chapter 1 surds

1Number and Algebra

SurdsWhen applying Pythagoras theorem, we have found lengthsthat cannot be expressed as an exact rational number.Pythagoras encountered this when calculating the diagonalof a square of side length 1 unit. A surd is a square root(p), cube root (

3p

), or any type of root whose exactdecimal or fraction value cannot be found.

n Chapter outlineProficiency strands

1-01 Surds and irrationalnumbers* U F R C

1-02 Simplifying surds* U F R1-03 Adding and subtracting

surds* U F R1-04 Multiplying and dividing

surds* U F R1-05 Binomial products

involving surds* U F R C1-06 Rationalising the

denominator* U F R C

*STAGE 5.3

nWordbankirrational number A number such as p or

2p

that cannotbe expressed as a fraction a

b

rational number Any number that can be written in theform a

b; where a and b are integers and b 6 0

rationalise the denominator To simplify a fractioninvolving a surd by making its denominator rational(that is, not a surd)

real number A number that is either rational or irrationaland whose value can be graphed on a number line

simplify a surd To write a surd

xp

in its simplest form sothat x has no factors that are perfect squares

surd A square root (or other root) whose exact valuecannot be found

Shut

ters

tock

.com

/tot

ojan

g197

7

9780170194662

NEW CENTURY MATHS ADVANCEDfor theA u s t r a l i a n C u r r i c u l u m1010A

n In this chapter you will: (STAGE 5.3) define rational and irrational numbers and perform operations with surds (STAGE 5.3) describe real, rational and irrational numbers and surds (STAGE 5.3) add, subtract, multiply and divide surds (STAGE 5.3) expand and simplify binomial products involving surds (STAGE 5.3) rationalise the denominator of expressions of the form a

bp

c

dp

SkillCheck

1 Simplify each expression.a (5y)2 b (4m)3 c (3x)2

2 Expand each expression.a 5(x 2) b 4(y 3) c 3(1 2w)d 2(5 y) e 5(2a 3) f k(1 2k)

3 Select the square numbers from the following list of numbers.44 81 25 100 75 72 16 50 64 32

4 Expand and simplify each expression.a (m 3)(m 7) b (y 1)(y 4) c (n 2)(n 3)d (2d 3)(1 3d) e (1 5p)(4 3p) f (3a 2f )(a 5f )g (x 4)2 h (y 3)2 i (2k 1)2j (a 5)(a 5) k (t 7)(t 7) l (3m 4)(3m 4)

1-01 Surds and irrational numbersA surd is a square root (

p), cube root (

3p

), or any type of root whose exact decimal or fractionalvalue cannot be found. As a decimal, its digits run endlessly without repeating (like p), so they areneither terminating nor recurring decimals.

7p

is read as the square root of 7 or simply root 7.Rational numbers such as fractions, terminating or recurring decimals, and percentages, can beexpressed in the form a

b; where a and b are integers (and b 6 0). Surds are irrational numbers

because they cannot be expressed in this form.

Worksheet

StartUp assignment 13

MAT10NAWK10091

Stage 5.3

4 9780170194662

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Surds

Recurring decimals

Rational numbers can be expressed in the forma

bIrrational numbers cannot be expressed in the form

a

b

2

3= 0.666 ...

5

6= 0.833 ...

= 0.3636 ...4

11

3

1= 26, = 3

Integers26

1

4

1= 4,

Non-surds whose decimal value also have no pattern and are non-recurring, for example, = 3.14159, cos 38 = 0.7880...,0.0097542

Transcendental numbers

Surds

5, 2, 6, 83

11

Terminating decimals

0.5, 7 = 7.125,

16% = 0.16, 1.32

18

Example 1

Select the surds from this list of square roots:

56p

135p

289p

99p

81p

Solution

56p 7:4833 . . . 135p 11:6189 . . . 289p 17

99p 9:9498 . . . 81p 9

So the surds are

56p

,

135p

and

99p

.

Example 2

Is each number rational or irrational?

a 37.5% b 2164p c 10p d 0:2 _6 e 48p

Solutiona 37:5% 37:5

100 3

8[ 37.5% is a rational number.

which is in the form of a fraction ab

b 2564p 4[ 2564p is a rational number.

which can be written as 41

c 10p 31.415 926 54 . . .[ 10p is an irrational number.

The digits run endlessly without repeating.

d 0:2 _6 0:26666 . . . 4

15

[ 0:2 _6 is a rational number.

which is a recurring decimal

which is a fraction

e

48p 6:928 203 23 . . .[

48p

is an irrational number.

The digits run endlessly without repeating.

Stage 5.3

59780170194662


Square rootsThe symbol

pstands for the positive square root of a number. For example,

4p 2 (not 2).

Furthermore, it is not possible to find the square root of a negative number. It is only possible to findthe square root of a positive number or zero, because the square of any real number is positive or zero.

Summary

For x > 0,

xp

is the positive square root of x.For x 0, xp is 0.For x < 0,

xp

is undefined.

Surds on a number lineThe rational and irrational numbers together make up the real numbers. Any real number can berepresented by a point on the number line.

3 2 1 0 1 2 3 4

3 10 5120%_35_23

103p 2:1544::: irrational (surd) 3

5 0:6 rational (fraction)

23 0:6666::: rational (fraction)

120% 1.2 rational (percentage)

5p 2:2360::: irrational (surd)p 3.1415 irrational (pi)

Example 3

Use a pair of compasses and Pythagoras theorem to estimate the value of

2p

on a number line.

SolutionStep 1Using a scale of 1 unit to 2 cm,draw a number line as shown.

0 1 32

Step 2Construct a right-angled triangle onthe number line with base length andheight 1 unit as shown. By Pythagorastheorem, show that XZ 2p units.

0 1 2 31

1

X

Z

2

Step 3With 0 as the centre, use compasses withradius XZ

2p

to draw an arc to meet thenumber line at A as shown. The point Arepresents the value of

2p

and should beapproximately 1.4142

0 1 2 31

1

XA

Z

2

Stage 5.3

Your calculator will tell you thatthere is a mathematical error ifyou enter, for example,

5p :

Worksheet

Surds on the numberline

MAT10NAWK10092

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Surds

Exercise 1-01 Surds and irrational numbers1 Which one of the following is a surd? Select the correct answer A, B, C or D.

A

64p

B

100p

C

250p

D

400p

2 Which one of the following is NOT a surd? Select the correct answer A, B, C or D.A

84p

B

196p

C

27p

D

160p

3 Select the surds from the following list of square roots.

32p

125p

625p

400p

4:9p

52p

169p

0:0009p

5625p

288p

4 Is each number rational (R) or irrational (I)?a 5: _6 b

8p

c

4p

d 317

e

273p

f 1:3 _5

g643p h 271

2% i

53 103p

j 311

k

50p

3l

4pp

5 Arrange each set of numbers in descending order.a 14

7;

2p

;p2

b

203p

; 2: _6; 279

6 Express each real number correct to one decimal place and graph them on a number line.a 14

5b 74% c 4

11d 12p

e 153p f 259

g p2

h 187%

7 Use the method from Example 3 to estimate the value of

2p

on a number line.

8 a Use the method from Example 3 to estimate the value of

5p

on a number line byconstructing a right-angled triangle with base length 2 units and height 1 unit.

b Use a similar method to estimate the following surds on a number line.

i

10p

ii

17p

Stage 5.3

Investigation: Proof that

2p

is irrational

A method of proof sometimes used in mathematics is to assume the opposite of what isbeing proved, and show that it is false. This is called a proof by contradiction.We will use this method to prove that

2p

is irrational.Firstly, assume that

2p

is rational. This means we assume that

2p

can be written in theform a

b; where b 6 0, and a and b are integers with no common factor.

2p

ab

2 a2

b2Squaring both sides

a2 2b22b2 is an even number because it is divisible by 2.[ a2 is even.

See Example 1

See Example 2

See Example 3

79780170194662


1-02 Simplifying surds

Summary

For x > 0 (positive):

xp 2 xp 3 xp x

x2p

x

Example 4

Simplify each expression.

a

12p 2 b 4

7p 2 c 5 2p 2

Solutiona

12p 2 12

b 4

7p 2

4

7p3 4

7p

4

7p

means 43

7p

423

7p 2

163 7 112

c 5

2p 2

523

2p 2

253 2 50

If a2 is even, then a is also even because any odd number squared gives another oddnumber.If a is even, then it is divisible by 2 and can be expressed in the form 2m, where m is aninteger.

) a2 2m2 2b24m2 2b22m2 b2b2 2m2

[ b2 is even[ b is even[ a and b are both even.

This contradicts the assumption that a and b have no common factor. Therefore, theassumption that

2p

is rational is false.[

2p

must be irrational.1 Use the method of proof just described to show that these surds are irrational.

a

3p

b

5p

2 Compare your proofs with those of other students.

Stage 5.3

Puzzle sheet

Simplifying surds

MAT10NAPS10093

Technology worksheet

Excel worksheet:Simplifying surds quiz

MAT10NACT00019


Excel spreadsheet:Simplifying surds

MAT10NACT00049

8 9780170194662

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Surds

Summary

The square root of a product

For x > 0 and y > 0:

xyp xp 3 yp

A surd

np

can be simplified if n can be divided into two factors, where one of them is a squarenumber such as 4, 9, 16, 25, 36, 49,

Example 5

Simplify each surd.

a

8p

b

108p

c 4

45p

d

288p

3

Solutiona

8p

4p3

2p

23

2p

2

2p

4 is a square number.

b Method 1

108p

36p

3

3p

63 3p

6

3p

Method 2

108p

4p3

27p

23

9p3

3p

23 33

3p

6 3p

Method 2 involves simplifying surds twice (

108p

and

27p

). Method 1 shows thatwhen simplifying surds, you should look for the highest square factor possible.

c 4

45p 43

9p3

5p

43 33

5p

12 5p

d

288p

3

144p

3

2p

3

12

2p

3

124 2p

31

4

2p

Stage 5.3

Video tutorial

Simplifying surds

MAT10NAVT10002

99780170194662


Stage 5.3 Exercise 1-02 Simplifying surds1 Simplify each expression.

a

2p 2

b

5p 2 c 3 3p 2 d 5 10p 2

e

0:09p 2 f 2 7p 2 g 3 5p 2 h 5 2p 2

2 Simplify each surd.a

50p

b

12p

c

28p

d

150p

e

700p

f

45p

g

48p

h

200p

i

96p

j

63p

k

288p

l

108p

m

75p

n

147p

o

32p

p

242p

q

162p

r

245p

s

125p

t

512p

3 Simplify each expression.a 3

20p

b 4

32p

c 8

72p

d

40p

2e

243p

9

f

28p

6g 3

24p

h 9

68p

i

3125p

10j 1

2

72p

k 34

48p

l 10

160p

m 3

75p

n 7

68p

o

52p

6

4 Which one of the following is equivalent to 4

50p

? Select A, B, C or D.A 8

5p

B 20

2p

C 8

2p

D 20

5p

5 Which one of the following is equivalent to

250p

10? Select A, B, C or D.

A

5p10

B

10p

2C 2

10p

D 5

10p

6 Decide whether each statement is true (T) or false (F).a 3

5p 15p b 18p 9 c 9:4p 2 9:4 d 75p 5 3p

e

3p 1:7 f The exact value of 10p is 3.162 277 8

Just for the record Unreal numbers are imaginary!There exist numbers that are neither rational nor irrational, so they are also not real numbers.For example,

2p is not a real number, because there is no real number which, if squared,equals 2. Numbers such as 2p ; 10p and 174p are called unreal or imaginary numbersand cannot be graphed on a number line (that is, their values cannot be ordered).Imaginary numbers were first noticed by Hero of Alexandria in the 1st century CE. In 1545,the Italian mathematician Girolamo Cardano wrote about them, but believed negativenumbers did not have a square root. Imaginary numbers were largely ignored until the 18thcentury when they were studied by Leonhard Euler and the Carl Friedrich Gauss.1p is defined to be the imaginary number i, so 1p i.

)36p 363 1 p 36p 3 1p 6i:

Imaginary numbers are useful for solving physics and engineering problems involving heatconduction, elasticity, hydrodynamics and the flow of electric current.

Simplify each imaginary number.

a100p b 25p c 164p d 646p

See Example 4

See Example 5

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Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Surds

Stage 5.31-03 Adding and subtracting surds

Just as you can only add or subtract like terms in algebra, you can only add or subtract likesurds. You may first need to express all the surds in their simplest forms.

Example 6


a 5

11p 7 11p b 8 5p 3 5p c 3 6p 4 2p 5 6p

d

80p 20p e 8p 27p 18p f 5 20p 3 125p

Solutiona 5

11p 7 11p 12 11p b 8 5p 3 5p 5 5p

c 3

6p 4 2p 5 6p 8 6p 4 2p

d

80p

20p

16p

5p

4p

5p

4 5p 2 5p

6 5p

Simplifying each surd.

e

8p

27p

18p

4p

2p

9p

3p

9p

2p

2

2p

3 3p 3

2p

5

2p

3

3p

f 5

20p

3 125p 5

4p

5p 3 25p 5p

53 2

5p

33 5

5p

10 5p 15 5p

5 5p

Exercise 1-03 Adding and subtracting surds1 Simplify each expression.

a 9

3p 2 3p b 11 2p 8 2p c 5 6p 6p

d

5p 3 5p e 5 17p 5 17p f 3 10p 2 10p

g 4

15p 3 15p 7 15p h 5 6p 2 6p 4 6p i 3 3p 4 3p 5 3p

j 4

5p 7 5p 5p k 8 10p 5 10p 3 10p l 10 3p 3 3p 12 3p

2 Simplify each expression.a 5

3p 9 2 3p b 11 10p 7 2p 4 10p

c 4 3p 5 2p 5 3p d 3 15p 3 2p 4 15p 5 2p

e

7p 3 5p 4 7p 5p f 4 6p 3 3p 2 6p 5 3p

g 10

11p 5 3p 3 11p 4 3p h 13p 8 7p 7 13p 3 7p

i 2

5p 3 7p 2 5p 3 7p j 4 10p 3 5p 4 10p

3 For each expression, select the correct simplified answer A, B, C or D.a

3p 12p

A 5

3p

B

15p

C 2

6p

D 3

3p

b 4

5p 2 125p

A 6 5p B 5p C 45p D 46 5p

Puzzle sheet

Surds code puzzle

MAT10NAPS10094

See Example 6

119780170194662


Stage 5.3 4 Simplify each expression.a

8p 32p b 108p 27p c 20p 80p d 28p 63p

e 3

6p 24p f 2 5p 125p g 40p 90p h 5 11p 99p

i 3

2p 18p j 27p 5 3p k 200p 7 2p l 50p 32p

m 5

3p 2 27p n 3 20p 245p o 7 12p 5 48p p 4 27p 2 243p

q 3

63p 2 28p r 2 98p 3 162p s 5 6p 2 150p t 4 50p 3 18p

u 5

27p 6 75p v 3 112p 2 252p w 32p 8p 12p

x

27p 54p 243p y 98p 3 20p 2 8p z 3 96p 2 150p 24p

1-04 Multiplying and dividing surds

Summary

The square root of products and quotients

For x > 0 and y > 0: xyp xp 3 yp

xy

r

xp

yp

Example 7


a

7p3

5p

b

6p3

14p

c 4

3p3 10

3p

d 12

90p

3

10p e 54p 4 2p f 5 27p 3 3 6p

Solutiona

7p3

5p 35p b 6

p3

14p

84p

4p3

21p

2

21p

c 4

3p3 10

3p

43 103

3p3

3p

403 3 120

d 12

90p

3

10p 43

9p

43 3 12

e

54p

4

2p

54p

2p

27p

9p3

3p

3

3p

f 5

27p

3 3

6p

53 33

27p

3

6p

15

162p

153

81p

3

2p

153 9

2p

135

2p

Worksheet

Multiplying and dividingsurds

MAT10NAWK10095

Puzzle sheet

Surds

MAT10NAPS00043


Excel worksheet:Simplifying surds quiz

MAT10NACT00019


Excel spreadsheet:Simplifying surds

MAT10NACT00049

12 9780170194662

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Surds

Example 8

Simplify 5

2p3 4

12p

10

8p :

Solution

5

2p3 4

12p

10

8p 20

24p

10

8p 2 3p

Exercise 1-04 Multiplying and dividing surds1 Simplify each expression.

a

10p

3

3p

b 6p 3 11p c 5p 3 8p

d

2p3

18p

e

5p3 5p f 8 3p 3 6 3p

g 5

10p

3 3

3p

h 2 7p 3 5 3p i 7 5p 3 4 5p

j 2

3p3 5 6p k 4 3p 3 27p l 3 5p 3 4 10p

m 7 2p 3 4 8p n 18p 3 8 3p o 10 2p 3 2 8p

p 3

18p

3 5

12p

q 3

44p

3 2 99p r 5 8p 3 4 40p

s 8

3p3 3

54p

t 8 32p 3 27p u 90p 3 72p

v 5 20p 3 3 8p w 7 18p 3 3 24p x 3 48p 3 2 12p

2 Simplify each expression.

a

24p

4

8p

b

30p

4 5p c 40

14p

5

2p

d 10

54p

4 5

27p

e 3 98p 4 6 14p f 7

18p

2p

g 2

24p

4 4

6p

h

128p

2p i 15 18p 4 3 6p

j 20

10p

4 5p k 36

24p

4 9

8p

l 16

30p

4 8

5p

m 12

14p

4 6 n 3

2p12 o

80p

4 4

5p

p 5

60p

4

15p

q 6

8p4 3

2p

r 42

54p

6

3p

s 12

63p

4 3

7p

t 8

50p

2

200p u 6 3p 4 243p

3 Simplify the expressions below.

a

6p3

6p

b

7p3

7p

c 2

3p3

3p

d 5 yp 3 3 yp e xp 3 xp f

a2p

3

ap

4 Simplify 3

2p3

6p

: Select the correct answer A, B, C or D.A 6 B 6

2p

C 6

3p

D 12

2p

Stage 5.3

See Example 7

139780170194662


5 Simplify 20

10p

4 5

2p

: Select A, B, C or D.A 4

5p

B 15

5p

C 10 D 20


a 3

5p3 4

2p

3

40p b 3

12p

3 8

6p

4

27p c 5

8p3 2

90p

10

24p

d 4

5p

2

15p

3 5

27p e 10

686p

3 3

12p

5

28p

3

18p f 8

80p

3 3

2p

4

5p3 6

8p

Stage 5.3

Mental skills 1 Maths without calculators

Percentage of a quantityLearn these commonly-used percentages and their fraction equivalents.

Percentage 50% 25% 12.5% 75% 20% 10% 33 13 % 6623 %

Fraction 12

14

18

34

15

110

13

23

Now we will use them to find a percentage of a quantity.

1 Study each example.

a 20%3 25 153 25

5b 50%3 120 1

23 120

60c 12:5%3 32 1

83 32

4d 75%3 56 3

43 60

143 60

3 3

153 3 45

e 3313%3 27 1

33 27

9f 66

23%3 60 2

33 60

133 60

3 2

203 2 40

2 Now simplify each expression.a 25% 3 44 b 33 13 %3 120 c 20% 3 35 d 66

23 %3 36

e 10% 3 230 f 12 12 %3 48 g 50% 3 86 h 20% 3 400i 75% 3 24 j 33 13 %3 45 k 25% 3 160 l 10% 3 650m 12.5% 3 88 n 66 23 %3 21 o 20% 3 60 p 75% 3 180

See Example 8

14 9780170194662

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Surds

1-05 Binomial products involving surdsSurd expressions involving brackets can be expanded in the same way as algebraic expressions ofthe form a(b c) and (a b)(c d).

Example 9

Expand and simplify each expression.

a

3p

5p 7p

b 2

11p

3

11p 5 2p

Solution

a 3

3 5 + 3 7

75 +

=

15 + 21=

b

2

2

11 3 11 5

2222

11 2= 2

= 6 11 10 = 66 10

11 3 11 52

Example 10


a

7p 5p 3 2p 3p b 3 2 10p 5p 3 2p

Solution

a

7p

5p

3

2p

3p

7p

3

2p

3p

5p 3

2p

3p

7p3 3

2p

7p3

3p 5p 3 3

2p

5p 3 3p

3

14p

21p

3

10p

15p

b 3 2

10p

5p 3

2p

3 5p 3

2p

2

10p

5p 3

2p

33 5p 33 3

2p

2

10p

3

5p 2

10p

3 3

2p

3

5p

9

2p

2

50p

6

20p

3 5p 9

2p

2 5

2p

6 2 5p

3

5p

9

2p

10

2p

12

5p

15

5p

19

2p

Summary

(a b)2 a2 2ab b2(a b)2 a2 2ab b2(a b)(a b) a2 b2

Stage 5.3

159780170194662


Stage 5.3 Example 11


a

7p 5p 2 b 2 3p 3 5p 2

c

5p 2p 5p 2p d 3 11p 4 3 11p 4

Solutiona

7p

5p 2

7p 2

23

7p3

5p 5p

2

7 2

35p

5 12 2 35p

Using (a b)2 a2 2ab b2

b 2

3p

3

5p 2

2

3p 2

232

3p33

5p

3

5p 2

433 12 15p 935 12 12

15p

45 57 12 15p

Using (a b)2 a2 2ab b2

c

5p

2p

5p

2p

5p 2

2p 2

5 2 3

Using (a b)(a b) a2 b2

d 3

11p

4

3

11p

4

3

11p 2

42

93 11 16 83

Using (a b)(a b) a2 b2

Exercise 1-05 Binomial products involving surds1 Expand and simplify each expression.

a

5p

3p 2p b 6p 2p 1 c 2p 3p 7p

d

5p

3

2p 5p e 3 2p 2p 2 3p f 11p 4 5p

g 2

7p

3

7p 4 h 5 5p 1 3 5p i 3 2p 4 2p 3p

2 Which expression is equivalent to

3p 2 5p 5 2p 3p ? Select the correct answer A, B,

C or D.A 20

10p

B 2

15p 5 6p

C 5

6p 3 10 10p 2 15p D 5 5p 3p 7 7p 4 2p

3 Expand and simplify each expression.a

5p 3 2 5p 2p b 7p 3p 7p 2

c 7

3p 2 4 2p 3p d 3 2p 5p 5 2p 2 5p

e

7p 2 11p 3 7p 4 11p f 5 3p 2 2p 4 3p 3 2p

g 6 2 10p 3 10p 1 h 7p 2 5p 3 5p 2 7p

Note that because of thedifference of two squares,the answer is not a surd buta rational number.

See Example 9

See Example 10

16 9780170194662

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Surds

4 Which expression is equivalent to 5 7p 2? Select A, B, C or D.A 12 B 32 C 32 10 7p D 32 5 7p


5p 3p 2 b 7p 2p 2 c 5p 2 2 d 3 10p 2

e 5

2p 3 3p 2 f 5 7p 2 2 g 3 2p 2 5p 2 h 2 5p 3p 2


3p 2p 3p 2p b 5 3p 5 3p

c 6 2 7p 6 2 7p d 5p 3p 5p 3p

e

11p 10p 11p 10p f 5 7p 3 5 7p 3

g 3

2p 5p 3 2p 5p h 4 2p 5 3p 4 2p 5 3p

7 Which expression is equivalent to 5

2p 4 3p 5 2p 4 3p ? Select A, B, C or D.

A 25

2p 16 3p B 10 2p 10 6p C 2 D 26

8 Expand and simplify each expression.a 3

7p 5 2 b 5 2p 4 2p 5

c 2

7p 3 5p 5p 7p d 4 3p 5 2

e 4

2p 3p 4 2p 3p f 3 10p 2p 2

Stage 5.3

Investigation: Making the denominator rational

If

2p 1:4142; what is the value of 1

2p ? Fractions containing surds in the denominator are

difficult to work with. When approximating the value of 12

p ; it is difficult to mentally divideby 1.4142. We can overcome this by making the denominator rational (that is, not a surd).1 What happens when we multiply the numerator and denominator of a fraction by the

same number?

a Simplify 12

p 3

2p

2p :

b Mentally approximate the value of

2p2

; given that

2p 1:4142:

c Check, using a calculator, that 12

p

2p2

: Why is this true?

2 a Is it true that 37

p 37

p 3

7p

7p ? Why?

b Simplify 37

p 3

7p

7p : Compare your answer with those of other students.

c Check, using a calculator, that 37

p 3

7p7

:

d Explain why 3

5p

2p 3

5p

2p 3

2p

2p :

e Show that 3

5p

2p 3

10p

2p :

See Example 11

179780170194662


Stage 5.31-06 Rationalising the denominator

Surds of the form 15

p ; 32

7p ;

3p

2p ; 5

7p

3p ; have denominators that are irrational. These expressions may

be rewritten with a rational denominator by multiplying both the numerator and denominator by the surdthat appears in the denominator. This method is called rationalising the denominator.

Example 12

Rationalise the denominator of each surd.

a 32

p b 54

3p c 8

2p

3

5p d

2p 1

3p

Solution

a3

2p 3

2p 3

2p

2p

3

2p

2

because

2p

2p 1

Because 32

p 3

2p2

; it is easier to approximate 32

p by mentally multiplying 32

by 1.4142

than by dividing 3 by 1.4142.

b5

4

3p 5

4

3p 3

3p

3p

5

3p

43 3

5

3p

12

c8

2p

3

5p 8

2p

3

5p 3

5p

5p

8

10p

33 5

8

10p

15

d

2p 1

3p

2p 1

3p 3

3p

3p

6p 3p

3

Exercise 1-06 Rationalising the denominator1 By rationalising the denominator, which surd is equivalent to 2

6p ? Select the correct answer

A, B, C or D.

A 2

6p

B

6p3

C

6p6

D 2

6p3

2 Rationalise the denominator of each surd.a 1

2p b 1

7p c 1

3p d 3

2p e 2

7p f 1

3

2p

g 12

3p h 1

4

7p i 7

3

5p j

2p3

5p k 3

2p

2

6p l 5

3p

4

5p

3 Which surd is equivalent to

3p2

5p ? Select A, B, C or D.

A

15p10

B 2

15p

C

15p

3D

5p

Worksheet

Rationalising thedenominator

MAT10NAWK10201

See Example 12

18 9780170194662

Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Surds

4 Which surd is equivalent to

27p3

18p ? Select A, B, C or D.

A 12

B

2p2

C

5p6

D

6p6

5 Rationalise the denominator of each expression.

a

2p 1

2p b 1

5p

5p c 5

3p

2

2p d

2p 3p

3

6p

6 Simplify each expression, giving the answer with a rational denominator.

a 17

p 12

p b

2p

5p 3

3p c 3

2

3p 1

2p

Power plus

1 a Is it true that 13 2p

13 2p 3

3 2p3 2p ? Explain.

b Simplify 13 2p 3

3 2p

3 2p : Is the denominator rational?

c Use a calculator to check that the value of your answer to part b is equal to the

value of 13 3p :

2 The conjugate of 3 2p is 3 2p : Find the conjugate of:a 5 2 3p b 2 3p c 5p 1 d 7p 3p

3 The process shown in question 1 involves rationalising a surd with a binomialdenominator. By first finding the conjugate of the denominator, rationalise thedenominator of each expression below.

a 12 3p b

2p

5p 1 c

1

7p 3p d

25 2 3p

4 The largest cube that can fit into a sphere must have its eightvertices touching the surface of the sphere. Express the sidelength, s, of the cube in terms of the diameter, D, of the sphere.

5 Squares are formed inside squares by joining the midpointsof the sides of the squares as shown. If AB 4 cm, find theexact length of the side of the shaded square.

D C

A B

6 Six stormwater pipes, each 2 mm in diameter, are stackedas shown in the diagram. Find the exact height, h, ofthis stacking.

h

Stage 5.3

199780170194662


Chapter 1 review

n Language of maths

approximate binomial denominator difference of two squares

expand irrational number perfect square product

Pythagoras theorem quotient rational number rationalise

real number root simplify square number

square root surd undefined

1 Why do you think a rational number has that name?

2 What is the difference between a rational number and a real number?

3 What is a surd?

4 How do you simplify a surd?

5 Why is p an example of an irrational number that is not a surd?

6 How do you rationalise the denominator of a surd expression?

n Topic overview

Copy and complete this mind map of the topic, adding detail to its branches and using pictures,symbols and colour where needed. Ask your teacher to check your work.

Adding andsubtracting surds

Multiplying anddividing surds

Binomial productsinvolving surds

Rationalising thedenominator

Simplifyingsurds

Surds and irrationalnumbers

Surds

Puzzle sheet

Surds crossword

MAT10NAPS10202

978017019466220

1 Which one of the following is a rational number? Select the correct answer A, B, C or D.

A 1 25 B

93p

C 2

5p

D 2p

2 Is each number rational (R) or irrational (I)?

a

8p

b 227

c 0:5 _6 d 3

5p

e

8p

1 f

1253p

g83p h 5 3p

3 Simplify each surd.

a

72p

b

98p

c

275p

d

128p

e 3

150p

f 7

28p

g 4

288p

h 5

45p

i 7

48p

j 12

24p

k 23

3125p

l 12

32p

m 2

44p

n 25

225p

o 23

162p


a

200p 18p b 3 5p 50p 2 125p

c 7

32p 27p 2 98p 4 75p d 4 45p 3 63p 5 80p

e

800p 2 243p 3 72p 2 27p f 7 44p 2 99p


a

3p3

7p

b

8p3

5p

c

6p3

8p

d

5p3

11p

e

72p

4

12p

f

98p

3

7p

g 8

42p

4 2

7p

h

125p

4 5

5p

i

75p3

3p

j

18p

3

3p

12p k

6p3

24p

27p

3 2

3p l 4

90p

3 7

8p

5

32p

3 6

10p

6 Expand and simplify each expression.

a 3 2p 2 2p 3 b 10p 1 5 2p c 3 5p 2 7p 3 7p 5p d 7p 42e 5 3p 2p 2 f 3 7p 2 5p 3 7p 2 5p g 3 4 7p 4 7p 3 h 5 10p 3 2p 2 10p 5 2p

7 Rationalise the denominator of each surd.

a 110

p b 32

p c 15

7p

d 34

3p e 5

3p

3

2p f 4

2p

3

2p

See Exercise 1-01

See Exercise 1-01

See Exercise 1-02

See Exercise 1-03

See Exercise 1-04

See Exercise 1-05

See Exercise 1-06

9780170194662

Chapter 1 revision

21

Chapter 1: SurdsSkillCheck1-01 Surds and irrationalnumbers*Investigation: Proof that 2 is irrational1-02 Simplifying surds*1-03 Adding and subtractingsurds*1-04 Multiplying and dividingsurds*Mental skills 1: Percentageof a quantity1-05 Binomial productsinvolving surds*Investigation: Making the denominator rational1-06 Rationalising thedenominator*Power plusChapter 1 review

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