Chapter 01 Surds.pdf

1

Number and Algebra

Surds When applying Pythagoras’ theorem, we have found lengths that cannot be expressed as an exact rational number. Pythagoras encountered this when calculating the diagonal of a square of side length 1 unit. A  surd  is a square root ( ), cube root ( 3 ), or any type of root whose exact decimal or fraction value cannot be found.

p  ffi

p  ffi

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    7     7     9     1    g    n    a     j    o     t    o     t      /    m    o    c  .      k    c    o     t    s    r    e     t     t    u      h      S

n Chapter outline

n Wordbank Proficiency strands

1-01 Surds and irrational numbers* 1-02 Simplifying surds* 1-03 Adding and subtracting surds* 1-04 Multiplying and dividing surds* 1-05 Binomial products involving surds* 1-06 Rationalising the denominator* *STAGE 5.3

U U

F F

R R

C

irrational number  A number such as p or be expressed as a fraction  a b

written n in the rational number  Any number that can be writte  a form ; where  a  and  b  are integers and  b 0

6¼

b

U

F

R

U

F

R

U

F

R

C

U

F

R

C

p  ffi2ffi that cannot

rationalise the denominator  To simplify a fraction involving a surd by making its denominator rational (that is, not a surd) real number  A number that is either rational or irrational and whose value can be graphed on a numbe numberr line

p ffi

simplest form form so simplify a surd  To write a surd  x in its simplest that  x  has no factors that are perfect squares exact value surd A square root (or other root) whose exact cannot be found

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Chapte Chap ter r 1 2345678910111213141516 Surds

n In this chapter you will: • • • • •

(STAGE 5.3) define rational (STAGE rational and irrationa irrationall numbers numbers and perform perform operations operations with surds surds (STAGE (STAG E 5.3) describe describe real, real, rational rational and irrationa irrationall numbers numbers and surds surds (STAGE (STAG E 5.3) add, add, subtrac subtract, t, multiply multiply and and divide divide surds surds (STAGE (STAG E 5.3) expand expand and simplify simplify binomial binomial product productss involving involving surds surds  a b (STAGE (STAG E 5.3) rationali rationalise se the denomin denominator ator of express expressions ions of the the form

p  ffiffi p  ffiffi c d 

SkillCheck Worksheet StartUp assignment 13

Simplify each expression.

1

a (5 y) 2

MAT10NAWK10091

c ( 3 x) 2

b 4( y 3) e  5(2a 3)

c 3(1 (1 f  k (1



Expand each expression.

2

a 5( x d 2(5

þ 2)   y)





þ

Select the square square numbers from the following following list of numbers. numbers.

3

44

81

25

100

75

72

16

50

þ 2w) þ 2k ) 64

32

Expand and simplify simplify each expression. expression.

4

a d g  j

Stage 5.3

b (4m) 3

(m 3)(m 7) (2d  3)(1 3d ) ( x 4) 2 (a  5)(a  5)

b e h k

þ þ þ þ þ  þ

( y (1 ( y (t 

þ 1)( y  4)   5 p)(4 þ 3 p)  3) þ 7)(t   7) 2

c f  i l

(n 2)(n 3) (3a 2 f   )(a  5 f   ) (2k  1) 2 (3m 4)(3m 4)

  þ þ þ þ 

1-01  Surds and irrational numbers p  ffi

p  ffi

A surd  is a square root ( ), cube root ( 3 ), or any type of root whose exact decimal or fractional value cannot be found. As a decimal, its digits run endlessly  without repeating  (like  (like p) ,  so they are neither terminating nor recurring decimals. 7 is read as ‘the square square root of 7’ or simply ‘root ‘root 7’. Rational numbers  such as fractions, terminating or recurring decimals, and percentages, can be expressed in the form  a ;  where  a  and  b  are integers (and  b 0). Surds are  irrational numbers b because they cannot be expressed in this form.

p  ffiffi

6¼

4

 

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Stage 5.3 Rational numbers can be expressed in the form

a

Irrational numbers cannot  be  be expressed in the form

b

Integers 26 4 –3 = –3 = 4, = 26, 1 1 1

√ 5, – √2,

2

0.5, 71 = 7.125,

3  5

16% = 0.16, 1.32

6

8

4 11

Example

b

Surds

Recurring decimals Terminating decimals

a

√11 3

, 8 √6

Transcendental numbers

= 0.666 ...

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

= 0.833 ... = 0.3636 ...

1

p  ffiffi ffi p ffi135 ffi ffi  ffi p ffi289ffi ffi  ffi p 99ffi ffi ffi p 81ffi ffi ffi

Select the surds from this list of square roots:  56

Solution

p  ffi 56ffi ffi ¼ 7:4833 . . . p  ffi135 ffi ffi ffiffi ¼ 11:6189 . . . p  ffi99ffi ffi ¼ 9:9498 . . . p  ffi81ffi ffi ¼ 9 p  ffiffi ffi p  ffiffi ffi ffi ffi p  ffiffi ffi So the surds are  56, 135 and 99. Example

p  ffi289 ffi ffi ffiffi ¼ 17

2

Is each number rational or irrational? a   37.5%

Solution  5% a 37: 5 [

b

4

 5 ¼ 3 : 5 ¼ 37 100 8

ffi ffi ffi ffi ¼ 4 p  ffi256 ffi ffi ffiffi is a rational number. p  ffi256 10 ¼ 31.415 926 54 . . . 4

d 0:26_

p  ffi48ffi ffi

e

 which is in the form of a fraction a

b

 which can be written as

4 1

The digits run endlessly without repeating.

 10p  is an irrational number.

d 0:26_

[

c 10p

4

p

[

e

ffi ffi ffiffi p  ffi216

 37.5% is a rational number.

[

c

b

¼ 0:26666 . . . ¼ 154

 which is a recurring decimal  which is a fraction

 0:26_ is a rational number.

p  ffi48ffi ffi ¼ 6:92820323 . . . p  ffi48ffi ffi is an irrational number.

The digits run endlessly without repeating.

[

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Chapte Chap ter r 1 2345678910111213141516 Surds

Stage 5.3

Square roots

p  ffiffi ¼

p  ffiffi

The symbol stands for the positive square root of a number. For example, example, 4 2 (not 2). Furt Fu rthe herm rmor ore, e, it is no nott po poss ssib ible le to fin find d th thee sq squa uare re ro root ot of a ne nega gati tive ve nu numb mber er.. It is on only ly po posssi sibl blee to fin find d thee sq th squa uarre ro root ot of a po posi siti tive ve nu numb mber er or ze zero ro,, be beca caus usee th thee sq squa uare re of an anyy re real al nu numb mber er is po posi siti tive ve or ze zero ro..



Summary 

p ffi ffi ¼ p  p ffi

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

Worksheet Surds on the number line

Your calculator calculator will tell you that there is a mathem mathematical atical error if  you enter, for example,  5:

p  ffiffi ffi ffi

Surds on a number line The rational and irrational numbers together make up the  real numbers . Any real number can be represented by a point on the number line.

MAT10NAWK10092

–

3

–3

p  ffi10ffi ffi  2:1544:::  53 ¼ 0:6 2  0:6666::: 3 3

Example

– 53_

10 –2

–1

2_ 3

0

irrational (surd) rational (fraction) rational (fraction)

120%

5

1

π

2

3

4

120%

1.2 p  ffi 5ffi  2¼:2360 :::

p

rational (percentage) irrational (surd)

 3.1415…

irrational (pi)

3

p  ffiffi

Use a pair of compasses and Pythagoras’ theorem to estimate the value of  2 on a number number line. line.

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

0

Step 2 Construct a right-angled triangle on the number line with base length and height 1 unit as shown. By Pythagoras’ theorem, show that  XZ  2  units.

 ¼ p  ffiffi  ¼

p  ffiffi 

6

p  ffiffi

 

2

3

 Z 

2

1

 X 

0

Step 3 With 0 as the centre, use compasses with radius  XZ  2  to draw an arc to meet the number line number  line at  A  as shown. The point  A represents the value of  2 and should be approximately 1.4142…

1

1

1

2

3

2

3

 Z 

2

1

 X 

0

1

1

 A

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NEW CENTURY MATHS ADVANCED for the

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Stage 5.3

p  ffi64ffi ffi

B

p  ffi84ffi ffi

B

ffi ffi ffiffi p  ffi100

C

ffi ffi ffiffi p  ffi196

C

ffi ffi ffiffi p  ffi250

p  ffiffi ffi ffiffi D 400

p  ffi27ffi ffi

D

See  Example  Example 1

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

3

þ

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

2

10 10A

Surds and irrational numbers

Exercise 1-01 1

A u s tr a l i an C u r ri c u l um

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

p  ffi32ffi ffi p  ffi 52ffi ffi

p 125 ffi ffi  ffiffi ffi p 169 ffiffiffiffi

   

ffiffiffi p ffi625 ffiffiffiffiffi p ffi0ffi:0009

   

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

b

p  ffiffi ffi64ffi ffi

p  ffi8ffi

c

   

p  ffi4ffi p  ffiffi ffi ffi ffi ffi ffi 3ffi

p  ffiffi ffi ffiffi

160

p ffi400 ffiffiffi p  5625 ffiffiffiffiffi

1 d 3 7 3 j 11

p 4ffi ffi:ffi9ffi p ffi288 ffiffiffi

    e

p  ffi27ffi ffi p  ffi 50ffi ffi 3

See  Example  Example 2

f  1:3 5_

p  ffip ffi ffiffi4ffi

7

1  5 3 10 g h 27 % i k l 2 3 Arrange each set of numbers numbers in descending order. p 4 7 a 1 ; 2; b 3 20;  2: _6; 2 7 2 9 Express each real number number correct to one decimal decimal place and graph graph them on a number line. line. 4 14 12 a b 74% c d  5 11  5 p 3 15 e f  2 g h   187% 9 2 Use the method method from Example 3  to estimate the value of  2 on a number line. line.

8

a

3

5 6

p  ffiffi

p  ffiffi ffi

p  ffiffi ffi

 p  ffiffi ffi

p  ffiffi p  ffiffi number line by  Use the method method from Example 3  to estimate the value of   5 on a number by 

 Example 3 See  Example

constructing a right-angled triangle with base length 2 units and height 1 unit. surds on a number line. b Use a similar method to estimate the following surds

p  ffi10ffi ffi

i

ii

p  ffi17ffi ffi

 ffi ffi p  Investigation: Proof that  2  is irrational A method of proof sometimes used in mathematics is to assume the opposite of what is being proved, and show that it is false. This is called a  proof by contradiction. We will use this method to prove that 2  is irrational. Firstly, assume that 2 is  rational . This means we assume that 2 can be written in the form  a ; where  b 0, and  a  and  b  are integers with no common factor.

p  ffiffi

2

6¼

b

¼ ab

p  ffiffi

p  ffiffi

p  ffiffi

2

¼ ab2 a2 ¼ 2b2 2

Squaring both sides

2b 2 is an even number because it is divisible by 2. 2 is even. [  a

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Stage 5.3

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

¼ ð2mÞ2 ¼ 2b2 4m2 ¼ 2b2 2m2 ¼ b2 b2 ¼ 2m2

)

a2

 b 2 is even [  b  is even [  a  and  b  are both even. [

This contradicts the assumption that  a  and  b  have no common factor. Therefore, the assumption that 2 is rational rational is false. 2 must be irrational. irrational. [ irrational. 1 Use the method of proof just described to show that these surds are irrational. 3 a b  5 with those of other students. 2 Compare your proofs with

p  ffiffi

Puzzle sheet Simplifying surds

p  ffiffi

p  ffiffi

p  ffiffi

1-02   Simplifying surds

MAT10NAPS10093 Technology worksheet Excel worksheet: Simplifying surds quiz MAT10NACT00019

Summary  For  x  >  0 (positive):

p  ffiffi ¼ p  ffiffi p  ffiffi ¼ p  ffi ffiffi ffi ¼  x

Technology worksheet Excel spreadsheet: Simplifying surds MAT10NACT00049

2

 x2

 x 3  x

 x

 x

Example

4

Simplify each expression. a

p  ffi12ffi ffi 

2

b

Solution

p  ffi12ffi ffi  ¼ 12  p  ffiffi  p  ffiffi p  ffiffi b 4 7 ¼4 7 4 7  p  ffiffi  ¼4 7 a

2

 5p  ffi2ffi 

2

c

2

2

3

2

p  ffiffi

2

3

p  ffiffi

4 7  means 4 3 7

3

¼ 16 ¼ 112

8

4p  ffi7ffi 

7

 

  p  ffiffi  p  ffiffi  2 c  5 2 ¼ ð 5Þ 2

2

¼ 25 ¼  50

3

2

3

2

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Stage 5.3

Summary  The square root of a product  For  x  >  0 and y  >  0:

p  ffi xyffi ffi ¼ p  ffi xffi p  ffi yffi 3

p ffi ffi

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

Example

Video tutorial

5

Simplifying surds

Simplify each surd. a

p  ffi8ffi

b

Solution a

p  ffiffi p  ffiffi p  ffiffi 8¼ 4 p  ffiffi 2 ¼ 2p  ffiffi 2 ¼2 2

p  ffi108 ffi ffi ffiffi

p  ffiffi ffi

c 4 45

d

ffi ffi ffiffi p  ffi288

MAT10NAVT10002

3

4 is a square number.

3

3

b Me Meth thod od 1

Method 2

p  ffiffi ffi ffiffi p  ffiffi ffi p  ffiffi 108 ¼ 36 p  ffiffi 3 ¼ 6p  ffiffi 3 ¼6 3

p  ffiffi ffi ffiffi p  ffiffi p  ffiffi ffi 108 ¼ 4 p  ffiffi 27p  ffiffi ¼ 2 9 p  ffiffi 3 ¼ 2p  ffiffi3 3 ¼ 6 p  ffi3ffi ffi  ffi ffi ffi ffi ffi p  Method 2 involves simplifying surds  twice  ( 108 and 27). Method Method 1 shows that that 3

3

3

3

3

3

3

 when simplifying surds, you should look for the highest square factor possible.

p  ffiffi ffi

c 4 45

9780170194662

p  ffiffi p  ffiffi ¼ 4 9 p  ffiffi  5 ¼ 4 p  ffi3ffi  5 ¼ 12  5 3 3

3

3

d

ffi ffi ffiffi p  ffi144 p  ffi288 ffi ffi ffiffi p  ffi2ffi ¼ 3 3 p  ffiffi 12 2 ¼ 3 4 p ffi ffi 12 2 ¼ 1 p  ffi3ffi ¼4 2 3

9

Chapte Chap ter r 1 2345678910111213141516 Surds

Stage 5.3 See  Example  Example 4

Exercise 1-02 Simplify each expression.

1

p  ffi2ffi  p  ffi 5ffi   p  ffiffi   p  ffiffi ffi  b c 3 3 d 5 10 p  ffi0ffi ffi09ffi ffi  f  2p  ffi7ffi   p  ffiffi   p  ffiffi  e g 3  5 h  5 2 Simplify each surd. p  ffi12ffi ffi p  ffi28ffi ffi p  ffi 50ffi ffi p  ffi150 ffi ffi ffiffi e p  ffi700 ffi ffi ffiffi a p  b c d p  ffi63ffi ffi ffi ffi ffiffi i p  ffi96ffi ffi p  ffi200  ffi45ffi ffi g p  ffi48ffi ffi f  p  h j  ffi288 ffi ffi ffiffi l p  ffi108 ffi ffi ffiffi p  ffi147 ffi ffi ffiffi o p  ffi32ffi ffi p  ffi75ffi ffi k p  m n ffi ffi ffiffi  ffi242 ffi ffi ffiffi q p  ffi162 ffi ffi ffiffi s p  ffi125 ffi ffi ffiffi t  p  ffi 512 ffi ffi ffiffi p  ffi245 p r Simplify each expression. p  ffi40ffi ffi p  ffi243 ffi ffi ffiffi p   ffi ffi ffi p   ffi ffi ffi  ffi ffi ffi p  a 3 20 b 4 32 c 8 72 d e 2 9 p  ffi28ffi ffi p  ffi3125 ffi ffi ffi ffiffi p  ffiffi p  ffiffi ffi p   ffi ffi ffi 1  ffi72 f  g 3 24 h 9 68 i j p  6p  2 ffiffi ffi  ffi ffi ffi p  ffiffi ffi ffiffi p  ffi10ffi ffi p   ffi ffi ffi  52 3 48 k l 10 160 m 3 75 n 7 68 o 4 6 p  ffiffi ffi Which one of the following following is equivalent to 4  50 ? Select A,  B ,  C  or  D . p  ffiffi p  ffiffi p  ffiffi p  ffiffi A 8  5 B 20 2 C 8 2 D 20  5 ffi ffi ffiffi p  ffi250 Which one following is equivalent to to ? Select A,  B ,  C  or  D . 10 p  ffi 5ffi one of the following p  ffi10ffi ffi p  ffiffi ffi p  ffiffi ffi A B C 2 10 D 5 10 10 2 Decide whether whether each statement statement is true (T) or false (F). p  ffi18ffi ffi ¼ 9  ffi ffi  ffi ffi ffi p  p  p  ffi9ffi ffi4ffi ¼ 9 4 d p  ffi75ffi ffi ¼ 5p  ffi3ffi a 3  5 ¼ 15 b c p  ffiffi ffi p  ffi3ffi  1 7 exact value of  10 is 3.162 277 8 e f  The exact 2

a

2

2

3

4

5

6

2

2

2

:

See  Example  Example 5

Simplifying Simplifyin g surds

2

2

2

:

2

:

:

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, 2 is not a real number, number, because there is no real number which, which, if squared, squared, 4 equals 2. Numbers such as 2; 10 and 17 are called unreal  or  imaginary numbers and 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 negative numbers did not have a square root. Imaginary numbers were largely ignored until the 18th century when they were studied by Leonhard Euler and the Carl Friedrich Gauss.



p  ffiffi ffi ffi

p  ffiffi ffi ffi p  ffiffi ffi ffi ffi

p  ffiffi ffi ffi ffi

p  ffiffi ffi1ffi is defined to be the imaginary number i, so p  ffiffi ffi1ffi ¼ i.  ffi36ffi ffi ffi ffi ffi ðffi ffi ffi1ffi Þffi ¼ p  ffi36ffi ffi p  ffiffi ffi1ffi ¼ 6i: p  ffiffi ffi36ffi ffi ¼ p  3

)

3

Imaginary numbers are useful for solving physics and engineering problems involving heat conduction, elasticity, hydrodynamics and the flow of electric current. Simplify each imaginary number. a

10

p  ffiffi ffi100 ffiffiffi

b

p  ffiffi ffi25ffi ffi

p  ffiffi ffi16ffi ffi 4

c

 

d

p  ffiffi ffi64ffi ffi 6

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NEW CENTURY MATHS ADVANCED for the

A u s tr a l i an C u r ri c u l um

10 10A

þ

Stage 5.3

1-03  Adding and subtracting surds

Puzzle sheet

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

Example

p  ffiffi ffi þ 7p  ffi11ffi ffi p  ffi80ffi ffi þ p  ffi20ffi ffi

p  ffiffi  3p  ffi 5ffi p  ffi8ffi  p  ffi27ffi ffi þ p  ffi18ffi ffi

b 8  5

c 3 6

d

e

f 

Solution

p  ffiffi ffi þ 7p  ffi11ffi ffi ¼ 12p  ffi11ffi ffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi c 3 6  4 2 þ 5 6 ¼ 8 6  4 2 p  ffiffi ffi p  ffiffi ffi p  ffiffi ffi p ffi ffi p  ffiffi p ffi ffi d 80 þ 20 ¼ 16  5 þ 4  5 p  ffiffi p  ffiffi ¼ 4p  ffi 5ffi þ 2  5 ¼p  ffi6ffi ffi  5 p  ffiffi p ffi ffi p  ffiffi p ffi ffi p  ffiffi p ffi ffi  ffi ffi p  p   ffi ffi ffi e 8  27 þ 18 ¼ 4 2  9 3 þ 9 2 p  ffiffi p  ffiffi p  ffiffi ¼ 2p  ffi2ffi  3p  ffi3ffi þ 3 2 ¼  5 2  3 3 a 5 11

Simplify each expression. d g  j

p  ffiffi p  ffiffi 9 3þ2 3 p  ffi 5ffi þ 3p  ffi 5ffi p  ffiffi ffi p  ffiffi ffi p  ffiffi ffi 4 15  3 15 þ 7 15 p  ffiffi p  ffiffi p  ffiffi 4  5 þ 7  5   5

Simplify each expression.

Simplifying each surd.

b e h k

p  ffi ffiffi  9 þ 2p  ffi3ffi p  ffi4ffip  ffi3ffi þp  ffi 5ffip  ffi2ffi p  ffi 5ffip  ffi3ffip  ffiffi 7  3  5  4 7 þ  5 p  ffiffi ffi p  ffiffi p  ffiffi ffi p  ffiffi 10 11  5 3 þ 3 11 þ 4 3 p  ffiffi p  ffiffi p  ffiffi p  ffiffi 2  5  3 7  2  5  3 7

3

 5  5

d f  h j

c f  i l

p  ffiffi  p  ffiffi 5 6 6  ffi ffi ffi p  p  ffiffi ffi 3 10  2 10 p  ffiffi p  ffiffi p  ffiffi 3 3 þ 4 3  5 3 p  ffiffi p  ffiffi p  ffiffi 10 3  3 3  12 3

See  Example  Example 6

p  ffiffi ffi  7p  ffi2ffi  4p  ffi10ffi ffi p  ffiffi ffi p  ffiffi p  ffiffi ffi p  ffiffi 3 15 þ 3 2 þ 4 15 þ 5 2 p  ffiffi p  ffiffi p  ffiffi p  ffiffi 4 6  3 3  2 6  5 3 p  ffi13ffi ffi þ 8p  ffi7ffi  7p  ffi13ffi ffi þ 3p  ffi7ffi p  ffiffi ffi p  ffiffi p  ffiffi ffi 4 10  3  5  4 10

c

i

3

p  ffiffi p  ffiffi 11 2  8 2 p  ffiffi ffi p  ffiffi ffi 5 17  5 17 p  ffiffi p  ffiffi p  ffiffi 5 62 64 6 p  ffiffi ffi p  ffiffi ffi p  ffiffi ffi 8 10  5 10 þ 3 10 b 11 10

g

p  ffiffi ffi p  ffiffi ffi ffiffi p  ffiffi p ffi ffi p  ffiffi ffi p ffi ffi  3 125 ¼  5 4 p   5  3 25  5  ffiffi 3  5p  ffi 5ffi ¼  5 p  ffi2ffi  5 p   ffiffi ¼ 10 p  ffi 5ffi 15  5 ¼

f  5 20

a 5 3 e

3

p  ffiffi  3p  ffi 5ffi ¼  5p  ffi 5ffi

b 8  5

Adding and subtracting surds

Exercise 1-03

2

p  ffiffi  4p  ffi2ffi þ 5p  ffi6ffi p  ffiffi ffi p  ffiffi ffi ffiffi 5 20  3 125

a 5 11

a

MAT10NAPS10094

6

Simplify each expression.

1

Surds code puzzle

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

p  ffi3ffi þ p  ffi12ffi ffi p  ffiffi A 5 3 p  ffiffi p  ffiffi ffi ffiffi 4  5  2 125 p  ffi ffiffi A 6  5

9780170194662

p  ffiffi

B

p  ffi15ffi ffi

C 2 6

B

p  ffi 5ffi

C

p  ffi45ffi ffi

p  ffiffi

D 3 3 D

46p  ffi 5ffi

11

Chapte Chap ter r 1 2345678910111213141516 Surds

Stage 5.3

4

Simplify each expression.

p  ffi8ffi þ p  ffi32ffi ffiffi b  ffi ffi p  p   ffi ffi ffi e 3 6 þ 24 f   ffi ffi p  p   ffi ffi ffi i 3 2 þ 18 j p   ffi ffi ffi p   ffi ffi m 5 3 þ 2 27 n p  p   ffi ffi ffi  ffi ffi ffi q 3 63  2 28 r p   ffi ffi ffi  ffi ffi ffi p  p  ffi6ffi ffi 75 p  ffiffi ffi ffiffi 5 ffiffi 27 u p  ffi 27 þ  54 þ 243 x a

Worksheet Multiplying and dividing surds

p  ffi108 p  ffi20ffi ffi  p  ffi80ffi ffiffi p  ffi28ffi ffi  p  ffi63ffi ffi ffi ffi ffiffi  p  ffi27ffi ffi c d p  ffiffi ffi p  ffiffi ffi p  ffi40ffi ffi  p  ffi90ffi ffi p  ffiffi p  ffiffi ffi ffiffi 2  5 þ 125 g h 5 11 þ 99 p  ffi200 ffi ffi ffiffi  7p  ffi2ffi p  ffi27ffi ffi þ 5p  ffi3ffi p  ffi 50ffi ffi þ p  ffi32ffi ffi k l p  ffiffi ffi p  ffiffi ffi p  ffiffi ffi p  ffiffi ffi ffiffi p  ffiffi ffi p  ffiffi ffi ffiffi 3 20  245 o 7 12  5 48 p 4 27 þ 2 243  ffi ffi ffi ffi ffi  ffi ffi p  p  p  p  ffiffi ffi ffiffi t  4p  ffi 50ffi ffi þ 3p  ffi18ffi ffi  ffi ffi ffi 2 98 þ 3 162 s  5 6 þ 2 150 p  ffiffi ffi ffiffi  2p  ffi252 ffi ffi ffiffi w p  ffi32ffi ffi þ p  ffi8ffi þ p  ffi12ffi ffi 3 ffiffi 112 v p  ffi p  ffiffi ffi p  ffiffi z 3p  ffi96ffi ffi  2p  ffi150 ffi ffi ffiffi þ p  ffi24ffi ffi 98  3 20  2 8 y

1-04  Multiplying and dividing surds

MAT10NAWK10095 Puzzle sheet

Summary 

Surds MAT10NAPS00043 Technology worksheet

The square root of products and quotients

  p ffi ffi ffi ¼ p  ffiffi p  ffiffi r  ffiffi p  ffiffi ¼ p  ffiffi

For  x  >  0 and  y  >  0:

Excel worksheet: Simplifying surds quiz MAT10NACT00019

 xy

 x

 x  y

 x  y

3

 y

Technology worksheet Excel spreadsheet: Simplifying surds MAT10NACT00049

Example

7

Simplify each expression. a d

p  ffi7ffi p  ffi 5ffi p  ffi90ffi ffi 12p   ffiffi ffi 3 10

b

3

e

Solution a

p  ffi7ffi p  ffi 5ffi ¼ p  ffi35ffi ffi p  ffiffi

p  ffiffi

f 

4

3

p  ffiffi p  ffiffi ffi p  ffiffi ffi 6 14 ¼ 84 p  ffiffi p  ffiffi ffi ¼ p 4 ffiffi ffi 21 ¼ 2 21 p  ffiffi ffi p  ffiffi 12 90 p   ffi ffi ffi ¼ 4 9 3 10 ¼4 3 ¼ 12 p  ffiffi ffi p  ffiffi p  ffiffi ffi p  ffiffi 5 27 3 6 ¼  5 3 p  ffiffi ffi ffiffi 27 6 ¼ 15 162 p  ffiffi ffi p  ffiffi ¼ 15 p 81 2  ffi ffi ¼ 15 p  ffi9ffi 2 ¼ 135 2 3

3

p  ffiffi p  ffiffi ¼ 4 10 3 3 ¼ 40 3 ¼ 120 p  ffi 54ffi ffi p  ffiffi ffi p  ffiffi  ffi2ffi  54 ð 2Þ ¼  p  p  ffiffi ffi ¼ p  ffi27ffi p  ffiffi ¼  p 9 ffiffi 3 ¼ 3 3 3

3

3

d

3

4

3

12

p  ffiffi p  ffiffi p  ffiffi ffiffi p  ffiffi 5 27 3 6

c 4 3 3 10 3

3

b

3

c 4 3 3 10 3

e

p  ffi6ffi p  ffi14ffi ffi p  ffi 54ffi ffi p  ffi2ffi 

 

3 3

f 

3

3

3

3

3

3

3

9780170194662

NEW CENTURY MATHS ADVANCED for the

Example

A u s tr a l i an C u r ri c u l um

10 10A

þ

Stage 5.3

8

p  ffiffi p  ffiffi ffi p  ffiffi

5 2 3 4 12 Simplify  : 10 8

Solution

p  ffiffi p  ffiffi ffi 20p  ffi24ffi ffi p  ffiffi p  ffiffi ¼ 10p  ffi8ffi ¼ 2 3

 5 2 3 4 12 10 8

Exercise 1-04 1

Simplify each expression.

p  ffi10ffi ffi p  ffi3ffi a p  ffi2ffi p  ffi18ffi ffi d p  ffiffi ffi p  ffiffi g 5 10 3 3 p  ffiffi  5p  ffi6ffi   j 2 3 p  ffiffi p  ffiffi m 7 2 4 8 p  ffiffi ffi p  ffiffi ffi p 3 18  5 12 p  ffiffi p  ffiffi ffi s 8 3 3  54 p  ffiffi ffi p  ffiffi v  5 20 3 8 3

3

3

3

3

3

3

3

2

p  ffi24ffi ffi p  ffi8ffi 4

p  ffiffi ffi p  ffiffi ffi p  ffiffi ffi p  ffiffi g 2 24 4 6 p  ffi10ffi ffi  20p   j 4  ffi 5ffi p  ffiffi ffi m 12 14 6 d 10  54 4 5 27 4

4

p  ffiffi ffi p  ffiffi ffi

p 5 60 4 15

p  ffiffi ffi p  ffiffi

s 12 63 4 3 7

e h k n q t  w

b e h k n q t 

Simplify the expressions below.

 ffi6ffi p  ffi11ffi ffi p  ffip  ffi p  ffi 5ffi   5 p  ffiffi p  ffiffi p  2 7  5 3  ffiffi p  ffi27ffi ffi 4 3 p  ffi18ffi ffi 8p  ffi3ffi p  ffiffi ffi 2p  ffi99ffi ffi  3 44 p  ffiffi ffi p  ffi27ffi ffi p  8 32  ffiffi ffi p  ffiffi ffi 7 18 3 24 3

3

3

9780170194662

3

c f  i

3

l

3

o

3

3

3

p  ffi30ffi ffi p  ffi 5ffi  3p  ffi98ffi ffi 6p  ffi14ffi ffi ffi ffi ffiffi p  ffi128 p  ffi2ffi p  ffiffi ffi p  ffiffi 36 24 9 8 p  ffiffi 3 2 12 p  ffiffi p  ffiffi 6 8 3 2 p  ffi 50ffi ffi 8p   ffiffi ffi ffiffi 2 200 4

4

4

4

p  ffiffi p  ffiffi p  ffi7ffi p  ffi7ffi b p ffi p  ffi xffi p ffi p ffi e  x p  ffiffi p  ffi6ffi : Select the correct answer  A, B, C  or  D. Simplify 3 2 p  ffiffi p  ffiffi A 6 B 6 2 C 6 3 63 6 a d 5  y 3 3  y

4

b

Simplify each expression. a

3

Multiplying Multiplyin g and dividing surds

3

3

r u x

p  ffi 5ffi p  ffi8ffi p  ffiffi p  ffiffi 8 3 6 3 p  ffiffi p  ffiffi 7  5 4  5 p  ffi 5ffi 4p  ffi10ffi ffi 3p   ffiffi p  ffiffi 10 2 2 8 p  ffiffi p  ffiffi ffi 5 8 4 40 p  ffi90ffi ffi p  ffi72ffi ffi p  ffiffi ffi p  ffiffi ffi 3 48 2 12

 Example 7 See  Example

3

3 3

3

3

3 3

3

p  ffiffi ffi p  ffiffi p  ffiffi ffi p  ffiffi p  ffiffi ffi p  ffiffi 15 18 3 6

40 14 c  5 2 7 18 f  2 i l o r u

4

p  ffiffi ffi p  ffiffi

16 30 4 8  5

p  ffi80ffi ffi 4p  ffi 5ffi ffiffi 42p  ffip  ffi 54 ffi 6 3 ffi ffi ffiffi p  ffiffi p  ffi243 6 3 4

4

p  ffiffi p  ffi3ffi p  ffiffi ffi p  ffiffi a

c 2 33 f  a2 3

p  ffiffi

D 12 2

13

Chapte Chap ter r 1 2345678910111213141516 Surds

Stage 5.3

p  ffiffi ffi p  ffiffi

Simplify 20 10 4 5 2:  Select  A,  B,  C  or  D .

5

p  ffiffi

A 4  5  Example 8 See  Example

p  ffiffi

B 15  5

C 10

Simplify each expression.

6

p  ffiffi p  ffiffi p  ffiffi ffi p  ffi 5ffi 4 p  ffiffi ffi p  ffiffi ffi 2 15  5 27

p  ffiffi ffi p  ffiffi p  ffiffi ffi p  ffiffi ffi ffi ffi p  ffiffi ffi 10 p  686 3 ffiffi ffi12  ffi ffi ffi p   5 28 18

3  5 3 4 2 a 3 40

3 12 3 8 6 b 4 27

d

e

3

D 20

3

f 

3

Mental ski killls 1

p  ffiffi p  ffiffi ffi p  ffiffi ffi p  ffi80ffi ffi 3p  ffi2ffi 8 p   ffiffi p  ffiffi 4  5 6 8

5 8 3 2 90 c 10 24 3

3

Maths without calcu cula lattors

Percentage of a quantity Learn these commonly-used percentages and their fraction equivalents. Percentage Fraction

50% 1 2

25% 1 4

1 2 .5 % 1 8

7 5% 3 4

20% 1  5

33 13 % 1 3

10% 1 10

66 23 % 2 3

Now we will use them to find a percentage of a quantity. 1

Study each example. 1 1 a 20% 3 25 b 50% 3 120 3 25 3 120  5 2 60  5 3 1 1 d 75% 3 56 e 33 % 3 27 3 60 3 27 4 3 3 1 9 3 60 33 4 15 3 3 45

¼ ¼ ¼  ¼ ¼ ¼

2

14

¼ ¼ ¼ ¼



¼ 18 ¼4 2 60 ¼ 3 ¼ 31 ¼ 20 ¼ 40

c 12: 5  5% 3 32

3

32

2 f  66 % 3 3

3

60

3

3



60

3

2

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% 3 400

i 75% 3 24

j

k 25% 3 160

l

m   12.5% 3 88

n 66 23 % 3 21

o 20% 3 60

p 75% 3 180

33 13 % 3 45

 

10% 3 650

9780170194662

NEW CENTURY MATHS ADVANCED for the

A u s tr a l i an C u r ri c u l um

10 10A

1-05  Binomial products involving surds

þ

Stage 5.3

Surd expressions involving brackets can be expanded in the same way as algebraic expressions of  the form  a (b c) and (a b)(c d ). ).

þ

þ

Example

þ

9

Expand and simplify each expression. a

p  ffiffi ffi  p  ffiffi ffi  5p  ffi2ffi 

p  ffi3ffi p  5ffi ffi þ p  ffi7ffi 

b 2 11 3 11

Solution a

3

b 2 11 3 11 – 5 2

 5 + 7

= 3 ×  5 + 3 × 7 = 15 + 21

Example

= 2 11 × 3 11 – 2 11 × 5 2 = 6 × 11 – 10 × 22 = 66 – 10 22

10

Expand and simplify each expression. a

p  ffi7ffi þ p  ffi 5ffi 3p  ffi2ffi  p  ffi3ffi 

b

Solution

3  2p  ffi10ffi ffiffi p  5ffi ffi  3p  ffi2ffi 

 p  ffiffi p  ffiffi  p  ffiffi p  ffiffi  p  ffiffi  p  ffiffi p  ffiffi  p  ffiffi  p  ffiffi p  ffiffi  7 þ  5 3 2  3 ¼ 7 3 2  3 þ  5 3 2  3 a p  ffiffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi ¼ p 7 ffiffi ffi 3 p  2 7 p  ffiffi ffi 3 þp  ffiffi 5ffi 3 2   5 3  ffi ffi ffi ¼ 3 14  21 þ 3 10  15  p  ffiffi ffi p  ffiffi p  ffiffi  p  ffiffi p  ffiffi  p  ffiffi ffi p ffi ffi p  ffiffi   5  3 2 ¼ 3  5  3 2  2 10  5  3 2 b 3  2 10 p  ffiffi p  ffiffi ffi p  ffiffi p  ffiffi ffi p  ffiffi p  ¼ 3p  ffiffi  ffi 5ffi p  ffi3ffi 3 p  ffi2ffi ffi 2 p  10  5 þ 2 10 3 2  ffi ffi ffi ¼ 3  5  9 2  2 50 þ6 20  p  ffiffi p  ffiffi ¼ 3p  ffi 5ffi  9 2  2 5 2 þ 6 2p  ffi 5ffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi ¼ 3 p   5  9 2  10 2 þ 12  5  ffiffi p  ffiffi ¼ 15  5  19 2 3

3

3

3

3

3

3

3

Summary  (a (a (a

2

2

2

2

2

2

þ b) ¼ a þ 2ab þ b  b) ¼ a  2ab þ b þ b)(a  b) ¼ a  b

9780170194662

2

2

15

Chapte Chap ter r 1 2345678910111213141516 Surds

Stage 5.3

Example

11

Expand and simplify each expression. a c

p  ffi7ffi  p  ffi 5ffi  p  ffi 5ffi  p  ffi2ffi p  5ffi ffi þ p  ffi2ffi  2

2p  ffi3ffi þ 3p  ffi 5ffi  3p  ffi11ffi ffi þ 43p  ffi11ffi ffi  4 2

b d

Solution

 p  ffiffi  p  ffiffi  ¼ p  ffiffi   p  ffiffi p  ffiffi þ p  ffiffi  7  5 7 2 7  5  5 a p  ffiffi ffi ¼ 7  2 p 35 þ 5  ffi ffi ffi ¼ 12  2 35  p  ffiffi p  ffiffi   p  ffiffi  p  ffiffi p  ffiffi  p  ffiffi  b 2 3 þ 3  5 ¼ 2 3 þ 2 2 3 3  5 þ 3  5 ¼ ð4 3Þ þp  ffi12ffi ffi p  ffi15ffi ffi þ ð9  5Þ ¼ 12 þ 12p  ffi15ffi ffi þ 45 ¼ 57 þ 12 15  p  ffiffi p  ffiffi p ffi ffi p  ffiffi  p  ffiffi  p  ffiffi   5  2  5 þ 2 ¼  5  2 c 2

2

2

3

2

3

2

2

3

3

3

2

¼  5  2 ¼3

 p  ffiffi ffi  p  ffiffi ffi   p  ffiffi ffi  d 3 11 þ 4 3 11  4 ¼ 3 11  4 2

¼ ð9 ¼ 83

2

3

2

2

þ b) ¼ a þ 2ab þ b

Using (a

þ b)(a  b) ¼ a  b

3

11

2

2

2

Note that because of the ‘difference of two squares’, the answer is not a surd but a rational number.

Using (a

2

þ b)(a  b) ¼ a  b

2

Þ  16

p  ffi6ffi p  ffi2ffi  1 p  ffi2ffi p 3ffi ffi þ p  ffi7ffi  c p  ffiffi p ffi ffi p  ffiffi  p  ffiffi ffi  p  ffiffi  d e 3 2 2þ2 3 f   11 4   5 p  ffiffi  p  ffiffi p  ffiffi  p  ffiffi  p  ffiffi  g h 5  5 1 þ 3  5 i 3 2 4 2þ 3 p  ffi3ffi þ 2p  ffi 5ffi 5p  ffi2ffi þ p  ffi3ffi ? Select the correct answer  A, B, Which expression is equivalent equivalent to C or  D. p  ffiffi ffi p  ffiffi ffi p  ffiffi A 20 10 B 2 15 þ 5 6 p   ffi  ffi ffi p   ffi ffi ffi  ffiffi p   ffi ffi p  p  p  p   ffi ffi ffi  ffi ffi  ffi ffi þ þ þ þ þ þ 3 7 7 4 2 C 5 6 3 10 10 2 15 D 5  5 p  ffi 5ffi p 3ffi ffi þ p  ffi2ffi  p  ffi 5ffi 3p  ffi2ffi  p  ffi 5ffi  p  ffiffi  p  ffiffi  2 7 3 74

b

Expand and simplify simplify each expression. expression. a c e g

16

2

Using (a

Expand and simplify simplify each expression. expression. a

See  Example  Example 10

2

Binomial products involvin involving g surds

Exercise 1-05 1

2

 b) ¼ a  2ab þ b

3

2

 Example 9 See  Example

2

Using (a

p  ffi 5ffi  32p  ffi 5ffi þ p  ffi2ffi  7p  ffi3ffi þ 24p  ffi2ffi þ p  ffi3ffi  p  ffi7ffi þ 2p  ffi11ffi ffi 3p  ffi7ffi þ 4p  ffi11ffi ffi  6 þ 2p  ffi10ffi ffi 3p  ffi10ffi ffi  1

p  ffi7ffi  p  ffi3ffi p 7ffi ffi þ 2  p  ffiffi p  ffiffi  p  ffiffi p  ffiffi  d 3 2   5 5 2 þ 2  5  ffi ffi  ffi ffi p  p  p  p    ffi ffi    ffi ffi    f  5 3 2 2 4 3 3 2 p  ffi7ffi  2p  ffi 5ffi 3p  ffi 5ffi þ 2p  ffi7ffi  h b

 

9780170194662

NEW CENTURY MATHS ADVANCED for the

4 5

8

2

2

2

7

10 10A

 þ p  ffi7ffi  ? Select A, B, C or D. p  ffiffi D 32 þ 5p  ffi7ffi A 12 B 32 C 32 þ 10 7 Expand and and simplify simplify each expression. expression.  ffi7ffi þ p  ffi2ffi  c p  ffi 5ffi  2 p  p  p    ffi ffi  ffi ffi    p  ffiffi ffi   5  3 a b d 3 þ 10  p  ffiffi p  ffiffi  f  5p  ffi7ffi  2 g 3p  ffi2ffi þ 2p  ffi 5ffi  h 2p  ffi 5ffi þ p  ffi3ffi  e 5 2þ3 3 Expand and and simplify simplify each expression. expression. p  p   ffi ffi  ffi ffi p  p    ffi ffi   ffi ffi   p  ffiffi  p  ffiffi  3 2 3þ 2 a b 5þ 3 5 3  p  ffiffi  p  ffiffi  p  ffi 5ffi  p  ffi3ffi p  5ffi ffi þ p  ffi3ffi  c 6þ2 7 62 7 d p  ffi11ffi ffi  p  ffi10ffi ffi  p 11ffi ffi ffi þ p  ffi10ffi ffi   p  ffi ffiffi  p  ffiffi  e f  5 7 þ 3 5 7  3  p  ffiffi p  ffiffi  p  ffiffi p  ffiffi   p  ffiffi p  ffiffi  p  ffiffi p  ffiffi  g 3 2 þ  5 3 2   5 h 4 2  5 3 4 2 þ 5 3  p  ffiffi p  ffiffi  p  ffiffi p  ffiffi  Which Whic h expre expressio ssion n is equi equivalen valentt to 5 2  4 3 5 2 þ 4 3 ? Select  A, B ,  C  or  D. p  ffiffi p  ffiffi B 10p  ffi2ffi þ 10p  ffi6ffi C 2 A 25 2  16 3 D 26 Expand and and simplify simplify each expression. expression.  ffi ffi p     p  ffiffi p 2ffi ffi þ 5 a 3 7  5 b 5 24  p  ffiffi p  ffiffi p  5ffi ffi þ p  ffi7ffi   p  ffiffi  c 2 7 þ 3  5 d 4 3 þ 5  p  ffiffi p  ffiffi  p  ffiffi p  ffiffi   p  ffiffi ffi p  ffiffi  e 4 2þ 3 4 2 3 f  3 10  2 Which Whic h expre expressio ssion n is equi equivalen valentt to 5

2

6

A u s tr a l i an C u r ri c u l um

þ

Stage 5.3  Example 11 See  Example

2

2

2

2

2

2

2

2

Investigation: Making the denominator rational

p  ffiffi  1:4142; what is the value of  p  ffi1 ffi ? Fractions containing surds in the denominator are

If  2

2

difficult to work with. When approximating the value of  1 ;  it is difficult to mentally divide 2 by  1.4142.   1.4142. We can overcome this by making the denominator  rational  (that is,  not  a   a surd). denominator of a fraction by the 1 What happens when we multiply the numerator and denominator same number? 1 3 2: a   Simplify  2 2 2 approximate the value value of  ;  given that 2 1:4142: b Mentally approximate 2 2 using a calculator, calculator, that 1 c Check, using :  Why is this true? 2 2

p  ffiffi

p  ffiffi p  ffiffi p  ffiffi

p  ffiffi p  ffiffi ¼ p  ffiffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi

p  ffiffi p  ffiffi p  ffiffi ¼

p  ffiffi 

3 3 7  ? Why? it true true that 3 2 a Is it 7 7 7 3 3 7 :  Compare your answer with those of other students. b   Simplify  7 7 3 7 3 using a calculator, calculator, that c Check, using : 7 7

p  ffiffi ¼ p  ffi 5ffi 3p  ffi 5ffi p  ffi2ffi  3p   ffi2ffi p  ffi2ffi : d   Explain why   ffiffi ¼ p  2 p  ffi 5ffi 3p  ffi10ffi ffi  3p   ffi2ffi : e   Show that  ffiffi ¼ p  2

p  ffiffi

3

9780170194662

17

Chapte Chap ter r 1 2345678910111213141516 Surds

Stage 5.3

1-06  Rationalising the denominator

Worksheet Rationalising the denominator MAT10NAWK10201

p  ffiffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi p  ffiffi

3  5 7 Surd Su rdss of th thee fo form rm 1 ; 3 ; havee den denomi ominat nators ors tha thatt are irr irrati ationa onal.l. The These se expr express essions ions may  ; ; … hav  5 2 7 2 3 be re rewr writ itte ten n wi with th a ra ratio tiona nall de deno nomi mina nator tor by mu mult ltip iply lyin ingg bo both th the nu nume mera rator tor an and d de deno nomi mina nator tor by the su surd rd that th at ap appe pear arss in th thee de denom nomin inat ator. or. Th This is me metho thod d is ca calllled ed rat rationa ionalisi lising ng the deno denomin minator ator.

Example

12

Rationalise the denominator of each surd. 8 2 3 5 a b c 4 3 3  5 2

p  ffiffi

p  ffiffi p  ffiffi

p  ffiffi

Solution

d

p  ffi2ffi þ 1 p  ffi3ffi

p  ffi2ffi p  ffi2ffi  ffi2ffi ¼ 1  ffiffi ¼ p  ffiffi p  ffi2ffi because p  a p  p  ffiffi ¼ p  ffi2ffi 3 3  ffi2ffi ¼ 2 ; it is easier to approximate p  ffi32ffi by mentally Because p  mentally multiplying multiplying  3  by 1.4142 2 3 2

3 3 2 3 2 2

than by dividing 3 by 1.4142.

b

p  ffi3ffi p  ffiffi ¼ p  ffi3ffi p  ffi3ffi 4 3 4 p   ffiffi 5 3 ¼4 3 p  ffiffi  5 3 ¼ 12 5

5

p  ffiffi 8p  ffi2ffi p  ffi 5ffi p  ffiffi ¼ 3p  ffi 5ffi p  ffi 5ffi p  ffiffi ffi 8 10 ¼ 3  5 p  ffiffi ffi 8 10 ¼ 15

8 2 c 3  5

3

3

Exercise 1-06 See  Example  Example 12

1

2

3

3

3

Rationalising Rationalisin g the denominato denominatorr

p  ffiffi

p  ffiffi p  ffiffi

p  ffiffi

p  ffiffi

p  ffiffi

p  ffiffi

p  ffiffi

p  ffiffi p  ffiffi p  ffiffi

p  ffi3ffi  ffi 5ffi  ? Select A, B, C or D. Which surd is equivalent to p  2 p  ffi15ffi ffi p  ffi15ffi ffi  ffi ffi ffi p  A B 2 15 C 10

18

d

By rationalising rationalising the denominator, denominator, which surd surd is equivalent equivalent to 2 ? Select the correct answer 6 A,  B ,  C  or  D . 6 6 2 6 A 2 6 B C D 3 6 3 Rationalise the denominator denominator of each surd. surd. 1 1 1 3 2 1 a b c d e f  3 7 7 2 2 3 2 5 3 2 3 2 1 1 7 g h i j k l 2 3 4 7 3  5 3  5 4  5 2 6

p  ffiffi

3

3

p  ffi2ffi þ 1 p  ffi2ffi þ 1 p  ffi3ffi p  ffi3ffi ¼ p  ffi3ffi p  ffi3ffi p  ffiffi p  ffiffi ¼ 6þ 3

3

 

p  ffiffi

p  ffiffi

p  ffiffi p  ffiffi p  ffiffi

p  ffiffi p  ffiffi p  ffiffi

D

p  ffi 5ffi 9780170194662

NEW CENTURY MATHS ADVANCED for the

A u s tr a l i an C u r ri c u l um

p  ffi27ffi ffi  ffiffi ffi ? Select A, B, C or D. Which surd is equivalent to p  3 18 p  ffi2ffi p  ffi 5ffi 1

4

10 10A

þ

Stage 5.3

p  ffi6ffi

A

B C D 2 2 6 6 Rationalise the denominator denominator of each expression. expression. 1  5 5 3 2 3 2 1 a b c d  5 2 2 2 3 6 Simplify each expression, giving giving the answer answer with a rational rational denominator. 2 1 1 3 3 1 a b c  5 7 3 2 3 2 2

5

p  ffiffi  p  ffiffi

6

 ffiffi p  ffip  ffi

p  ffiffi p  ffiffi þ p  ffiffi

p  ffiffi þ p  ffiffi

p  ffiffi  p  ffiffi p  ffiffi

þp  ffip  ffiffi ffi

p  ffiffi  p  ffiffi

Power plus 1

a Is itit true that

1

1

3 3

 ffi2ffi  p   p  ffi2ffi  ? Explain.

p  ffiffi þ p  ffi2ffi ¼ p  3þ 2  ffi2ffi  3 1 p  ffiffi p  ffiffi : Is the denominator rational? b   Simplify  3þ 2 3 2 3

3

3

check that the value of your answer to part part b  is equal to the c Use a calculator to check value of  1 : 3 3

2 3

þ p  ffiffi

þ p  ffiffi p   ffiffip  ffiffi 

The conjugate of 3 2 is 3 2:  Find the conjugate of: 3 7 3 a 5 2 3 b 2 c  5 1 d The process shown in in question question 1  involves rationalising a surd with a binomial denominator. By first finding the conjugate of the denominator, rationalise the denominator of each expression below.

þ p  ffiffi

a

1

p  ffiffi 2 3

b

p  ffi2ffi p  ffi 5ffi þ 1

p  ffiffi  p  ffiffi

p  ffiffi þ

c

p  ffi7ffi 1 p  ffi3ffi

4

The largest largest cube that can fit into a sphere sphere must have have its eight vertices touching the surface of the sphere. Express the side length, s, of the cube in terms of the diameter,  D, of the sphere.

5

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

d

 5

2 2 3

þ p  ffiffi

 D

C 

 A

B

¼

6

Six stormwater pipes, each 2 mm in diameter, diameter, are stacked stacked as shown in the diagram. Find the exact height,  h , of  this stacking.

9780170194662

h

19

Chapter 1 review Language ge of maths n Langua Puzzle sheet Surds crossword MAT10NAPS10202

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

do you think think a rational number  has that name? 1 Why do between a rational number  and a  real number ? 2 What is the difference between 3 What is a  surd?

you simplify  a surd? 4 How do you 5   Why is p an example of an  irrational number  that is not a  surd?

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

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. Surds and irrational numbers

Simplifying surds

Adding and subtracting surds

Multiplying and dividing surds

Surds

Rationalising the denominator

20

Binomial products involving surds

9780170194662

Chapter 1 revision 1

Which one of the the following is a rational number? number? Select the correct answer A , B , C  or  D . A 1 52

2

3

B

p  ffiffi p  ffiffi

p  ffi72ffi ffi p  ffiffi ffi ffiffi e 3 150 p  ffiffi ffi i 7 48 p  ffiffi ffi m 2 44

e

j n

 5

225

c g k o

p  ffi200 ffi ffi ffiffi þ p  ffi18ffi ffi p  ffiffi ffiffi p  ffiffi ffi p  ffiffi ffi p  ffiffi ffi 7 32  27  2 98 þ 4 75 ffi ffi ffiffi  2p  ffi243 p  ffi800 ffi ffi ffiffi þ 3p  ffi72ffi ffi  2p  ffi27ffi ffi

d g  j

p  ffi3ffi p  ffi7ffi p  ffi 5ffi p  ffi11ffi ffi p  ffiffi ffi p  ffiffi 8 42 2 7 p  ffi18ffi ffi p  ffi3ffi p  ffi12ffi ffi 3

b

3

e

4

3

h k

p  ffi8ffi p  ffi 5ffi p  ffi72ffi ffi p  ffi12ffi ffi p  ffi125 ffi ffi ffiffi  5p  ffi 5ffi p  ffi6ffi p  ffi24ffi ffi p  ffi27ffi ffi 2p  ffi3ffi 3

4

4

3

3

d f 

c f  i l

Expand and simplify each expression. a c e g

3p  ffip  ffi ffiffi2ffi ð2p  ffip  ffi2ffi ffi 3Þp  ffiffi p  ffiffi ð3p  ffi 5ffi  2p  ffiffi 7Þð3 7 þ  5Þ ð 5 3 þp  ffiffi 2Þ2p  ffiffi ð3 þ 4 7Þð4 7  3Þ

Rationalise the d denominator enominator of each each surd. surd. 1 3 a b 10 2 d

p  ffiffi ffi 3 ffiffi p  4 3

9780170194662

e

p  ffiffi p  ffi3ffi 5p   ffiffi 3 2

1-01

p  ffiffi p  ffiffi 5þ 3

See  Exercise   Exercise

1-02

See  Exercise   Exercise

1-03

See  Exercise   Exercise

1-04

See  Exercise   Exercise

1-05

See  Exercise   Exercise

1-06

d 3  5

p  ffiffi ffi8ffi 3

h

p  ffi275 ffi ffi ffiffi p  ffiffi ffi ffiffi 4 288  ffi3125 ffi ffi ffi ffiffi 2 p  3  ffiffi ffi ffiffi 2 p  3

See  Exercise   Exercise

d h l

p  ffi128 ffi ffi ffiffi p  ffiffi ffi 5 45  ffiffi ffi 1 p  2

32

162

p  ffiffi þ p  ffi 50ffi ffi  2p  ffi125 ffi ffi ffiffi p  ffiffi ffi p  ffiffi ffi p  ffiffi ffi 4 45  3 63 þ 5 80 p  ffiffi ffi p  ffiffi ffi 7 44  2 99

b 3  5

Simplify each expression. a

7

f 

p  ffi98ffi ffi p  ffiffi ffi 7 28  ffi24ffi ffi 1 p  2  ffiffi ffi ffiffi 2 p 

g

1-01

D 2p

 56_ c 0: 5

Simplify each expression.

c

6

b

p  ffiffi

C 2  5

p  ffiffi ffi ffiffi

Simplify each surd.

a

5

3

Is each number rational ((R) R) or irrational (I)? 22 8 a b 7 81 e f  3 125

a

4

p  ffi9ffi

See  Exercise   Exercise

b d f  h

c f 

p  ffi6ffi p  ffi8ffi p  ffi98ffi ffi p  ffi7ffi p  ffi75ffi ffi p  ffiffi 3 3 p  ffiffi ffi p  ffi8ffi 4 90 7p  p  ffiffi ffi  ffiffi ffi  5 32 6 10 3

3

3

3

p  ffi10ffi ffi ð1  5p  ffi2ffi Þ ðp  ffip 7 ffiffi ffi 4Þ2p  ffiffi p  ffiffi p  ffiffi ð3p  ffi7ffi ffi 2 p   5Þð3 7 þ 2  5Þ ð 5 10  3  ffi2ffi Þð2p  ffi10ffi ffi þ 5p  ffi2ffi Þ 1 ffiffi p   5 7 p  ffi2ffi 4 þp   ffiffi 3 2

21