lie. A quick look at a newspaper will

give plenty o examples. For example in the sports section, a newspaper may

report that 50 000 ans attended a ootball match.An accurate number is not always necessary. There may have been exactly

49 976 people at the ootball game, but 50 000 gives an idea o the size o the

crowd.

EXAMPLES

1. Round o 24 629 to the nearest thousand.

Solution

This number is between 24 000 and 25 000, but it is closer to 25 000.

24 629 25 000` = to the nearest thousand

CONTINUED

MEMORY KEYS

Use STO to store a number in memory.

There are several memories that you can use at the same timeany letter rom

A to F, or X, Y and M on the keypad.To store the number 50 in, say, A press 50 STO A

To recall this number, press ALPHA A =

To clear all memories press SHIFT CLR

X-1 KEY

Use this key to fnd the reciprocal o x. For example, to evaluate

7.6 2.1

1

#-

0.063= -

press ( ( ) 7.6 2.1 ) x1

#- =-

(correct to 3 decimalplaces)

Different calculators use

different keys so check

the instructions for your

calculator.


5/42


2. Write 850 to the nearest hundred.

Solution

This number is exactly halway between 800 and 900. When a number is

halway, we round it o to the larger number.

850 900` = to the nearest hundred

In this course you will need to round o decimals, especially when using

trigonometry or logarithms.

To round a number o to a certain number o decimal places, look at the

next digit to the right. I this digit is 5 or more, add 1 to the digit beore it and

drop all the other digits ater it. I the digit to the right is less than 5, leave thedigit beore it and drop all the digits to the right.

EXAMPLES

1. Round o 0.6825371 correct to 1 decimal place.

Solution

.. .

0 68253710 6825371 0 7 correct to1 decimal place` =

#

2. Round o 0.6825371 correct to 2 decimal places.

Solution

.

. .

0 6825371

0 6825371 0 68 correct to 2 decimal places` =#

3. Evaluate . .3 56 2 1'

correct to 2 decimal places.

Solution

. . . 5

.

3 56 2 1 1 69 238095

1 70 correct to 2 decimal places

' =

=

#

Drop off the 2 and all digits

to the right as 2 is smaller

than 5.

Add 1 to the 6 as the 8 is

greater than 5.

Check this on your

calculator. Add 1 to the

69 as 5 is too large to just

drop off.


6/42


While using a fxed number o decimal places on the display, the

calculator still keeps track internally o the ull number o decimal places.

EXAMPLE

Calculate . . . .3 25 1 72 5 97 7 32#' + correct to 2 decimal places.

Solution

. . . . . . .

. .

.

3 25 1 72 5 97 7 32 1 889534884 5 97 7 32

11 28052326 7 32

18 60052326

18.60 correct to 2 decimal places

' # #+ = +

= +

=

=

I the FIX key is set to 2 decimal places, then the display will show

2 decimal places at each step.

3.25 1.72 5.97 7.32 1.89 5.97 7.32

. .

.

11 28 7 32

18 60

' # #+ = +

= +

=

I you then set the calculator back to normal, the display will show theull answer o 18.60052326.

Dont round off at

each step of a series of

calculations.

The calculator does not round o at each step. I it did, the answer might

not be as accurate. This is an important point, since some students round

o each step in calculations and then wonder why they do not get the same

answer as other students and the textbook.

1.1 Exercises

FIX KEY

Use MODE or SET UP to fx the number o decimal places (see the

instructions or your calculator). This will cause all answers to have a fxed number

o decimal places until the calculator is turned o or switched back to normal.

1. State which numbers are rational

and which are irrational.

(a) 169

0.546(b)

(c) 17-

(d)3

r

(e) .0 34

() 218

(g) 2 2

(h)271

17.4%(i)

(j)5

1


7/42


2. Evaluate

(a) 20 8 4'-

(b) 3 7 2 5# #-

(c) 4 27 3 6# ' '] g (d) 17 3 2#+ -

(e) . .1 9 2 3 1#-

()1 3

14 7'- +

(g) 253

51

32

#-

(h)

65

143

81

-

(i)

41 81

85

65

'

+

(j)1

41

21

351

107

-

-

3. Evaluate correct to 2 decimal

places.

(a) 2.36 4.2 0.3'+

(b) . . .2 36 4 2 0 3'+] g (c) 12.7 3.95 5.7# '

(d) 8.2 0.4 4.1 0.54' #+

(e) . . . .3 2 6 5 1 3 2 7#- +] ]g g()

4.7 1.31

+

(g)4.51 3.28

1

+

(h)5.2 3.60.9 1.4

-

+

(i)1.23 3.15

5.33 2.87

-

+

(j) 1.7 8.9 3.942 2 2

+ -

4. Round o 1289 to the nearest

hundred.

5. Write 947 to the nearest ten.

6. Round o 3200 to the nearest

thousand.

7. A crowd o 10 739 spectators

attended a tennis match.

Write this fgure to the nearest

thousand.

8. A school has 623 students. What

is this to the nearest hundred?

9. A bank made loans to the value

o $7 635 718 last year. Round this

o to the nearest million.

10. A company made a proft o

$34 562 991.39 last year. Write

this to the nearest hundred

thousand.

11. The distance between two cities

is 843.72 km. What is this to the

nearest kilometre?

12. Write 0.72548 correct to

2 decimal places.

13. Round o 32.569148 to the

nearest unit.

14. Round o 3.24819 to 3 decimal

places.

15. Evaluate 2.45 1.72# correct to

2 decimal places.

16. Evaluate 8.7 5' correct to

1 decimal place.

17. I pies are on special at 3 or

$2.38, fnd the cost o each pie.

18. Evaluate 7.48 correct to

2 decimal places.

19. Evaluate8

6.4 2.3+

correct to

1 decimal place.

20. Find the length o each piece

o material, to 1 decimal place,

i 25 m o material is cut into

7 equal pieces.


8/42


DID YOU KNOW?

In building, engineering and other industries where accurate measurements are used, the

number o decimal places used indicates how accurate the measurements are.

For example, i a 2.431 m length o timber is cut into 8 equal parts, according to the

calculator each part should be 0.303875 m. However, a machine could not cut this accurately.

A length o 2.431 m shows that the measurement o the timber is only accurate to the nearest

mm (2.431 m is 2431 mm). The cut pieces can also only be accurate to the nearest mm (0.304 m

or 304 mm).

The error in measurement is related to rounding o, as the error is hal the smallest

measurement. In the above example, the measurement error is hal a millimetre. The length o

timber could be anywhere between 2430.5 mm and 2431.5 mm.

Directed Numbers

Many students use the calculator with work on directed numbers (numbers

that can be positive or negative). Directed numbers occur in algebra and

other topics, where you will need to remember how to use them. A good

understanding o directed numbers will make your algebra skills much better.

-^ h KEYUse this key to enter negative numbers. For example,

press ( ) 3- =

21. How much will 7.5 m2 o tiles

cost, at $37.59 per m2?

22. Divide 12.9 grams o salt into

7 equal portions, to 1 decimal

place.

23. The cost o 9 peaches is $5.72.

How much would 5 peaches cost?


places.

(a) 17.3 4.33 2.16#-

(b) . . . .8 72 5 68 4 9 3 98# #-

(c)5.6 4.35

3.5 9.8

+

+

(d)7.63 5.12

15.9 6.3 7.8

-

+ -

(e)

6.87 3.21

1

-

25. Evaluate.

. ..

5 39

9 68 5 479 912

-- ] g

correct to 1 decimal place.


9/42


Adding and subtracting

To add: move to the right along the number line

To subtract: move to the let along the number line

AddSubtract

-4 -3 -2 -1 0 1 2 3 4

Same signs

Dierent signs

= +

+ + = +

- =

= -

+ - = -

- + = -

- +

EXAMPLES

Evaluate

1. 4 3- +

Solution

Start at 4- and move 3 places to the right.

-4 -3 -2 -1 0 1 2 3 4

4 3 1- + = -

2. 1 2- -

Solution

Start at 1- and move 2 places to the let.

-4 -3 -2 -1 0 1 2 3 4

1 2 3- - = -

Multiplying and dividing

To multiply or divide, ollow these rules. This rule also works i there are two

signs together without a number in between e.g. 32 - -

You can also do these on a

calculator, or you may have

a different way of working

these out.


10/42


EXAMPLES

Evaluate

1. 2 7#-

Solution

Dierent signs ( 2 7and- + ) give a negative answer.

2 7 14#- = -

2. 12 4'- -

Solution

Same signs ( 12 4and- - ) give a positive answer.

12 4 3'- - =

3. 1 3- - -

Solution

The signs together are the same (both negative) so give a positive answer.

1 3

2

= - +

=

1 3- - -

1. 2 3- +

2. 7 4- -

3. 8 7# -

4. 37 -- ] g5. 28 7' -

6. . .4 9 3 7- +

7. . .2 14 5 37- -

8. . .4 8 7 4# -

9. . .1 7 4 87- -] g10.

53

132

- -

11. 5 3 4#-

12. 2 7 3#- + -

13. 4 3 2#- -

14. 1 2- - -

15. 7 2+ -

16. 2 1- -] g17. 2 15 5'- +

18. 2 6 5# #- -

19. 28 7 5#'- - -

20. 3 2-] g

1.2 Exercises

Evaluate

Start at 1- and move 3

places to the right.


11/42


Fractions, Decimals and Percentages

EXAMPLES

1. Write 0.45 as a raction in its simplest orm.

Solution

.0 4510045

55

20

9

'=

=

2. Convert83 to a decimal.

Solution

.

.

.

8 3 0000 375

8

30 375So =

g

3. Change 35.5% to a raction.

Solution

. %.

35 5100

35 5

22

20071

#=

=

4. Write 0.436 as a percentage.

Solution

. . %

. %

0 436 0 436 100

43 6

#=

=

5. Write 20 g as a raction o 1 kg in its simplest orm.

Solution

1 1000kg g=

1

20

1000

20

501

kg

g

g

g=

=

Multiply by 100% to

change a fraction or

decimal to a percentage.

Conversions

You can do all theseconversions on your

calculator using the

ac

bor S D+ key.

83 means3 8.'


12/42


Sometimes decimals repeat, or recur.

Example

. 0.31

0 33333333 3

f= =

There are dierent methods that can be used to change a recurring

decimal into a raction. Here is one way o doing it. Later you will discover

another method when studying series. (See HSC Course book, Chapter 8.)

EXAMPLES

1. Write .0 4

as a rational number.

Solution

. ( )

. ( )

( ) ( ):

n

n

n

n

0 44444 1

10 4 44444 2

2 1 9 4

94

Let

Then

f

f

=

=

- =

=

2. Change .1 329

to a raction.

Solution

. ( )

. ( )

( ) ( ): .

.

n

n

n

n

1 3292929 1

100 132 9292929 2

2 1 99 131 6

99131 6

1010

990

1316

1495

163

Let

Then

#

f

f

=

=

- =

=

=

=

A rational number is

any number that can be

written as a fraction.

Check this on your

calculator by dividing

4 by 9.

Try multiplying n by 10.

Why doesnt this work?

6. Find the percentage o people who preer to drink Lemon Fuzzy, i 24

out o every 30 people preer it.

Solution

% %3024

1100 80# =

CONTINUED


13/42


1. Write each decimal as a raction

in its lowest terms.0.64(a)

0.051(b)

5.05(c)

11.8(d)

2. Change each raction into a

decimal.

(a)52

(b) 1

8

7

(c)12

5

(d)117

3. Convert each percentage to a

raction in its simplest orm.

2%(a)

37.5%(b)

0.1%(c)

109.7%(d)

4. Write each percentage as a decimal.

27%(a)

109%(b)

0.3%(c)

6.23%(d)

5. Write each raction as a

percentage.

(a)207

(b)31

(c) 2

15

4

(d)1000

1

6. Write each decimal as a

percentage.

1.24(a)

0.7(b)

0.405(c)

1.2794(d)

7. Write each percentage as a

decimal and as a raction.

52%(a)

7%(b)

16.8%(c)

109%(d)

43.4%(e)

() %1241

8. Write these ractions as recurring

decimals.

(a)65

(b)7

99

(c)99

13

(d)61

(e)32

1.3 Exercises

Another method

Let .

. ( )

. ( )

( ) ( ):

n

n

n

n

n

1 3292929

10 13 2929292 1

1000 1329 292929 2

2 1 990 1316

990

1316

1495

163

Then

and

f

f

f

=

=

=

- =

=

=

This method avoids decimals

in the fraction at the end.


14/42


Investigation

Explore patterns in recurring decimals by dividing numbers by 3, 6, 9, 11,

and so on.

Can you predict what the recurring decimal will be i a raction has 3 in

the denominator? What about 9 in the denominator? What about 11?

Can you predict what raction certain recurring decimals will be? What

denominator would 1 digit recurring give? What denominator would you

have or 2 digits recurring?

Operations with fractions, decimals and percentages

You will need to know how to work with ractions without using a calculator,

as they occur in other areas such as algebra, trigonometry and surds.

()33

5

(g)71

(h) 1112

9. Express as ractions in lowest

terms.

(a) .0 8

(b) .0 2

(c) .1 5

(d) .3 7

(e) .0 67

() .0 54

(g) .0 15

(h) .0 216

(i) .0 219

(j) .1 074

10. Evaluate and express as a decimal.

(a)3 6

5

+

(b) 8 3 5'-

(c)12 34 7

+

+

(d) 199

31-

(e)7 413 6

+

+

11. Evaluate and write as a raction.

(a) . . .7 5 4 1 7 9' +] g(b)

4.5 1.3

15.7 8.9

-

-

(c)12.3 8.9 7.6

6.3 1.7

- +

+

(d). .

.

11 5 9 7

4 3

-

(e)8100

64

12. Angel scored 17 out o 23 in a

class test. What was her score as apercentage, to the nearest unit?

13. A survey showed that 31 out o

40 people watched the news on

Monday night. What percentage

o people watched the news?

14. What percentage o 2 kg is 350 g?

15. Write 25 minutes as a percentage

o an hour.


15/42


DID YOU KNOW?

Some countries use a comma or the decimal pointor example, 0,45 or 0.45.

This is the reason that our large numbers now have spaces instead o commas between

digitsor example, 15 000 rather than 15,000.

EXAMPLES

1. Evaluate 1 .52

43

-

Solution

152

43

57

43

2028

2015

20

13

- = -

= -

=

2. Evaluate 221

3' .

Solution

221

325

1

3

25

31

56

' '

#

=

=

=

3. Evaluate . .0 056 100#

Solution

. .0 056 100 5 6# =

Move the decimal point

2 places to the right.

The examples on ractions show how to add, subtract, multiply or divide

ractions both with and without the calculator. The decimal examples will

help with some simple multiplying and the percentage examples will be useul

in Chapter 8 o the HSC Course book when doing compound interest.

Most students use their calculators or decimal calculations. However, it

is important or you to know how to operate with decimals. Sometimes thecalculator can give a wrong answer i the wrong key is pressed. I you can

estimate the size o the answer, you can work out i it makes sense or not. You

can also save time by doing simple calculations in your head.


16/42


4. Evaluate . . .0 02 0 3#

Solution

. . .0 02 0 3 0 006# =

5. Evaluate10

8.753.

Solution

. .8 753 10 0 8753' =

6. The price o a $75 tennis racquet increased by %.521

Find the new

price.

Solution

% $ . $

$ .

5 75 0 055 75

4 13

o` #=

=

% . % $ . $

$ .

521

0 055 10521

75 1 055 75

79 13

21

or o #= =

=

So the price increases by $4.13 to $79.13.

7. The price o a book increased by 12%. I it now costs $18.00, what did

it cost beore the price rise?

Solution

The new price is 112% (old price 100%, plus 12%)

1%$ .

100%$ .

$16.07

112

18 00

112

18 00

1100

`

#

=

=

=

So the old price was $16.07.

1.4 Exercises

1. Write 18 minutes as a raction o

2 hours in its lowest terms.

2. Write 350 mL as a raction o

1 litre in its simplest orm.

3. Evaluate

(a)53

41

+

(b) 352

2107

-

(c)43

152

#

(d)73

4'

(e) 153

232

'

Multiply the numbers

and count the number

of decimal places in

the question.

Move the decimal

point 1 place to

the left.


17/42


4. Find53

o $912.60.

5. Find75

o 1 kg, in grams correct

to 1 decimal place.

6. Trinh spends 31 o her day

sleeping,247

at work and121

eating. What raction o the day

is let?

7. I get $150.00 a week or a casual

job. I I spend101

on bus ares,

152

on lunches and31

on outings,

how much money is let over orsavings?

8. John grew by20017

o his height

this year. I he was 165 cm tall

last year, what is his height now,

to the nearest cm?

9. Evaluate

(a) 8.9 3+

(b) 9 3.7-

(c) .1 9 10# (d) .0 032 100#

(e) .0 7 5#

() . .0 8 0 3#

(g) . .0 02 0 009#

(h) .5 72 1000#

(i)1008.74

(j) . .3 76 0 1#

10. Find 7% o $750.

11. Find 6.5% o 845 mL.

12. What is 12.5% o 9217 g?

13. Find 3.7% o $289.45.

14. I Kaye makes a proft o $5 by

selling a bike or $85, fnd the

proft as a percentage o the

selling price.

15. Increase 350 g by 15%.

16. Decrease 45 m by %.821

17. The cost o a calculator is now

$32. I it has increased by 3.5%,

how much was the old cost?

18. A tree now measures 3.5 m, which

is 8.3% more than its previous

years height. How high was the

tree then, to 1 decimal place?

19. This month there has been a

4.9% increase in stolen cars. I

546 cars were stolen last month,

how many were stolen this

month?

20. Georges computer cost $3500. I

it has depreciated by 17.2%, what

is the computer worth now?


18/42


Powers and Roots

A power(or index) o a number shows how many times a number is

multiplied by itsel.

PROBLEM

I both the hour hand and minute hand start at the same position at

12 oclock, when is the frst time, correct to a raction o a minute, that

the two hands will be together again?

EXAMPLES

1. 4 4 4 4 643 # #= =

2. 2 2 2 2 2 2 325 # # # #= =

In 43 the 4 is called the base

number and the 3 is called

the index or power.

A root o a number is the inverse o the power.

EXAMPLES

1. 36 6= since 6 362 =

2. 8 23 = since 2 83 =

3. 64 26 = since 2 646 =

DID YOU KNOW?

Many ormulae use indices (powers and roots).

For example the compound interest ormula that you will study in Chapter 8 o the HSC

Course book is 1A P rn

= +^ h Geometry uses ormulae involving indices, such as

3

4V r3r= . Do you know what this

ormula is or?

In Chapter 7, the ormula or the distance between 2 points on a number plane is

d x x y y ( ) ( )2 1

2

2 1

2= - + -

See i you can fnd other ormulae involving indices.


19/42


Proof

( )( )( )

a aa

a

a a aa a a m

na a a m n

a

1

timestimes

times

m nn

m

m n

'

# # #

# # #

# # #

f

f

f

=

=

=-

=-

Index laws

There are some general laws that simpliy calculations with indices.

a a am n m n# = +

Proof

( ) ( )a a a a a a a a

a a a

a

m n

m n

m n

m n

times times

times

# # # # # # # #

# # #

f f

f

=

=

=+

+

1 2 34444 4444 1 2 34444 4444

1 2 34444 4444

These laws work for any m

and n, including fractions and

negative numbers.

a a am n m n' = -

a=( )am n mn

Proof

( ) ( )

( )

a a a a a n

a n

a

times

times

m n m m m m

m m m m

mn

# # # #f=

=

=

f+ + + +

POWER AND ROOT KEYS

Use the x2

and x3

keys or squares and cubes.

Use the xy or ^ key to fnd powers o numbers.

Use the key or square roots.

Use the 3 key or cube roots.

Use the x or other roots.


20/42


( )ab a bn n n=

Proof

( ) ( )

( ) ( )

ab ab ab ab ab n

a a a b b b

a b

timesn

n n

n ntimes times

# # # #

# # # # # # #

f

f f

=

=

=

1 2 34444 4444 1 2 34444 4444

b

a

b

an

n

n

=c m

Proof

( )

( )( )

b

a

b

a

b

a

b

a

b

an

b b b b

a a a a nn

b

a

times

timestimes

n

n

n

# # # #

# # # #

# # # #

f

f

f

=

=

=

c m

EXAMPLES

Simpliy

1. m m m9 7 2# '

Solution

m m m m

m

9 7 2 9 7 2

14

# ' =

=

+ -

2.

3

( )y2

4

Solution

( ) ( )y y

y

y

2 2

2

8

4 3 3 4 3

3 4 3

12

=

=

=

#

CONTINUED


21/42


1. Evaluate without using a

calculator.

(a) 5 23 2#

(b) 3 84 2+

(c)41 3c m

(d) 273

(e) 164


place.

(a) 3.72

(b) 1.061.5

(c) 2.3 0.2-

(d) 193

(e) . . .34 8 1 2 43 13 #-

()0.99 5.61

13

+

3. Simpliy(a) a a a6 9 2# #

(b) y y y3 8 5# #-

(c) a a1 3#- -

(d) 2 2w w#1 1

(e) x x6 '

() p p3 7' -

(g)y

y5

11

(h) ( )x7 3

(i) (2 )x5 2

(j) (3 )y 2 4-

(k) a a a3 5 7# '

(l)y

x9

2 5f p (m)

w

w w3

6 7#

(n)( )

p

p p9

2 3 4#

(o)x

x x2

6 7'

(p)( )

a b

a b4 9

2 2 6

#

#

(q)( ) ( )

x y

x y1 4

2 3 3 2

#

#

-

-

4. Simpliy

(a) x x5 9

# (b) a a1 6#- -

(c)m

m3

7

(d) k k k13 6 9# '

(e) a a a5 4 7# #- -

() 5 5x x#2 3

(g)m n

m n4 2

5 4

#

#

1.5 Exercises

3.( )

y

y y5

6 3 4#

-

Solution

( )

y

y y

y

y y

y

y

y

y

y

( )

5

6 3 4

5

18 4

5

18 4

5

14

9

# #=

=

=

=

- -

+ -


22/42


(h)2 2

p

p p2

#

1 1

(i) (3 )x11 2

(j)( )

x

x3

4 6

5. Simpliy

(a) 5( )pq3

(b)b

a 8c m (c)

4

b

a4

3d n (7(d) a5b)2

(e)(2 )

m

m4

7 3

() ( )xyxy xy

3 2 4

#

(g)3

4

( )

( )

k

k

6

23

8

(h) yy

28

5 712

#_ i (i)

a

a a11

6 4 3#

-e o (j)

x y

xy58 3

9 3

#

f p 6. Evaluate a3b2 when 2a = and

43

b = .

7. I32

x = and91

,y= fnd the value

oxy

x y5

3 2

.

8. I21

,31

a b= = and41

,c=

evaluatec

a b4

2 3as a raction.

9. (a) Simpliya b

a b8 7

11 8

.

Hence evaluate(b)a b

a b8 7

11 8

when

52

a = and85

b = as a raction.

10. (a) Simpliyp q r

p q r4 6 2

5 8 4

.

(b) Hence evaluatep q r

p q r4 6 2

5 8 4

as a

raction when

8

7,

3

2p q= = and

43

r= .

11. Evaluate ( )a4 3 when6.a

32

=

1

c m

12. Evaluateb

a b4

3 6

when a21

= and

b32

= .

13. Evaluatex y

x y5 5

4 7

when x 31= and

y92

= .

14. Evaluatek

k9

5

-

-

when .k31

=

15. Evaluate( )a b

a b3 2 2

4 6

when a43

= and

b91

= .

16. Evaluatea b

a b5 2

6 3

#

#as a raction

when a91

= and b43

= .

17. Evaluatea b

a b3

2 7

as a raction in

index orm when a52 4

= c m andb

85 3

= c m .

18. Evaluate( )

( )

a b c

a b c2 4 3

3 2 4

as a raction

when ,a31

= b76

= and c97

= .


23/42


Proof

x x x

x

x xx

x

x

1

1

n n n n

n nn

n

0

0

'

'

`

=

=

=

=

=

-

Negative and zero indices

Class Investigation

Explore zero and negative indices by looking at these questions.

For example simpliy x x3 5' using (i) index laws and (ii) cancelling.

(i) x x x3 5 2' = - by index laws

(ii)x

xx x x x x

x x x

x

1

5

3

2

# # # #

# #=

=

xx

1So 2

2=

-

Now simpliy these questions by (i) index laws and (ii) cancelling.

(a) x x2 3'

(b) x x2 4'

(c) x x2 5'

(d) x x3 6'

(e) x x3 3'

() x x2 2'

(g) x x2'

(h) x x5 6'

(i) x x4 7'

(j) x x3'

Use your results to complete:

x

x

0

n

=

=-

x 10 =


24/42


1x

xn

n=-

Proofx x x

x

x xx

x

x

xx

1

1

n n

n

nn

n

nn

0 0

00

'

'

`

=

=

=

=

=

-

-

-

EXAMPLES

1. Simpliy .abc

ab c4

5 0e o Solution

1abc

ab c4

5 0

=e o

2. Evaluate .2 3-

Solution

22

1

81

3

3=

=

-

3. Write in index orm.

(a)1

x2

(b)3

x5

(c)51x

(d)x 1

1+

CONTINUED


25/42


1. Evaluate as a raction or whole

number.(a) 3 3-

(b) 4 1-

(c) 7 3-

(d) 10 4-

(e) 2 8-

6() 0

(g) 2 5-

(h) 3 4-

(i) 7 1-

(j) 9 2-

(k) 2 6-

(l) 3 2-

4(m) 0

(n) 6 2-

(o) 5 3-

(p) 10 5-

(q) 2 7-

(r) 20

(s) 8 2-

(t) 4 3-

2. Evaluate

(a) 20

(b)21 4-c m

(c)32 1-c m

(d)6

5 2-c m

(e)3

2

x y

x y 0

-

+f p ()

51 3-c m

(g)43

1-

c m

(h)71 2-c m

(i)32 3-c m

(j)21 5-c m

(k)73 1-c m

1.6 Exercises

Solution

(a)1

xx

2

2=

-

(b) x x

x

33

1

3

5 5

5

#=

=-

(c)x x

x

51

51 1

51 1

#=

=-

(d)( )x x

x

11

1

1

1

1

1

+=

+

= +-] g

4. Write a3 without the negative index.

Solution

aa

133

=-


26/42


(l)9

8 0c m

(m)76 2-c m

(n)

10

9 2-c m

(o)11

6 0c m

(p)41 2

-

-c m (q)

52 3

-

-c m (r) 3

72 1

-

-c m (s)

8

3 0-c m

(t) 1 412

-

-

c m

3. Change into index orm.

(a)1

m3

(b)1x

(c)1

p7

(d)1

d9

(e)1

k5

()1

x2

(g)2

x4

(h)3

y2

(i)2

1

z6

(j)53t8

(k)72x

(l)2

5

m6

(m)3

2

y7

(n)(3 4)

1

x 2+

(o)( )

1

a b 8+

(p)

2

1

x-

(q)( )p5 1

13

+

(r)(4 9)

2

t 5-

(s)( )x4 1

111

+

(t)9( 3 )

5

a b 7+

4. Write without negative indices.

(a) t 5-

(b) x 6-

(c) y 3-

(d) n 8-

(e) w 10-

() x2 1-

(g) 3m 4-

(h) 5x 7-

(i) 2x 3-] g (j) n4

1-

] g (k) x 1 6+ -] g (l) y z8 1+ -^ h (m) 3k 2- -] g (n) 3 2x y 9+ -^ h (o)

1x

5-b l

(p) y1 10-c m

(q)2p

1-

d n

(r)1

a b

2

+

-c m

(s) x y

x y 1

-

+-e o

(t)32

x y

w z 7

+

--e o


27/42


Proof

n

n

a a

a a

a a

by index lawsn

n n

n`

=

=

=

1

1

` ^^j hh

Fractional indices

Class Investigation

Explore ractional indices by looking at these questions.

For example simpliy (i) 2x21` j and (ii) .x 2^ h

2( ) x xx

i by index laws2

1=

=

1` ^j h

2

2

( ) x x

x x x

x x

ii

So

2

22

`

=

= =

=

1

1

^` ^

hj h

Now simpliy these questions.

(a) 2x21^ h

(b) x2

(c) 3x31` j

(d) 3x31^ h

(e) x33^ h

() x33

(g)4

x41` j

(h) 4x41

^ h (i) x4

4^ h (j) x44

Use your results to complete:

nx =1

na an=1


28/42


EXAMPLES

1. Evaluate

(a)2

491

(b) 3271

Solution

(a)2

49 497

=

=

1

(b)3

27 273

3=

=

1

2. Write x3 2- in index orm.

Solution

2( )x x3 2 3 2- = -1

3. Write 7( )a b+1

without ractional indices.

Solution

7( )a b a b7+ = +1

Proof

n n

n n

a a

a

a

a

m

n m

m

mn

=

=

a =

=

m

m

1

1

`^^

jhh

Putting the ractional and negative indices together gives this rule.

- na

a

1n

=

1

Here are some urther rules.

n

( )

a a

a

mn

n m

=

=

m


29/42


b

aabn n

=

-c bm l

EXAMPLES

1. Evaluate

(a)3

8

4

(b)-

31251

(c)32 3-c m

Solution

(a) 3 ( ) ( )

8 8 8

2

16

or3 4 43

4

=

=

=

4

(b)-

3

3

125

125

1

125

1

51

3

=

=

=

1

1

Proof

b

a

b

a

b

a

b

a

a

b

ab

ab

1

1

1

1

n

n

n

n

n

n

n

n

n

n

n

'

#

=

=

=

=

=

=

-c c

b

m m

l


30/42


(c)32

2

3

827

3 8

3

3 3

=

=

=

-c cm m


(a) x5

(b)( )x4 1

12 23

-

Solution

(a) 2x x5 =5

(b)

-

3

3

( ) ( )

( )

x x

x

4 1

1

4 1

1

4 1

2 23 2

2

-

=

-

= -

2

2

3. Write-

5r3

without the negative and ractional indices.

Solution

-5

5

r

r

r

1

135

=

=

3

3

DID YOU KNOW?

Nicole Oresme (132382) was the frst mathematician to use ractional indices.

John Wallis (16161703) was the frst person to explain the signifcance o zero, negative

and ractional indices. He also introduced the symbol 3 or infnity.Do an Internet search on these mathematicians and fnd out more about their work and

backgrounds. You could use keywords such as indices and infnity as well as their names to fnd

this inormation.


31/42


1. Evaluate

(a) 2811

(b) 3271

(c) 2161

(d) 381

(e) 2491

() 310001

(g) 4161

(h) 2641

(i) 3641

(j) 711

(k) 4811

(l) 5321

(m) 801

(n) 31251

(o) 33431

(p) 71281

(q) 42561

(r) 293

(s)-

381

(t)-

3642


places.

(a) 4231

(b) 45.84

(c) 1.24 4.327 +

(d)12.9

15

(e). .

. .

1 5 3 7

3 6 1 48

+

-

(). .

. .

8 79 1 4

5 9 3 74 #

-

3. Write without ractional indices.

(a) 3y1

(b) 3y2

(c) 2x-

1

(d) 2( )x2 5+1

(e)-

2( )x3 1-1

() 3( )q r6 +1

(g)-

5( )x 7+2


(a) t

(b) y5

(c) x3

(d) 9 x3 -

(e) s4 1+

()2 3

1

t+

(g)(5 )

1

x y 3-

(h) ( )x3 1 5+

(i)( 2)

1x 23 -

(j)2 7

1

y+

(k)4

5

x3 +

(l)y3 1

22

-

(m)5 ( 2)

3

x2 34 +

5. Write in index orm and simpliy.

(a) x x

(b) xx

(c)x

x3

(d)x

x3

2

(e) x x4

1.7 Exercises


32/42


6. Expand and simplify, and write in

index form.

(a) ( )x x 2+

(b) ( )( )a b a b3 3 3 3+ -

(c)1

pp

2

+

f p

(d) (1

)xx

2+

(e)( )

x

x x x3 13

2- +

7. Write without fractional or

negative indices.

(a)-

3( )a b2-1

(b) 3( )y 3--

2

(c)-

7( )a4 6 1+4

(d)

-4( )x y

3

+

5

(e)

-9( )x

7

6 3 8+2

Scientific notation (standard form)

Very large or very small numbers are usually written in scientific notation to

make them easier to read. What could be done to make the figures in the box

below easier to read?

DID YOU KNOW?

The Bay o Fundy, Canada, has the largest tidal changes in the world. About 100 000 000 000

tons o water are moved with each tide change.

The dinosaurs dwelt on Earth or 185 000 000 years until they died out 65 000 000 years ago.

The width o one plant cell is about 0.000 06 m.

In 2005, the total storage capacity o dams in Australia was 83 853 000 000 000 litres and

households in Australia used 2 108 000 000 000 litres o water.

A number in scientific notation is written as a number between 1 and 10

multiplied by a power of 10.

EXAMPLES

1. Write 320 000 000 in scientific notation.

Solution

.320 000 000 3 2 108#=

2. Write .7 1 10 5# - as a decimal number.

Solution

. .

.

7 1 10 7 1 10

0 000 071

5 5# '=

=

-

Write the number

between 1 and 10

and count the decimal

places moved.

Count 5 places to

the left.


33/42


SIGNIFICANT FIGURES

The concept of significant figures is related to rounding off. When we look

at very large (or very small) numbers, some of the smaller digits are not

significant.

For example, in a football crowd of 49 976, the 6 people are not really

significant in terms of a crowd of about 50 000! Even the 76 people are not

significant.

When a company makes a profit of $5 012 342.87, the amount of87 cents is not exactly a significant sum! Nor is the sum of $342.87.

To round off to a certain number of significant figures, we count from the

first non-zero digit.

In any number, non-zero digits are always significant. Zeros are not

significant, except between two non-zero digits or at the end of a decimal

number.

Even though zeros may not be significant, they are still necessary. For

example 31, 310, 3100, 31 000 and 310 000 all have 2 significant figures but

are very different numbers!

Scientific notation uses the significant figures in a number.

SCIENTIFIC NOTATION KEY

Use the EXP or 10 x# key to put numbers in scientifc notation.

For example, to evaluate 3.1 10 2.5 10 ,4 2

# ' #-

press 3.1 EXP 4 2.5 EXP ( ) 2

1240 000

' =-

=

DID YOU KNOW?

Engineering notation is similar to scientifc notation, except the powers o 10 are always

multiples o 3. For example,

3.5 103

#

15.4 10 6# -

EXAMPLES

. ( )

. . ( )

. . ( )

12 000 1 2 10 2

0 000 043 5 4 35 10 3

0 020 7 2 07 10 3

significant figures

significant figures

significant figures

4

5

2

#

#

#

=

=

=

-

-

When rounding off to significant figures, use the usual rules for rounding off.


34/42


EXAMPLES

1. Round o 4 592 170 to 3 signifcant fgures.

Solution

4 592 170 4 590 000= to 3 signifcant fgures

2. Round o 0.248 391 to 2 signifcant fgures.

Solution

. .0 248 391 0 25= to 2 signifcant fgures

3. Round o 1.396 794 to 3 signifcant fgures.

Solution

. .1 396 794 1 40= to 3 signifcant fgures

1. Write in scientifc notation.

3 800(a)

1 230 000(b)61 900(c)

12 000 000(d)

8 670 000 000(e)

416 000()

900(g)

13 760(h)

20 000 000(i)

80 000(j)


0.057(a)0.000 055(b)

0.004(c)

0.000 62(d)

0.000 002(e)

0.000 000 08()

0.000 007 6(g)

0.23(h)

0.008 5(i)

0.000 000 000 07(j)

3. Write as a decimal number.

(a) .3 6 104#

(b) .2 78 10

7#

(c) .9 25 103#

(d) .6 33 106#

(e) 4 105#

() .7 23 10 2# -

(g) .9 7 10 5# -

(h) .3 8 10 8# -

(i) 7 10 6# -

(j) 5 10 4# -

4. Round these numbers to

2 signifcant fgures.235 980(a)

9 234 605(b)

10 742(c)

0.364 258(d)

1.293 542(e)

8.973 498 011()

15.694(g)

322.78(h)

2904.686(i)

9.0741(j)

1.8 Exercises

Remember to putthe 0s in!


35/42


5. Evaluate correct to 3 signifcant

fgures.

(a) . .14 6 0 453#

(b) .4 8 7'

(c) 4. . .47 2 59 1 46#+

(d) . .3 47 2 71-

6. Evaluate . . ,4 5 10 2 9 104 5# # #

giving your answer in scientifc

notation.

7. Calculate.

.

1 34 10

8 72 107

3

#

#-

and write

your answer in standard ormcorrect to 3 signifcant fgures.

Investigation

A logarithm is an index. It is a way o fnding the power (or index) to

which a base number is raised. For example, when solving ,3 9x = the

solution is .x 2=

The 3 is called the base number and the x is the index or power.

You will learn about logarithms in the HSC course.

Ia yx = then log y xa

=

The expression log1.7

49 means the power o 7 that gives 49.

The solution is 2 since .7 492 =

The expression log2.2

16 means the power o 2 that gives 16.

The solution is 4 since .2 164 =

Can you evaluate these logarithms?

log1.3

27

log2.5

25

log3.10

10 000

log4.2

64

log5.4

4

log6.7

7

log7.3

1

log8.4

2

9.31

log3

10.41

log2

The a is called the base

number and the x is the

index or power.


36/42


Absolute Value

Negative numbers are used in maths and science, to show opposite directions.

For example, temperatures can be positive or negative.

But sometimes it is not appropriate to use negative numbers.For example, solving 9c2 = gives two solutions, c 3!= .

However when solving 9,c2 = using Pythagoras theorem, we only use

the positive answer, 3,c = as this gives the length o the side o a triangle. The

negative answer doesnt make sense.

We dont use negative numbers in other situations, such as speed. In

science we would talk about a vehicle travelling at 60k/h going in a negative

direction, but we would not commonly use this when talking about the speed

o our cars!

Absolute value defnitions

We write the absolute value ox as x

xx x

x

0when

when x 01

$=

-

)

EXAMPLES

1. Evaluate .4

Solution

4 4 04 since $=

We can also defne

x as the distance

o x rom 0 on the

number line. We will

use this in Chapter 3.

CONTINUED


37/42


2. Evaluate .3-

Solution

3 3 3 0

3

since 1- = - - -

=

] g

The absolute value has some properties shown below.

Properties o absolute value

a 9= = =

| | | | | | | | | | | |

| | | || | | |

| | | | | | | |

| | | | | | | |

| | | | | | | | | | | | | | | | | |

ab a b

aa a

a a

a b b a

a b a b

2 3 2 3 6

3 35 5 57 7 7

2 3 3 2 1

2 3 2 3 3 4 3 4

e.g.

e.g.e.g.e.g.

e.g.

e.g. but

2 2 2 2

2 2

# # #

1#

= - = - =

- -

= = =

- = - = =

- = - - = - =

+ + + = + - + - +

] g

EXAMPLES

1. Evaluate 2 1 32

- - + - .

Solution

2 1 3 2 1 3

2 1 9

10

22- - + - = - +

= - +

=

2. Show that a b a b#+ + when a 2= - and 3b = .

Solution

a b

2 3

1

1

LHS = +

= - +

=

=

LHS means Let Hand Side.


38/42


a b

2 3

2 3

5

RHS = +

= - +

= +

=

a b a b

1 5Since 1#+ +

3. Write expressions or 2 4x - without the absolute value signs.

Solution

1

x x x

x

x

x x x

x x

x

2 4 2 4 2 4 0

2 4

2

2 4 2 4 2 4 0

2 4 2 4

2

when

i.e.

when

i.e.

1

1

$

$

$

- = - -

- = - - -

= - +] g

Class Discussion

Are these statements true? I so, are there some values or which the

expression is undefned (values ox or ythat the expression cannot

have)?

1.x

x1=

2. 2 2x x=

3. 2 2x x=

4. x y x y + = +

5. x x2 2=

6. x x3 3=

7. x x1 1+ = +

8.x

x

3 2

3 21

-

-=

9.x

x1

2=

10. x 0$

Discuss absolute value and its defnition in relation to these statements.

RHS means Right Hand Side.


39/42


1. Evaluate

(a) 7

(b) 5-

(c) 6- (d) 0

(e) 2

() 11-

(g) 2 3-

(h) 3 8-

(i) 52

-

(j) 5 3-

2. Evaluate

(a) 3 2+ -

(b) 3 4- - (c) 5 3- +

(d) 2 7# -

(e) 3 1- + -

() 5 2 62

#- -

(g) 2 5 1#- + -

(h) 3 4-

(i) 2 3 3 4- - -

(j) 5 7 4 2- + -

3. Evaluate a b- i

(a) 5 2a band= =

(b) 1 2a band= - =

(c) 2 3a band= - = -

(d) 4 7a band= =

(e) .a b1 2and= - = -

4. Write an expression or

(a) a a 0when 2

(b) 0a awhen 1

(c) 0a awhen =

(d) 0a a3 when2

(e) 0a a3 when 1

() 0a a3 when =

(g) a a1 1when 2+ -

(h) 1a a1 when 1+ -

(i) 2x x2 when 2-

(j) 2x x2 when 1- .

5. Show that a b a b#+ +

when

(a) 2 4a band= =

(b) 1 2a band= - = -

(c) 2 3a band= - =

(d) 4 5a band= - =

(e) .a b7 3and= - = -

6. Show that x x2 = when

(a) 5x =

(b) x 2= - (c) x 3= -

(d) 4x =

(e) .x 9= -

7. Use the defnition o absolute

value to write each expression

without the absolute value signs

(a) x 5+

(b) 3b -

(c) 4a +

(d) 2 6y- (e) 3 9x +

() 4 x-

(g) k2 1+

(h) 5 2x -

(i) a b+

(j) p q-

8. Find values ox or which .x 3=

9. Simpliy nn

where .n 0!

10. Simpliy2

2

x

x

-

-and state which

value x cannot be.

1.9 Exercises


40/42


1. Convert

0.45 to a raction(a)

14% to a decimal(b)

(c)85

to a decimal

78.5% to a raction(d)

0.012 to a percentage(e)

()1511

to a percentage

2. Evaluate as a raction.(a) 7 2-

(b) 5 1-

(c) 29-

1

3. Evaluate correct to 3 signifcant fgures.

(a) . .4 5 7 62 2+

(b) 4.30.3

(c)5.7

23

(d)..

3 8 101 3 10

6

9

#

#

(e)-

362

4. Evaluate

(a) | | | |3 2- -

(b) |4 5 |-

(c) 7 4 8#+

(d) [( ) ( ) ]3 2 5 1 4 8# '+ - -

(e) 4 3 9- + -

() 12- - -

(g) 24 6'- -

5. Simpliy

(a) x x x5 7 3# '

(b) (5 )y3 2

(c)( )

a b

a b9

5 4 7

(d)3

2x63d n

(e)

a b

ab5 6

4 0

e o

6. Evaluate

(a) 153

87

-

(b)76

332

#

(c) 943

'

(d)52

2101

+

(e) 1565#

7. Evaluate

(a) 4-

(b) 2361

(c) 5 2 32- -

(d) 4 3- as raction

(e) 382

() 2 1- -

(g) 249-

1

as a raction

(h) 4161

(i) 3 0-] g (j) 4 7 2 32- - - -

8. Simpliy

(a) a a14 9'

(b) x y5 36_ i

(c) p p p6 5 2# '

(d) 2b9 4

^ h (e)

(2 )

x y

x y10

7 3 2


(a) n

(b)1

x5

(c)1

x y+

(d) x 14 +

Test Yourself 1


41/42


(e) a b7 +

() 2x

(g)2

1

x3

(h) x43

(i) (5 3)x 97 +

(j)1

m34

10. Write without ractional or negative

indices.

(a) a 5-

(b) 4n1

(c) 2( )x 1+1

(d) ( )x y 1- -

(e) (4 7)t 4- -

() 5( )a b+1

(g) 3x-

1

(h) 4b3

(i)3

( )x2 3+

4

(j)

-2x3

11. Show that a b a b#+ + when 5a =

and 3b = - .

12. Evaluate a2b4 when259

a = and 132

b = .

13. I31

a4

= c m and43

,b = evaluate ab3 as a

raction.

14. Increase 650 mL by 6%.

15. Johan spends31

o his 24-hour day

sleeping and41

at work.

How many hours does Johan spend(a)

at work?

What raction o his day is spent at(b)

work or sleeping?

I he spends 3 hours watching TV,(c)

what raction o the day is this?What percentage o the day does he(d)

spend sleeping?

16. The price o a car increased by 12%. I

the car cost $34 500 previously, what is

its new price?

17. Rachel scored 56 out o 80 or a maths

test. What percentage did she score?

18. Evaluate ,2118 and write your answer in

scientifc notation correct to 1 decimalplace.


(a) x

(b)1y

(c) 3x6 +

(d)(2 3)

1

x 11-

(e) y73


0.000 013(a)

123 000 000 000(b)

21. Convert to a raction.

(a) .0 7

(b) .0 124

22. Write without the negative index.

(a) x 3-

(b) ( )a2 51

+

-

(c)b

a 5-c m

23. The number o people attending a

ootball match increased by 4% rom last

week. I there were 15 080 people at the

match this week, how many attended

last week?

24. Show that | |a b a b#+ + when

2a = - and 5.b = -


42/42


1. Simpliy 843

332

4 1 .52

87

'+ -c cm m

2. Simpliy .53

125

180149

307

+ + -

3. Arrange in increasing order o size: 51%,

0.502, . ,0 5

.9951

4. Mark spends31 o his day sleeping,

121

o the day eating and201

o the day

watching TV. What percentage o the day

is let?

5. Write-

3642


6. Express . .3 2 0 01425 ' in scientifc

notation correct to 3 signifcant fgures.

7. Vinh scored 17 2

1

out o 20 or a mathstest, 19 out o 23 or English and 55

21

out o 70 or physics. Find his average

score as a percentage, to the nearest

whole percentage.

8. Write .1 3274


9. The distance rom the Earth to the moon

is .3 84 105# km. How long would it take

a rocket travelling at .2 13 10 km h4# to

reach the moon, to the nearest hour?

10. Evaluate. . .

. .0 2 5 4 1 3

8 3 4 13

'

#

+correct to

3 signifcant fgures.

11. Show that ( ) ( ) .2 2 1 2 2 2 1k k k1 1- + = -+ +

12. Find the value ob c

a3 2

in index orm i

.,a b c52

31

53

and4 3 2

= = - =c c cm m m

13. Which o the ollowing are rational

numbers: , . , , , . , ,3 0 34 2 3 1 5 0

7

3r- ?

14. The percentage o salt in 1 L o water is

10%. I 500 mL o water is added to this

mixture, what percentage o salt is there

now?

15. Simpliy| |

x

x

1

12

-

+

or .x 1!!

16. Evaluate2.4 3.31

4.3 2.93 2

1.3

6

+

-correct to

2 decimal places.

17. Write 15 g as a percentage o 2.5 kg.

18. Evaluate . .2 3 5 7 10.1 8 2#+ - correct to

3 signifcant fgures.

19. Evaluate( . )

. .

6 9 10

3 4 10 1 7 105 3

3 2

#

# #- +- -

and

express your answer in scientifc notation

correct to 3 signifcant fgures.

20. Prove | | | | | |a b a b#+ +

or all real a, b.

Challenge Exercise 1

maths in focus - margaret grove - ch1

Documents