number system 3 of 7

7/29/2019 Number System 3 of 7

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Mathematic KFP00105

Topic 1 : Number System

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TOPIC : 1.0 NUMBER SYSTEM

SUBTOPIC : 1.3 Surd

LEARNING : At the end of the lesson, student should be able to

OUTCOMES

(a) Explain the meaning of surd and its conjugate and to carry outalgebraic operations on surds.

(b) Solve equation involving surds.

CONTENT

A number expressed in terms of root sign is radical or a surd. For example, 24 = and 3273 =

are called radical expression. However, radical expressions in the form of 7 or 3 71 whichequal to irrational numbers are known as surds.

Some examples of surds are 2 , 5 and 7 . They cannot be evaluated exactly, i.e. no

approximation is precise. For example 414213562.12 = to ten significant figures. This is not

the exact value because 2 is irrational, it can never be represented exactly as a fraction or as

terminating decimals.

RULES OF SURDS

ab = a b a, b 0 PROPERTY 1

b

a=

b

aa, 0 , b 0 PROPERTY 2

a b + c b = (a + c) b PROPERTY 3

a b - c b = (a - c) b PROPERTY 4

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Mathematic KFP00105


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Additional rules

1) aaa =

2) a + a = 2 a Remark: ba+ ( a + b )

3) a b =b

a

4) ( a + b ) 2 = a + b+ 2 ab

5) ( a + b ) ( a - b ) = a2 b2

Example

Simplify:

a) 45 b) 24 c) 6 727 + d) 5 273

Solution

a) 45 = 59

= 59

= 3 5

b) 24 = 64

= 64

= 2 6

c) 6 727 + = (6 + 2) 7

= 8 7

d) 5 273 = 5 3 39

= 5 3 ( 39 )

= 5 3 3 3 = 2 3

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Multiplying Radicals

There are 2 types.

1) a x b = ab

2) ba x dc = bdac

Example

Multiply:

1. 3 6 x 5 7

2. 7 8 x 10 6

3. 3 (4 7 - 3 )

4. 8 2 (5 6 + 2 )

5. (2 3 +4 2 )(6 3 +2 2 )

Solution

1. 3 6 x 5 7 = 15 42

2. 7 8 x 10 6 = 70 48

= 70 316x

= 70(4 3 )

= 280 3

3. 3 (4 7 - 3 ) = 4 21 - 9

= 4 21 -3

4. 8 2 (5 6 + 2 ) = 40 12 + 8 4

= 40 34x + 8(2)

= 80 3 +16

5. (2 3 +4 2 )(6 3 +2 2 ) =12(3) + 4 6 + 24 6 + 8(2)

= 36 + 4 6 + 24 6 +16

= 52 + 28 6

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Example

Expand and simplify ( )38 ( )38 + .

Solution

( )38 ( )38 + = 8 ( )38 + - 3 ( )38 +

= 2)8( + 38 - 83 - (2)3

= 2)8( - ( 2)3

= 8 3 = 5

The conjugate of a + b is a b where ( a + b)( a b) = a b is a rational

number.

For example, ( 8 + 3 )( 8 - 3 ) = ( 8 ) 2 - ( 3 ) 2 = 8 3 = 5

Rationalising the denominator

When the denominator of a fraction contains a square root, we generally simplify the expression

by rationalising the denominator.

To rationalize a denominator, multiply the numerator and the denominator of the fraction thatwill result in the denominator to become a rational number.

If Denominator To Obtain Denominator

Contains the Factor Multiply by Free from surds

3 3 ( ) 33 2 =13 + 13 ( ) 21313 22 ==32 32 + ( ) 79232 22 ==

35 35 +

( ) ( ) 2353522

==

In rationalizing the denominator of a quotient, be sure to multiply both the numerator and the

denominator by the same expression.

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Example

Rationalise:

(a)3

5(b)

32

3(c)

27

1

(d)35

32

Solution

a)3

5=

3

5

3

3

=3

35.

(b)

32

3

3

3

32

3x=

)3(2

33=

2

3=

(c)27

1

= (

27

1

)

)27(

)27(

+

+

= 22 )2()7(

)27(

+

=249

27

+

=47

27 +.

(d)35

32

= (35

32

)(35

35

+

+)

= 2

2

)3(353525

)3(2310

+

+

=22

6310 +

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Mathematic KFP00105


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Example

Rationalise the denominator and simplify.

(a) 517

517

+

(b) 21

31

21

21

+

+

+

(c) 223

23+

+

Solution

(a)517

517

+

=517

517

+

)517(

)517(

=( )( )

22 )5()17(

517517

= 517

)5(175517)17( 22

+

=12

1752517 +

=12

85222

=12

)8511(2

=6

8511.

(b)21

21

21

21

+

+

+=

)21)(21(

)21)(21()21)(21(

+

+++

=[ ] ]

2

22

)2(221

)2(221)2(221

+

++++

=1

6

= -6

(c) 223

23+

+= 2

23

23

23

23+

+

+

+x

= 2)2(66)3(

)2(66)3(22

22

+

+

+++

= 21

625+

+

= 2625 ++

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Surd Equation

Example

Solve each of the following equation :(a) 0512 =x

(b) 13 +x + 1 =x

(c) x + 2+x = 2

Solution

(a) 0512 =x

512 =x

22 )5()12( =x

2x 1 = 252x = 26

x = 13

(b) 13 +x + 1 =x

13 +x =x 1

( 13 +x )2 = (x 1)2

3x + 1 =x2 2x + 1

x2 5x = 0

x(x 5) = 0

x = 0 orx = 5 but x 0

x = 5

(c) x + 2+x = 2

( x + 2+x )2 = 22

x + 2 x 2+x +x + 2 = 4

2 )2( +xx = 2 2x

xx 22 + = 1 x

(xx 2

2+ )

2 = (1 x)2

x2 + 2x = 1 2x +x2

4x = 1

x =4

1

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number system 3 of 7

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