MQ 10 RATIONAL AND IRRATIONAL NUMBERS

Manning’s formula is a

formula used to estimate the

flow of water down a river in

a flood event, measured in

metres per second. The

formula is v = , where

R is the hydraulic radius, S is

the slope of the river and n is

the roughness coefficient.

What will be the flow of

water in the river if R = 8,

S = 0.0025 and n = 0.625?

To find the answer to this

problem you will need to

have an understanding of

fractional indices, which in

this chapter we will be

relating to your work on

surds from last year.

R

2

3---

S

1

2---

n-----------

1Rational and

irrational

numbers

2 M a t h s Q u e s t 8 f o r V i c t o r i a

READY?areyou

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be

obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon

next to the question on the Maths Quest 10 CD-ROM or ask your teacher for a copy.

Identifying surds

1 Which of the following are surds?

a b c d 4

Simplifying surds

2 Simplify each of the following.

a b c 5 d 3

Adding and subtracting surds


a 2 – 4 + 7 – 5 b 2 – 5 – 4 + 10

Multiplying and dividing surds


a × b 2 × 4 c d

Rationalising denominators

5 Rationalise the denominator of each of the following.

a b c d –

Evaluating numbers in index form

6 Evaluate each of the following.

a 72 b 34 c (2.5)6 d (0.3)4

Using the index laws


a x3 × x7 b 4y3 × 5y8 c 24a3b ÷ 6ab5 d (2m4)2

1.1

7 10 49 2

1.2

48 98 12 72

1.3

6 3 3 6 32 45 180 8

1.4

7 10 3 66

2

-------5 6

10 3

-------------

1.5

1

5

-------2

3

-------3

6

-------7 5

5 7

----------

1.9

1.10

C h a p t e r 1 R a t i o n a l a n d i r r a t i o n a l n u m b e r s 3

Classifying numbersThe number systems used today evolved from a basic and practical need of primitive

people to count and measure magnitudes and quantities such as livestock, people,

possessions, time and so on.

As societies grew and architecture

and engineering developed, number

systems became more sophisticated.

Number use developed from solely

whole numbers to fractions, decimals

and irrational numbers.

We shall explore these different

types of numbers and classify them

into their specific groups.

The Real Number System contains

the set of rational and irrational

numbers. It is denoted by the symbol

R. The set of real numbers contains a

number of subsets which can be classi-

fied as shown in the chart below.

The relationship which exists between the

subsets of the Real Number System can be

illustrated in a Venn diagram as shown on

the right.

We can say N ⊂ Z, Z ⊂ Q, and so on,

where ⊂ means ‘is a subset of’.

Rational numbers (Q)

A rational number (ratio-nal) is a number

that can be expressed as a ratio of two whole

numbers in the form , where b ≠ 0.

Rational numbers are given the symbol Q. Examples are:

, , , , 7, –6, 0.35, 1.4.

Real numbers R

Irrational numbers I

(surds, non-terminating

and non-recurring

decimals, ,e)

Rational numbers Q

Integers

Z

Non-integer rationals

(terminating and

recurring decimals)

Zero

(neither positive

nor negative)

Positive

(Natural

numbers N)

Z +Negative

Z –

π

(Irrationalnumbers)

Q (Rational numbers)

(Integers)

(Naturalnumbers)

= R

I

N

Z

ε

a

b---

1

5---

2

7---

3

10------

9

4---

4 M a t h s Q u e s t 1 0 f o r N e w S o u t h W a l e s 5 . 3 P a t h w a y

Integers (Z)Rational numbers may be expressed as integers. Examples are:

= 5, = −4, = 27, – = −15

The set of integers consists of positive and negative whole numbers and 0 (which is

neither positive nor negative). They are denoted by the letter Z and can be further

divided into subsets. That is:

Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}

Z+ = {1, 2, 3, 4, 5, 6, . . .}

Z− = {−1, −2, −3, −4, −5, −6, . . .}

Positive integers are also known as natural numbers (or counting numbers) and are

denoted by the letter N. That is:

N = {1, 2, 3, 4, 5, 6, . . .}

Integers may be represented on the number line as illustrated below.

Note: Integers on the number line are marked with a solid dot to indicate that they are

the only points in which we are interested.

Non-integer rationalsRational numbers may be expressed as terminating decimals. Examples are:

= 0.7, = 0.25, = 0.625, = 1.8

These decimal numbers terminate after a specific number of digits.

Rational numbers may be expressed as recurring decimals (non-terminating or

periodic decimals). For example:

= 0.333 333 . . . or 0.3.

= 0.818 181 . . . or 0.8.

1.

(or )

= 0.833 333 . . . or 0.83.

= 0.230 769 230 769 . . . or 0.2.

30769.

(or )

These decimals do not terminate, and the specific

digit (or number of digits) is repeated in a pattern.

Recurring decimals are represented by placing a dot

or line above the repeating digit or pattern.

Rational numbers are defined in set notation as: Q = rational numbers

Q = { , a, b ∈ Z, b ≠ 0} where ∈ means ‘an element of’.

Rational numbers may be represented on the number line as shown above.

5

1---

4–

1------

27

1------

15

1------

–3 –2 –1 3 Z210

The set of integers

N

The set of positive integersor natural numbers

1 2 3 4 5 6 Z–

The set of negative integers

–1–2–3–4–5–6

7

10------

1

4---

5

8---

9

5---

1

3---

9

11------ 0.81

5

6---

3

13------ 0.230 769

–3–4 –2 –1 3210 4

–3.7431.63 3.6–2

3–4

1–2

Q

a

b---


Irrational numbers (I)Numbers that cannot be expressed as a ratio between two whole numbers are called

irrational numbers. Irrational numbers are denoted by the letter I. Numbers such as

surds (for example , ), decimals that neither terminate nor recur, and π and e

are examples of irrational numbers. The numbers π and e are examples of transcen-

dental numbers; these will be discussed briefly later in this chapter.

Irrational numbers may also be represented on the number line with the aid of a ruler

and compass.

An irrational number (ir-ratio-nal) is a number that cannot be expressed as a

ratio of two whole numbers in the form , where b ≠ 0.

Irrational numbers are given the symbol I. Examples are:

, , , , π, e

Irrational numbers may be expressed as decimals. For example:

= 2.236 067 977 5 . . . = 0.173 205 080 757 . . .

= 4.242 640 687 12 . . . = 5.291 502 622 13 . . .

π = 3.141 592 653 59 . . . e = 2.718 281 828 46 . . .

These decimals do not terminate, and the digits do not repeat themselves in any par-

ticular pattern or order (that is, they are non-terminating and non-recurring).

Rational and irrational numbers belong to the

set of real numbers (denoted by the symbol R).

They can be positive, negative or 0. The real

numbers may be represented on a number line

as shown at right (irrational numbers above the

line; rational numbers below it).

To classify a number as either rational or irrational:

1. Determine whether it can be expressed as a whole number, a fraction or a

terminating or recurring decimal.

2. If the answer is yes, the number is rational; if the answer is no, the number is irrational.

Consider an isosceles right-angled triangle of side length 1 unit.

By Pythagoras’ theorem, (OB)2 = (OA)2 + (AB)2; therefore the length of the

hypotenuse is units.

By using a compass, we can transfer the length of the

hypotenuse OB to the number line (labelled C). This

distance can now be measured using a ruler. Although

this distance will be inaccurate due to the equipment used,

there is an exact point on the number line for each

irrational number.

This geometric model can be extended to any irrational number in surd form.

π (pi)The symbol π (pi) is used for a particular number; that is, the circumference of a circle

whose diameter length is 1 unit. It can be approximated as a decimal that is non-

terminating and non-recurring. Therefore, π is classified as an irrational number. (It is

also called a transcendental number and cannot be expressed as a surd.)

7 10

a

b---

7 13 5 217

9-------

5 0.03

18 2 7

–3–4 –2 –1 3210 4

1–2

– π

–4

π

– 12 – 5 2

R

2

R

1 unit

0O A

B

C

1 unit 1 2 2

2 units


In decimal form, π = 3.141 592 653 589 793 23 . . . It has been calculated to

29 000 000 (29 million) decimal places with the aid of a computer.

Specify whether the following numbers are rational or irrational.

a b c d 2π e 0.28 f g h

THINK WRITE

a is already a rational number. a is rational.

b Evaluate . b

The answer is an integer, so classify

.

is rational.

c Evaluate . c = 3.316 624 790 36 . . .

The answer is a non-terminating and

non-recurring decimal; classify .

is irrational.

d Use your calculator to find the value

of 2π.

d 2π = 6.283 185 307 18 . . .

The answer is a non-terminating and

non-recurring decimal; classify 2π.

2π is irrational.

e 0.28 is a terminating decimal; classify it

accordingly.

e 0.28 is rational.

f Evaluate . f = 4

The answer is a whole number, so

classify .

is rational.

g Evaluate . g = 2.802 039 330 66 . . .

The result is a non-terminating and

non-recurring decimal; classify .

is irrational.

h Evaluate . h =

The result is a number in a rational

form. is rational.

1

4--- 16 11 64

322

3 1

8---3

1

4---

1

4---

1 16 16 4=

2

16

16

1 11 11

2

11

11

1

2

1 643

643

2

643

643

1 223

223

2

223

223

11

8---

3 1

8---

3 1

2---

2 1

8---

3

1WORKEDExample


Classifying numbers

1 Specify whether the following numbers are rational (Q) or irrational (I).

a b c d e

f g 2 h i j 0.15

k −2.4 l m n o π

p q 7.32 r s t 7.216349157 . . .

u v 3π w x y

2 Specify whether the following numbers are rational (Q), irrational (I) or neither.

a b c d e −6

f g h i j

k l m n o

p q r s t

u v w x y

3

Which of the following best represents a rational number?

A π B C D

1. Rational numbers (Q) can be expressed in the form , where a and b are whole

numbers and b ≠ 0. They include whole numbers, fractions and terminating and

recurring decimals.

2. Irrational numbers (I) cannot be expressed in the form , where a and b are

whole numbers and b ≠ 0. They include surds, non-terminating and non-

recurring decimals, and numbers such as π and e.

3. Rational and irrational numbers together constitute the set of Real numbers (R).

a

b---

a

b---

remember

1AWORKED

Example

1 4 4

5---

7

9--- 2 7

0.04 1

2--- 5 9

4---

100 14.4 1.44

25

9------ 21– 1000

81– 623 1

16------ 0.0001

3

1

8--- 625 11

4------

0

8---

1

7---

813

11–1.44

4---------- π 8

0---

213 π

7--- 5–( )23

− 3

11------

1

100---------

64

16------

2

25------

6

2------- 27

3 1

4-------

22π

7--------- 1.728–

36 4 4 6 ( 2)

4

multiple choice

4

9---

9

12------ 3

3


4

Which of the following best represents an irrational number?

A − B C D

5

Which of the following statements regarding the numbers −0.69, , , is

correct?

A is the only rational number.

B and are both irrational numbers.

C −0.69 and are the only rational numbers.

D −0.69 is the only rational number.

6

Which of the following statements regarding the numbers 2 , , , is

correct?

A and are both irrational numbers.

B is an irrational number and is a rational number.

C and are both irrational numbers.

D 2 is a rational number and is an irrational number.

Surds

We have classified a particular group of numbers as irrational and will now further

examine surds — one subset of irrational numbers — and some of their associated

properties.

A surd is an irrational number that is represented by a root sign or a radical sign,

for example: , ,

Examples of surds include: , , ,

Examples that are not surds include:

, , ,

Numbers that are not surds can be simplified to rational numbers, that is:

, , ,

multiple choice

81 6

5--- 343

322

multiple choice

7 π

3--- 49

π

3---

7 49

49

multiple choice

1

2--- −

11

3------ 624 99

3

−11

3------ 624

624 993

624 993

1

2--- −

11

3------

3 4

7 5 113

154

9 16 1253

814

9 3= 16 4= 1253

5= 814

3=


Proof that a number is irrational

In Mathematics you are required to study a variety of types of proofs. One such method

is called proof by contradiction.

This method is so named because the logical argument of the proof is based on an

assumption that leads to contradiction within the proof. Therefore the original assump-

tion must be false.

An irrational number is one that cannot be expressed in the form (where a and b

are integers). The next worked example sets out to prove that is irrational.

Which of the following numbers are surds?

a b c d e f

THINK WRITE

a Evaluate .

The answer is rational (since it is a

whole number), so state your

conclusion.

a = 5

is not a surd.

b Evaluate .

The answer is irrational (since it is a

non-recurring and non-terminating

decimal), so state your conclusion.

b = 3.162 277 660 17 . . .

is a surd.

c Evaluate .

The answer is rational (a fraction);

state your conclusion.

c =

is not a surd.

d Evaluate .

The answer is irrational (a non-

terminating and non-recurring

decimal), so state your conclusion.

d = 2.223 980 090 57 . . .

is a surd.

e Evaluate .

The answer is irrational, so classify

accordingly.

e = 2.771 488 002 48 . . .

is a surd.

f Evaluate .

The answer is rational; state your

conclusion.

f = 7

is not a surd.

So b, d and e are surds.

25 10 1

4--- 11

359

4343

3

1 25

2

25

25

1 10

2

10

10

11

4---

2

1

4---

1

2---

1

4---

1 113

2

113

113

1 594

2

594

594

594

1 3433

2

3433

3433

2WORKEDExample

a

b---

2


The ‘dialogue’ included in the worked example should be present in all proofs and is an

essential part of the communication that is needed in all your solutions.

Note: An irrational number written in surd form gives an exact value of the number;

whereas the same number written in decimal form (for example, to 4 decimal places)

gives an approximate value.

Prove that is irrational.

THINK WRITE

Assume that is rational; that is, it

can be written as in simplest form.

We need to show that a and b have no

common factors.

= , where b ≠ 0

Square both sides of the equation. 2 =

Rearrange the equation to make a2 the

subject of the formula.

a2 = 2b

2 [1]

If x is an even number, then x = 2n. ∴ a2 is an even number and a must also

be even; that is, a has a factor of 2.

Since a is even it can be written as

a = 2r.

∴ a = 2r

Square both sides. a2 = 4r

2 [2]

But a2 = 2b

2 from [1]

Equate [1] and [2]. ∴ 2b2 = 4r

2

b2 =

= 2r2

∴ b2 is an even number and b must also be even;

that is, b has a factor of 2.

Repeat the steps for b as previously

done for a.

Both a and b have a common factor of 2.

This contradicts the original assumption that

= , where a and b have no common factor.

∴ is not rational.

∴ It must be irrational.

2

1 2

a

b---

2a

b---

2a

2

b2

-----

3

4

5

6

7

4r2

2--------

8

2a

b---

2

3WORKEDExample

A number is a surd if:

1. it is an irrational number (equals a non-terminating, non-recurring decimal)

2. it can be written with a radical sign (or square root sign) in its exact form.

remember


Surds

1 Which of the numbers below are surds?

a b c d e f

g h i j k l

m n o p q r 2π

s t u v w x

y z

2

The correct statement regarding the set of numbers { , , , , } is:

A and are the only rational numbers of the set.

B is the only surd of the set.

C and are the only surds of the set.

D and are the only surds of the set.

3 Prove that the following numbers are irrational, using a proof by contradiction:

a b c

4

Which of the numbers of the set { , , , , } are surds?

A only B only C and D and only

5

Which statement regarding the set of numbers {π, , , , } is not

true?

A is a surd. B and are surds.

C π is irrational but not a surd. D and are not rational.

6

Which statement regarding the set of numbers { , , , , , } is

not true?

A when simplified is an integer. B and are not surds.

C is smaller than . D is smaller than .

7 Complete the following statement by selecting appropriate words, suggested in

brackets:

is definitely not a surd, if a is . . . (any multiple of 4; a perfect square and cube).

8 Find the smallest value of m, where m is a positive integer, so that is not a surd.

1BWORKED

Example

2

SkillSHEET

1.1

Identifyingsurds

81 48 16 1.6 0.16 11

3

4---

3

27------

3 1000 1.44 4 100 2 10+

323

361 1003

1253

6 6+

1693 7

8--- 16

47( )

233

30.0001

325

80

multiple choice

6

9--- 20 54 27

39

273

9

6

9---

6

9--- 20

20 54

WORKED

Example

33 5 7

multiple choice

1

4---

1

27------3 1

8--- 21 8

3

21 1

8---

1

8--- 8

3 1

8--- 21

multiple choice

1

49------ 12 16 3 1+

12 12 16

12 3 1+

multiple choice

6 7 144

16--------- 7 6 9 2 18 25

144

16---------

144

16--------- 25

7 6 9 2 9 2 6 7

a6

16m3


Operations with surdsIn Year 9 we looked at surds and various operations with surds. In this section, we will

revise each of these operations.

Simplifying surdsTo simplify a surd means to make a number (or an expression) under the radical sign

( ) as small as possible. To simplify a surd (if it is possible), it should be rewritten as

a product of two factors, one of which is a perfect square, that is, 4, 9, 16, 25, 36, 49,

64, 81, 100 and so on.

We must always aim to obtain the largest perfect square when simplifying surds so

that there are fewer steps involved in obtaining the answer. For example, could be

written as = 2 ; however, can be further simplified to , so

= 2 × 2 ; that is = 4 . If, however, the largest perfect square had been

selected and had been written as = × = 4 , the same answer

would be obtained in fewer steps.

32

4 8× 8 8 2 2

32 2 32 2

32 16 2× 16 2 2

Simplify the following surds. Assume that x and y are positive real numbers.

a b c − d

THINK WRITE

a Express 384 as a product of two

factors where one factor is the

largest possible perfect square.

a =

Express as the product of

two surds.

=

Simplify the square root from the

perfect square (that is, = 8).=

b Express 405 as a product of two

factors, one of which is the largest

possible perfect square.

b =

Express as a product of two

surds.=

Simplify . =

Multiply together the whole numbers

outside the square root sign

(3 and 9).

=

384 3 4051

8--- 175 5 180x3 y5

1 384 64 6×

2 64 6× 64 6×

3

648 6

1 3 405 3 81 5×

2 81 5× 3 81 5×

3 81 3 9 5×

4 27 5

4WORKEDExample


Addition and subtraction of surds

Surds may be added or subtracted only if they are alike.

Examples of like surds include , and . Examples of unlike surds

include , , and .

In some cases surds will need to be simplified before you decide whether they are

like or unlike, and then addition and subtraction can take place. The concept of adding

and subtracting surds is similar to adding and subtracting like terms in algebra.

THINK WRITE

c Express 175 as a product of two factors

in which one factor is the largest

possible perfect square.

c − = −

Express as a product of 2

surds.= − ×

Simplify . = − ×

Multiply together the numbers outside

the square root sign.= −

d Express each of 180, x3 and y5 as a

product of two factors where one factor

is the largest possible perfect square.

d =

Separate all perfect squares into one

surd and all other factors into the other

surd.

=

Simplify . =

Multiply together the numbers and the

pronumerals outside the square root

sign.

=

11

8--- 175 1

8--- 25 7×

2 25 7× 1

8--- 25 7×

3 25 1

8--- 5 7

4 5

8--- 7

1 5 180x3y5 5 36 5 x2 x y4 y×××××

2 5 36x2y4 5xy××

3 36x2y4 5 6 x y2 5xy××××

4 30xy2 5xy

7 3 7 5 7–

11 5 2 13 2 3–

Simplify each of the following expressions containing surds. Assume that a and b are

positive real numbers.

a b c

Continued over page

THINK WRITE

a All 3 terms are alike because they

contain the same surd .

Simplify.

a =

=

3 6 17 6 2 6–+ 5 3 2 12 5 2 3 8+–+1

2--- 100a3b2 ab 36a 5 4a2b–+

6( )

3 6 17 6 2 6–+ 3 17 2–+( ) 6

18 6

5WORKEDExample


Multiplication and division of surds

Multiplying surds

To multiply surds, multiply together the expressions under the radical signs. For

example, , where a and b are positive real numbers.

When multiplying surds it is best to first simplify them (if possible). Once this has

been done and a mixed surd has been obtained, the coefficients are multiplied with each

other and then the surds are multiplied together. For example,

THINK WRITE

b Simplify surds where possible. b

=

=

=

Add like terms to obtain the

simplified answer.

=

c Simplify surds where possible. c

=

=

=

Add like terms to obtain the

simplified answer.

=

1 5 3 2 12 5 2 3 8+–+

5 3 2 4 3× 5 2 3 4 2×+–+

5 3 2 2 3 5 2 3 2 2×+–×+

5 3 4 3 5 2 6 2+–+

2 9 3 2+

11

2--- 100a3b2 ab 36a 5 4a2b–+

1

2--- 10 a2 a b2

×× ab 6 a 5 2 a b××–×+×

1

2--- 10 a b a ab 6 a 5 2 a b××–×+×××

5ab a 6ab a 10a b–+

2 11ab a 10a b–

a b× ab=

m a n b× mn ab=

Multiply the following surds, expressing answers in the simplest form. Assume that x and

y are positive real numbers.

a b c d

THINK WRITE

a Multiply the surds together, using

(that is, multiply

expressions under the square root sign).

Note: This expression cannot be

simplified any further.

a =

=

11 7¥ 5 3 8 5¥ 6 12 2 6¥ 15x5 y2 12x2 y¥

a b× ab=

11 7× 11 7×

77

6WORKEDExample


When working with surds, we sometimes need to multiply surds by themselves; that is,

square them. Consider the following examples:

We observe that squaring a surd produces the number under the radical sign. This is

not surprising, because squaring and taking the square root are inverse operations and,

when applied together, leave the original unchanged.

When a surd is squared, the result is the number (or expression) under the radical

sign; that is, , where a is a positive real number.

THINK WRITE

b Multiply the coefficients together and

then multiply the surds together.

b =

=

=

c Simplify . c =

=

=

Multiply the coefficients together

and multiply the surds together.

=

Simplify the surd. =

=

=

d Simplify each of the surds. d

= ×

= x2 × y × × 2 × x ×

= x2y × 2x

Multiply the coefficients together

and the surds together.

= x2y × 2x

= 2x3y

Simplify the surd. = 2x3y

= 2x3y × 3

= 6x3y

5 3 8 5× 5 8 3 5×××

40 3 5××

40 15

1 12 6 12 2 6× 6 4 3× 2 6×

6 2 3× 2 6×

12 3 2 6×

2 24 18

3 24 9 2×

24 3 2×

72 2

1 15x5y2 12x2y×

15 x4

× x× y2

× 4 3× x2

× y×

15 x× 3 y×

15x 3y

2 15x 3y×

45xy

3 9 5xy×

5xy

5xy

( 2)2

2 2× 4 2= = = ( 5)2

5 5× 25 5= = =

( a)2

a=


Dividing surdsTo divide surds, divide the expressions under the radical signs;

that is, , where a and b are whole numbers.

When dividing surds it is best to simplify them (if possible) first. Once this has been

done, the coefficients are divided next and then the surds are divided.

Simplify each of the following.

a b

THINK WRITE

a Use ( )2 = a, where a = 6. a = 6

b Square 3 and use ( )2 = a to

square .

b = 32 × ( )2

= 9 × 5

Simplify. = 45

6( )2

3 5( )2

a 6( )2

1 a

5

3 5( )2

5

2

7WORKEDExample

a

b

-------a

b---=

Divide the following surds, expressing answers in the simplest form. Assume that x and y

are positive real numbers.

a b c d

THINK WRITE

a Rewrite the fraction, using . a =

Divide the numerator by the

denominator (that is, 55 by 5).

=

Check if the surd can be simplified any

further.

b Rewrite the fraction, using . b =

Divide 48 by 3. =

Evaluate . = 4

55

5

----------48

3

----------9 88

6 99

-------------36xy

25x9 y11

------------------------

1a

b

-------a

b---=

55

5

----------55

5------

2 11

3

1a

b

-------a

b---=

48

3

----------48

3------

2 16

3 16

8WORKEDExample


Rationalising denominatorsIf the denominator of a fraction is a surd, it can be changed into a rational number. In

other words, it can be rationalised.

As we discussed earlier in this chapter, squaring a simple surd (that is, multiplying it

by itself) results in a rational number. This fact can be used to rationalise denominators

as follows.

, where = 1

If both numerator and denominator of a fraction are multiplied by the surd contained

in the denominator, the denominator becomes a rational number. The fraction takes on

a different appearance, but its numerical value is unchanged, because multiplying the

numerator and denominator by the same number is equivalent to multiplying by 1.

THINK WRITE

c Rewrite surds, using . c =

Simplify the fraction under the radical

by dividing both numerator and

denominator by 11.

=

Simplify surds. =

Multiply the whole numbers in the

numerator together and those in the

denominator together.

=

Cancel the common factor of 18. =

d Simplify each surd. d =

=

Cancel any common factors — in this

case .

=

1a

b

-------a

b---=

9 88

6 99

-------------9

6---

88

99------

29

6---

8

9---

39 2 2×

6 3×

-------------------

418 2

18-------------

5 2

136xy

25x9y11

-----------------------6 xy

5 x8 x y10 y×××

--------------------------------------------

6 xy

5x4y5 xy

------------------------

2

xy

6

5x4y5--------------

a

b

-------b

b

-------×ab

b----------=

b

b

-------


Express the following in their simplest form with a rational denominator.

a b c

THINK WRITE

a Write the fraction. a

Multiply both the numerator and denominator by

the surd contained in the denominator (in this case

). This has the same effect as multiplying the

fraction by 1, because .

=

=

b Write the fraction. b

Simplify the surds. (This avoids dealing with large

numbers.)

=

=

=

Multiply both the numerator and denominator by .

(This has the same effect as multiplying the fraction by

1, because .)

Note: We need to multiply only by the surd part of the

denominator (that is, by rather than by ).

=

=

Simplify . =

=

=

Divide both the numerator and denominator by 6

(cancel down).

=

6

13

----------2 12

3 54

-------------17 3 14–

7

-----------------------------

16

13

----------

2

13

13

13

---------- 1=

6

13

----------13

13

----------×

78

13----------

12 12

3 54

-------------

22 12

3 54

-------------2 4 3×

3 9 6×

-------------------

2 2 3×

3 3 6×

-------------------

4 3

9 6

----------

3 6

6

6

------- 1=

6 9 6

4 3

9 6

----------6

6

-------×

4 18

9 6×

-------------

4 184 9 2×

9 6×

-------------------

4 3 2×

54-------------------

12 2

54-------------

52 2

9----------

9WORKEDExample


Rationalising denominators using conjugate surdsThe product of pairs of conjugate surds results in a rational number. (Examples of pairs

of conjugate surds include and , and , and

.)

This fact is used to rationalise denominators containing a sum or a difference of

surds.

To rationalise the denominator that contains a sum or a difference of surds, we

multiply both numerator and denominator by the conjugate of the denominator.

Two examples are given below:

1. To rationalise the denominator of the fraction , multiply it by .

2. To rationalise the denominator of the fraction , multiply it by .

A quick way to simplify the denominator is to use the difference of two squares

identity:

=

= a − b

THINK WRITE

c Write the fraction. c

Multiply both the numerator and denominator by .

Use grouping symbols to make it clear that the whole

numerator must be multiplied by .

=

Apply the Distributive Law in the numerator.

a(b + c) = ab + ac

=

=

Simplify . =

=

=

117 3 14–

7

-----------------------------

2 7

7

( 17 3 14)–

7

----------------------------------7

7

-------×

317 7× 3 14– 7×

7 7×

----------------------------------------------------------

119 3 98–

7--------------------------------

4 98119 3 49 2×–

7------------------------------------------

119 3 7 2×–

7---------------------------------------

119 21 2–

7--------------------------------

6 11+ 6 11– a b+ a b– 2 5 7–

2 5 7+

1

a b+

--------------------a b–

a b–

--------------------

1

a b–

--------------------a b+

a b+

--------------------

( a b– )( a b+ ) ( a)2( b)

2–


Rationalise the denominator and simplify the following.

a b

THINK WRITE

a Write down the fraction. a

Multiply the numerator and denominator

by the conjugate of the denominator.

(Note that = 1.)

= ×

Apply the Distributive Law in the

numerator and the difference of two

squares identity in the denominator.

=

Simplify. =

=

b Write down the fraction. b

Multiply the numerator and denominator

by the conjugate of the denominator.

(Note that = 1.)

= ×

Multiply the expressions in grouping

symbols in the numerator, and apply

the difference of two squares identity

in the denominator.

=

Simplify. =

=

=

=

=

=

1

4 3–

----------------6 3 2+

3 3+

------------------------

1

1

4 3–

----------------

2

4 3+( )

4 3+( )

---------------------

1

4 3–( )

---------------------4 3+( )

4 3+( )

---------------------

34 3+

4( )2 3( )2

–

------------------------------

44 3+

16 3–----------------

4 3+

13----------------

1

6 3 2+

3 3+

------------------------

2

3 3–( )

3 3–( )

---------------------

6 3 2+( )

3 3+( )

-----------------------------3 3–( )

3 3–( )

---------------------

36 3 6+× 3– 3 2 3 3 2 3–×+×+×

3( )2 3( )2

–

-----------------------------------------------------------------------------------------------------------

43 6 18 9 2 3 6–+–

9 3–------------------------------------------------------------

18– 9 2+

6-------------------------------

9 2×– 9 2+

6-------------------------------------

3 2– 9 2+

6-------------------------------

6 2

6----------

2

10WORKEDExample


Operations with surds

1 Simplify the following surds.

a b c d

e f g h

i j k l

2 Simplify the following surds.

a b c d

e f g h

i j k l

1. To simplify a surd means to make a number (or an expression) under the

radical sign as small as possible.

2. To simplify a surd, write it as a product of two factors, one of which is the

largest possible perfect square.

3. Only like surds may be added and subtracted.

4. Surds may need to be simplified before adding and subtracting.

5. When multiplying surds, simplify the surd if possible and then apply the

following rules:

(a) × =

(b) m × n = mn , where a and b are positive real numbers.

6. When a surd is squared, the result is the number (or the expression) under the

radical sign: ( )2 = a, where a is a positive real number.

7. When dividing surds, simplify the surd if possible and then apply the following

rule:

÷ = =

where a and b are whole numbers, and b ≠ 0.

8. To rationalise a surd denominator, multiply the numerator and denominator by

the surd contained in the denominator. This has the effect of multiplying the

fraction by 1, and thus the numerical value of the fraction remains unchanged,

while the denominator becomes rational:

= × =

where a and b are whole numbers and b ≠ 0.

9. To rationalise the denominator containing a sum or a difference of surds,

multiply both the numerator and denominator of the fraction by the conjugate

of the denominator. This eliminates the middle terms and leaves a rational

number.

a b ab

a b ab

a

a ba

b

-------a

b---

a

b

-------a

b

-------b

b

-------ab

b----------

remember

1CSkillSHEET

1.2

Simplifyingsurds

WORKED

Example

4a 12 24 27 125

54 112 68 180

88 162 245 448

WORKED

Example

4b, c

Mathcad

Simplifyingsurds

2 8 8 90 9 80 7 54

6 75– 7 80– 16 481

7--- 392

1

9--- 162

1

4--- 192

1

9--- 135

3

10------ 175


3 Simplify the following surds. Assume that a, b, c, d, e, f, x and y are positive real

numbers.

a b c d

e f g h

i j k l

4 Simplify the following expressions containing surds. Assume that x and y are positive

real numbers.

a b

c d

e f

g h

5 Simplify the following expressions containing surds. Assume that a and b are positive

real numbers.

a b

c d

e f

g h

i j

k l

6 Simplify the following expressions containing surds. Assume that a and b are positive

real numbers.

a b

c d

e f

g h

i j

k l

7 Multiply the following surds, expressing answers in the simplest form. Assume that a,

b, x and y are positive real numbers.

a b c

d e f

g h i

EXCE

L Spreadsheet

Simplifyingsurds

GC pr

ogram– Casio

Surds

GC pr

ogram– TI

Surds

WORKED

Example

4d

16a2 72a2 90a2b 338a4

338a3b3 68a3b

5125x6y4 5 80x3y2

6 162c7d5 2 405c7d9 1

2--- 88ef

1

2--- 392e11 f 11

WORKED

Example

5a

3 5 4 5+ 2 3 5 3 3+ +

8 5 3 3 7 5 2 3+ + + 6 11 2 11–

7 2 9 2 3 2–+ 9 6 12 6 17 6 7 6––+

12 3 8 7– 5 3 10 7–+ 2 x 5 y 6 x 2 y–+ +

SkillSH

EET 1.3


Mat

hcad


WORKED

Example

5b

200 300– 125 150– 600+

27 3– 75+ 2 20 3 5– 45+

6 12 3 27 7 3– 18+ + 150 24 96– 108+ +

3 90 5 60– 3 40 100+ + 5 11 7 44 9 99– 2 121+ +

2 30 5 120 60 6 135–+ + 6 ab 12ab– 2 9ab 3 27ab+ +

1

2--- 98

1

3--- 48

1

3--- 12+ +

1

8--- 32

7

6--- 18– 3 72+

WORKED

Example

5c

7 a 8a– 9 9a 32a–+ 10 a 15 27a– 8 12a 14 9a+ +

150ab 96ab 54ab–+ 16 4a2 24a– 4 8a2 96a+ +

8a3 72a3 98a3–+1

2--- 36a

1

4--- 128a

1

6--- 144a–+

9a3 3a5+ 6 a5b a3b 5 a5b–+

ab ab 3ab a2b 9a3b3+ + a3b 5 ab 2 ab– 5 a3b+ +

32a3b2 5ab 8a– 48a5b6+ 4a2b 5 a2b 3 9a2b–+

SkillSH

EET 1.4


Mat

hcad


WORKED

Example

6

2 7× 6 7× 8 6×

10 10× 21 3× 27 3 3×

5 3 2 11× 10 15 6 3× 4 20 3 5×


j k l

m n o

p q r


a ( )2 b ( )2 c ( )2 d ( )2

e (3 )2 f (4 )2 g (2 )2 h (5 )2

9 Simplify the following surds, expressing answers in the simplest form. Assume that a,

b, x and y are positive real numbers.

a b c d

e f g h

i j k l

m n

10 Express the following in their simplest form with a rational denominator.

a b c d e

f g h i j

k l m n o

11 Express the following in their simplest form with a rational denominator.

a b c d

e f g h

i j k l

10 6 3 8×1

4--- 48 2 2×

1

9--- 48 2 3×

1

10------ 60

1

5---× 40 xy x3y2

× 3a4b2 6a5b3×

12a7b 6a3b4× 15x3y2 6x2y3

×1

2--- 15a3b3 3 3a2b6

×

WORKED

Example

7 2 5 12 15

2 5 7 8

WORKED

Example

8

15

3

----------8

2

-------60

10

----------128

8

-------------

18

4 6

----------65

2 13

-------------96

8

----------7 44

14 11

----------------

9 63

15 7

-------------2040

30

----------------x4y3

x2y5

---------------16xy

8x7y9

------------------

xy

x5y7

---------------12x8y12

x2y3

-----------------------×2 2a2b4

5a3b6

----------------------10a9b3

3 a7b

---------------------×

SkillSHEET

1.5

Rationalisingdenominators

WORKED

Example

9a, b

Mathcad

Rationalisingdenominators

5

2

-------7

3

-------4

11

----------8

6

-------12

7

----------

15

6

----------2 3

5

----------3 7

5

----------5 2

2 3

----------4 3

3 5

----------

5 14

7 8

-------------16 3

6 5

-------------8 3

7 7

----------8 60

28

-------------2 35

3 14

-------------

WORKED

Example

9c 6 12+

3

-----------------------15 22–

6

--------------------------6 2 15–

10

--------------------------2 18 3 2+

5

------------------------------

3 5 6 7+

8

---------------------------4 2 3 8+

2 3

---------------------------3 11 4 5–

18

------------------------------2 7 2 5–

12

---------------------------

7 12 5 6–

6 3

------------------------------6 2 5–

4 8

-----------------------6 3 5 5–

7 20

---------------------------3 5 7 3+

5 24

---------------------------


12 Rationalise the denominator and simplify.

a b c

d e f

g h i

1 Explain whether is rational or irrational.

2 Explain whether is a surd.

3 Simplify .

4 Simplify 6 + 11 .

5 Simplify + – .

6 Simplify 6 × 2 .

7 Simplify .

8 Simplify .

9 Express in simplest form with a rational denominator.


SkillSH

EET 1.6

Conjugate pairs

SkillSH

EET 1.7

Applying the difference of two squares rule to surds

WORKED

Example

101

5 2+

----------------1

8 5–

--------------------4

2 11 13–

-----------------------------

5 3

3 5 4 2+

---------------------------

WorkS

HEET 1.1

8 3–

8 3+

----------------12 7–

12 7+

-----------------------

3 1–

5 1+

----------------3 6 15–

6 2 3+

--------------------------5 3–

4 2 3–

-----------------------

1

3

1

4---

80

2 2

18 50 72

18 8

90

6

----------

2 24

3 3

-------------

6 3

5 2

----------

1 2+

1 2–

----------------


Fractional indices

Consider the expression . Now consider what happens if we square that expression.

( )2 = a (using the Fourth Index Law, (am)n = am × n)

Now, from our work on surds we know that ( )2 = a.

From this we can conclude that ( )2 = ( )2 and further conclude that = .

We can similarly show that = .

This pattern can be continued and generalised to produce = . On a Casio

calculator, the function can be used to find the value of expressions containing

fractional indices. Other models of calculator may have the function . Check with

your teacher if you are unsure.

a

1

2---

a

1

2---

a

a

1

2---

a a

1

2---

a

a

1

3---

a3

Evaluate each of the following without using a calculator.

a b

THINK WRITE

a Write as . a =

Evaluate. = 3

b Write as . b =

Evaluate. = 4

9

1

2---

64

1

3---

1 9

1

2---

9 9

1

2---

9

2

1 64

1

3---

643

64

1

3---

643

2

11WORKEDExample

a

1

n

---

an

x

x

1

y

---

Use a calculator to find the value of the following, correct to 1 decimal place.

a b

Continued over page

THINK WRITE/DISPLAY

10

1

4---

200

1

5---

12WORKEDExample


Now consider the expression . Using our work so far on fractional indices, we

can say = .

We can also say = using the index laws.

We can therefore conclude that = .

Such expressions can be evaluated on a calculator either by using the index function,

which is usually either ^ or xy and entering the fractional index, or by separating the

two functions for power and root.

We can also use the index law = to convert between expressions that involve

fractional indices and surds.

THINK WRITE/DISPLAY

a Press on a scientific

calculator or [ ] on a

Casio graphics calculator. (On a TI graphics calculator,

press (5: ) .)

a ≈ 1.8

b Press on a scientific

calculator or [ ]

on a Casio graphics calculator. (On a TI graphics calculator,

press (5: ) .)

b ≈ 2.9

4x

1 0 =

4 SHIFTx

1 0 EXE

4 MATH 5x

1 0 ENTER

10

1

4---

5x

2 0 0 =

5 SHIFTx

2 0 0 EXE

5 MATH 5x

2 0 0 ENTER

200

1

5---

am

( )

1

n

---

am

( )

1

n

---

amn

am

( )

1

n

---

a

m

n

----

a

m

n

----

amn

Evaluate , correct to 1 decimal place.

THINK WRITE

Method 1

Press on a scientific

calculator or

on a Casio graphics calculator. (On a TI graphics calculator,

press .)

≈ 1.4

Method 2

Press on a scientific

calculator or [ ] on a

Casio graphics calculator. (On a TI graphics calculator, press

(5: ) .)

3

2

7---

3 xy 2 ab/c 7 =

3 ^ ( 2 ab/c 7 ) EXE

3 ^ ( 2 ÷ 7 ) ENTER

3

2

7---

7x

3 xy 2 =

7 SHIFTx

3 ^ 2 EXE

7 MATH 5x

3 ^ 2 ENTER

13WORKEDExample

a

1

2---

a


In Year 9 you would have studied the index laws and all of these laws are valid for

fractional indices.

Write each of the following expressions in surd form.

a b

THINK WRITE

a Since an index of is equivalent to

taking the square root, this term can be

written as the square root of 10.

a =

b A power of means square root of

the number cubed.

b =

Evaluate 53. =

Simplify . = 5

10

1

2---

5

3

2---

1

2--- 10

1

2---

10

13

2--- 5

3

2---

53

2 125

3 125 5

14WORKEDExample

Simplify each of the following.

a b c

THINK WRITE

a Write the expression. a

Multiply numbers with the same base by

adding the indices.

=

b Write the expression. b

Multiply each index inside the grouping

symbols by the index on the outside.

=

Simplify the fractions. =

c Write the expression. c

Multiply the index in both the numerator

and denominator by the index outside the

grouping symbols.

=

m

1

5---

m

2

5---

× a2b3( )1

6--- x

2

3---

y

3

4---

-----

1

2---

1 m

1

5---

m

2

5---

×

2 m

3

5---

1 a2b3( )1

6---

2 a

2

6---

b

3

6---

3 a

1

3---

b

1

2---

1x

2

3---

y

3

4---

-----

1

2---

2x

1

3---

y

3

8---

-----

15WORKEDExample


Fractional indices

1 Evaluate each of the following without using a calculator.

a b c

d e f

2 Use a calculator to evaluate each of the following, correct to one decimal place.

a b c

d e f

3 Use a calculator to find the value of each of the following, correct to one decimal

place.

a b c

d e f

4 Write each of the following expressions in simplest surd form.

a b c

d e f

5 Write each of the following expressions with a fractional index.

a b c

d e f


a b c

d e f

g h i

1. Fractional indices are those that are expressed as fractions.

2. Numbers with fractional indices can be written as surds, using the following

identities:

3. All index laws are applicable to fractional indices.

a

1

n---

an= a

m

n----

amna

n( )m

= =

remember

1D

SkillSH

EET 1.8

Finding square roots, cube roots and other roots

Mat

hcad

Fractionalindices

WORKED

Example

1116

1

2---

251

2---

8112−

81

3---

271

3---

1251

3---

SkillSH

EET 1.9

Evaluating numbers in index form

WORKED

Example

1281

1

4---

161

4---

31

3---

51

2---

71

5---

81

9---

WORKED

Example

13

12

3

8---

100

5

9---

50

2

3---

0.6( )4

5---

3

4---

3

4---

4

5---

2

3---

WORKED

Example

147

1

2---

12

1

2---

72

1

2---

2

5

2---

3

3

2---

10

5

2---

5 10 x

m3

2 t 63

SkillSH

EET 1.10

Using the index laws

EXCE

L Spreadsheet

Index laws

WORKED

Example

15a4

3

5---

4

1

5---

× 2

1

8---

2

3

8---

× a

1

2---

a

1

3---

×

x

3

4---

x

2

5---

× 5m

1

3---

2m

1

5---

× 1

2---b

3

7---

4b

2

7---

×

4y2– y

2

9---

× 2

5---a

3

8---

0.05a

3

4---

× 5x3 x

1

2---

×



a b c

d e f


a b c

d e f

g h i


a b c

d e f


a ( ) b ( ) c ( )

d e ( ) f ( )

g 4( ) h ( ) i ( )


a ( ) b c ( )

d ( ) e ( ) f

g h i

a

2

3---

b

3

4---

a

1

3---

b

3

4---

× x

3

5---

y

2

9---

x

1

5---

y

1

3---

× 2ab

1

3---

3a

3

5---

b

4

5---

×

6m

3

7--- 1

3---m

1

4---

n

2

5---

× x3y

1

2---

z

1

3---

x

1

6---

y

1

3---

z

1

2---

× 2a

2

5---

b

3

8---

c

1

4---

4b

3

4---

c

3

4---

×

3

1

2---

3

1

3---

÷ 5

2

3---

5

1

4---

÷ 122 12

3

2---

÷

a

6

7---

a

3

7---

÷ x

3

2---

x

1

4---

÷ m

4

5---

m

5

9---

------

2x

3

4---

4x

3

5---

--------7n2

21n

4

3---

-----------25b

3

5---

20b

1

4---

-----------

x3y2 x

4

3---

y

3

5---

÷ a

5

9---

b

2

3---

a

2

5---

b

2

5---

÷ m

3

8---

n

4

7---

3n

3

8---

÷

10x

4

5---

y 5x

2

3---

y

1

4---

÷ 5a

3

4---

b

3

5---

20a

1

5---

b

1

4---

-----------------p

7

8---

q

1

4---

7 p

2

3---

q

1

6---

---------------

2

3

4---

3

5---

5

2

3---

1

4---

7

1

5---

6

a3( )1

10------

m

4

9---

3

8---

2b

1

2---

1

3---

p

3

7---

14

15------

x

m

n----

n

p---

3m

a

b---

b

c---

WORKED

Example

15b, c

a

1

2---

b

1

3---

1

2---

a4b( )3

4---

x

3

5---

y

7

8---

2

3a

1

3---

b

3

5---

c

3

4---

1

3---

x

1

2---

y

2

3---

z

2

5---

1

2---

a

3

4---

b-----

2

3---

m

4

5---

n

7

8---

------

2

b

3

5---

c

4

9---

-----

2

3---

4x7

2y

3

4---

--------

1

2---


12

Note: There may be more than one correct answer.

If ( ) is equal to , then m and n could not be:

A 1 and 3 B 2 and 6 C 3 and 8 D 4 and 9


a b c

d e f

g h i

Manning’s formulaAt the start of this chapter we looked at Manning’s formula, which is used to

calculate the flow of water in a river during a flood situation. Manning’s formula is

v = , where R is the hydraulic radius, S is the slope of the river and n is the

roughness coefficient. This formula is used by meteorologists and civil engineers to

analyse potential flood situations.

We were asked to find the flow of water in metres per second in the river if

R = 8, S = 0.0025 and n = 0.625.

1 Use Manning’s formula to find the flow

of water in the river.

2 To find the volume of water flowing

through the river, we multiply the flow

rate by the average cross-sectional area

of the river. If the average cross-

sectional area is 52 m2, find the volume

of water flowing through the river each

second. (Remember 1 m3 = 1000 L.)

3 If water continues to flow at this rate,

what will be the total amount of water to

flow through in one hour?

4 Use the Bureau of Meteorology’s

website, www.bom.gov.au, to find the

meaning of the terms hydraulic radius

and roughness coefficient.

GAM

E time

Rational and

irrational numbers

— 001

multiple choice

a

3

4---

m

n----

a

1

4---

WorkS

HEET 1.2

a8 b93m164

16x4 8y9316x8y124

27m9n15332 p5q105

216a6b183

R

2

3---

S

1

2---

n-----------

Simplify each of the followingto solve the code below.

Simplify each of the followingto solve the code below.

B R( )3

2

2

1

x31

32

33 yx ×

L E3

1

43

3

2

4

3

yxyx × 4 41216 yx

WFT( )3

x ( )252

3x 52

51

73

31

412 yxyx ÷

ZSC 52

43

34yxyx ÷ 3

121

xx ÷3

32

61

y

x

ONP 5

1

5

2

62

1

xx × ( )252

3 x ( )23

x

YIH( )5

125

yx5

4x83

85

2 yxxy ÷

5

2

xy yx3

2 35

1

15

2

3 yx3

2

x 5

4

3x

2

3

x 5

2

xy yx3

2 5

4

3x3

2

x 3

1

x yx3

2 yx2

3

5

3

3x 3

1

3

2

9 yx 2

5

2x2

2

1

y

x yx3

2 5

4

9x 3

2

x3

1

3

2

9 yx

5

3

3x5

2

xy 8

5

8

3

2 yx 6

1

x 2

5

2x5

13

4

13

yx6

1

x

How was Albert Einstein honoured

in 1921?



1 Simplify 4 .

2 Simplify 5 + 8 – 4 + 2 .

3 In simplest surd form, what is the product of 3 and 5 ?

4 Evaluate (3 )2.


6 State the value of .

7 Find correct to two decimal places.

8 Find correct to one decimal place.

9 Write in surd form.

10 Write using a fractional index.

Negative indicesIn Year 9 you would have looked at negative indices and discovered the rules

a–1 = and a

–n = .

When using a calculator to evaluate expressions that involve negative indices, we need

to familiarise ourselves with the keys needed. Many calculators will have a key for .

To evaluate 5−2 on a calculator using this function, press 5 2. Other calculators,

including the Casio and TI graphics calculators, have an x−1 function. To perform the

same calculation, first calculate 52 and then press x−1.

The following worked examples demonstrate how these calculations are performed

on a Casio calculator. Check with your teacher if you are unsure how to use your calcu-

lator for these questions.

2

63

3 27 3 147

30 6

5

7 5 6 7–

2 3

---------------------------

64

1

2---

90

1

3---

10

4

5---

6

1

2---

x53

1

a

---1

an

-----

1

xy

-----

1

xy

-----


Consider the index law a–1 = . Now let us look at the case in which a is fractional.

Consider the expression .

=

= 1 ×

=

We can therefore consider an index of –1 to be a reciprocal function.

Evaluate each of the following.

a 4−1 b 2−4

THINK WRITE/DISPLAY

a Press the following keys.

Scientific calculator: 4 x−1

Graphics calculator: 4 SHIFT [x–1]

a 4−1 = 0.25

b Press the following keys.

Scientific calculator: (2 xy 4) x−1

Graphics calculator: 2 ^ 4 SHIFT [x–1]

b 2−4 = 0.0625

16WORKEDExample

1

a---

a

b---

1–

a

b---

1– 1

a

b---

------

b

a---

b

a---

Write down the value of each of the following.

a ( )–1 b ( )–1 c (1 )–1

THINK WRITE

a To evaluate ( )–1 take the reciprocal of . a ( )–1 =

b To evaluate ( )–1 take the reciprocal of . b ( )–1 =

Write as a whole number. = 5

c Write 1 as an improper fraction. c (1 )–1 = ( )–1

Take the reciprocal of . =

2

3---

1

5---

1

4---

2

3---

2

3---

2

3---

3

2---

11

5---

1

5---

1

5---

5

1---

25

1---

11

4---

1

4---

5

4---

25

4---

4

5---

17WORKEDExample


Negative indices


a 5−1 b 3−1 c 8−1 d 10−1

e 2−3 f 3−2 g 5−2 h 10−4

2 Find the value of each of the following, correct to 3 significant figures.

a 6−1 b 7−1 c 6−2 d 9−3

e 6−3 f 15−2 g 16−2 h 5−4


a (2.5)−1 b (0.4)−1 c (1.5)−2 d (0.5)−2

e (2.1)−3 f (10.6)−4 g (0.45)−3 h (0.125)−4


a (−3)−1 b (−5)−1 c (−2)−2 d (−4)−2

e (−1.5)−1 f (−2.2)−1 g (−0.6)−1 h (−0.85)−2

5 Write down the value of each of the following.

a ( )–1 b ( )–1 c ( )–1 d ( )–1

e ( )–1 f ( )–1 g ( )–1 h ( )–1

i (1 )–1 j (2 )–1 k (1 )–1 l (5 )–1

6 Find the value of each of the following, leaving your answer in fraction form.

a ( )–2 b ( )–2 c ( )–3 d ( )–2

e (1 )–2 f (2 )–2 g (1 )–3 h (2 )–3

7 Find the value of each of the following.

a (– )–1 b (– )–1 c (– )–1 d (– )–1

e (– )–2 f (– )–2 g (–1 )–1 h (–2 )–2

8 Consider the expression 2–n. Explain what happens to the value of this expression as n

increases.

1. To evaluate an expression that involves negative indices, use the or the x–1

function.

2. An index of –1 can be considered as a reciprocal function and applying this to

fractions gives us the rule = .

1

xy

-----

a

b---

1– b

a---

remember

1EWORKED

Example

16

EXCE

L Spreadsheet

Negativeindices

Mat

hcad

Negativeindices

WORKED

Example

174

5---

3

10------

7

8---

13

20------

1

2---

1

4---

1

8---

1

10------

1

2---

1

4---

1

10------

1

2---

1

2---

2

5---

2

3---

1

4---

1

2---

1

4---

1

3---

1

5---

GAM

E time

Rational and irrational numbers — 002

2

3---

3

5---

1

4---

1

10------

2

3---

1

5---

1

2---

3

4---

WorkS

HEET 1.3


Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows.

1 A number that can be expressed in the form , where b ≠ 0, is called a

number.

2 An cannot be expressed in the form , where b ≠ 0, and

takes the form of a decimal that neither recurs nor terminates.

3 Examples of irrational numbers are and numbers such as π

and e.

4 A surd is when the number under the radical sign is as small

as possible.

5 Only surds can be added or subtracted.

6 Unlike surds can be using the rules × = and

m × n = mn .

7 When a surd is squared the result is a number.

8 A surd in fractional form can be simplified by the

denominator.

9 Raising a number to an index of is equivalent to finding the

of the number.

10 The rules connecting fractional indices to surds are = and

= .

11 An index of –1 is equivalent to a function.

summary

a

b---

a

b---

a b ab

a b ab

1

2---

a

1

n---

amn

W O R D L I S T

reciprocal

multiplied

rational

simplified

irrational

square root

like

whole

surds

rationalising

an

a

m

n----


1

Which of the given numbers, , , 5, −3.26, 0.5.

, , are rational?

2 For each of the following, state whether the number is rational or irrational and give the

reason for your answer:

a b c d 0.6.

e

3

Which of the numbers of the given set, { , , , , , }, are

surds?

A ,

B and only

C , and only

D , , and

4 Which of , , , , , are surds

a if m = 4? b if m = 8?


a b c d

6

The expression may be simplified to:

A B C D

7 Simplify the following surds. Give the answers in the simplest form.

A , 5, −3.26, 0.5.

and B and

C , and D 5, −3.26 and

a b

CHAPTERreview

1A multiple choice

6

12------ 0.81

π

5--- 3

12------

0.81 3

12------

6

12------

π

5---

6

12------ 0.81 3

12------

6

12------

1A

12 121 2

9--- 0.08

3

1B multiple choice

3 2 5 7 9 4 6 10 7 12 12 64

9 4 12 64

3 2 7 12

3 2 5 7 6 10

3 2 5 7 6 10 7 12

1B2m 25m

m

16------

20

m

------ m3

8m3

1C50 180 2 32 5 80

1C multiple choice

392x8y7

196x4y3 2y 2x4y3 14y 14x4y3 2y 14x4y3 2

1C4 648x7y9

2

5---

25

64------x5y11–


8 Simplify the following, giving answers in the simplest form.

a

b


a × b 2 × 3 c 3 × 5 d ( )2

10 Simplify the following, giving answers in the simplest form.

11 Simplify the following.

a b c d

12 Rationalise the denominator of each of the following.

a b c d

13 Evaluate each of the following, correct to 1 decimal place.

a b c d

14 Evaluate each of the following, correct to 1 decimal place.

a b c d

15 Write each of the following in simplest surd form.

a b c d

16 Evaluate each of the following, without using a calculator. Show all working.

a b


a 4−1 b 9−1 c 4−2 d 10−3


a 12−1 b 7−2 c (1.25)−1 d (0.2)−4

19 Write down the value of each of the following.

a ( )–1 b ( )–1 c ( )–1 d (3 )–1

a b

1C7 12 8 147 15 27–+

1

2--- 64a3b3 3

4---ab 16ab–

1

5ab--------- 100a5b5+

1C3 5 6 7 10 6 5

1C1

5--- 675 27× 10 24 6 12×

1C30

10

----------6 45

3 5

-------------3 20

12 6

-------------7( )

2

14--------------

1C2

6

-------3

2 6

----------2

5 2–

----------------3 1–

3 1+----------------

1D64

1

3---

20

1

2---

10

1

3---

50

1

4---

1D20

2

3---

2

3

4---

0.7( )3

5---

2

3---

2

3---

1D2

1

2---

18

1

2---

5

32−

8

4

3---

1D16

3

4---

81

1

4---

×

6 16

1

2---

×

---------------------- 125

2

3---

27

2

3---

–

1

2---

1E

1E

1E2

3---

7

10------

1

5---

1

4---


Non-calculator questions

20

The expression may be simplified to:

A B C D

21

When expressed in its simplest form, is equal to:

A B C D

22

When expressed in its simplest form, is equal to:

A B C D

23 Find the value of the following, giving your answer in fraction form.

a b

24 Find the value of each of the following, leaving your answer in fraction form.

a 2−1 b 3−2 c 4−3 d

multiple choice

250

25 10 5 10 10 5 5 50

multiple choice

2 98 3 72–

4 2– 4– 2 4– 4 2

multiple choice

8x3

32--------

x x

2----------

x3

4---------

x3

2---------

x x

4----------

2

5---

1– 2

3---

2–

testtest

CHAPTER

yourselfyourself

11

2---

1–

Documents

MQ 10 RATIONAL AND IRRATIONAL NUMBERS