Surds

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Contents

Simplifying a SurdRationalising a SurdConjugate PairsTrial & Improvement

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Starter Questions

Use a calculator to find the values of :

= 6

= 12

= 3

= 2

36 1443 8 4 16

2 3 2141.1 76.2

What is a Surd ?

These roots have exact values and are called rational

These roots do NOT have exact values and are called irrational OR

= 6

= 12

Surds

36 144

2 3 2141.1 76.2

Adding & Subtracting Surds

To add or subtract surds such as 2, treat as a single object.

Eg.

Note :

√2 + √3 does not

equal √5

210

2624

239

2372316

3437310 313

Multiplying Surds

4 6 24

a b ab

4 10 40

•Eg

•List the first 10 square numbers•1, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100

Simplifying Surds

Some square roots can be simplified by using this rule - 12

To simplify 12 we must split 12 into factors with at least one being a square number.

= 4 x 3

Now simplify the square root.

= 2 3

Have a go -

You need to look for square numbers

45 = 9 x 5= 35

32= 16 x 2= 42

72= 4 x 18

= 2 x 9 x 2= 2 x 3 x 2= 62

Simplifying Surds

Simplify the following square roots :

(1) 20 (2) 27 (3) 48

(4) 75 (5) 4500 (6) 3200

= 25

= 33

= 43

= 53

= 305 = 402

Starter Questions

Simplify :

= 2√5

= 3√2

= ¼

= ¼

√20 √18

1 x 12 2

1 x 1√4 √4

Second Rule

4 4 4

a a a

13 13 13

Examples

Rationalising SurdsRemember fractions –

Fractions can contain surds in the numerator, denominator or both:

1 Numerator

2 Denominator

53

345

5323

Rationalising Surds

a a a

Removing the surd form numerator or denominator

Remember the rules

This will help us to rationalise a surd fraction

a b ab

Rationalising Surds

Multiply top and bottom by the square root you are trying to remove:

Remember5 x 5 = 25 = 5 )

53 Multiply top and bottom by √5

55

53

553

Rationalising Surds

Remember multiply top and bottom by root you are trying to remove

7

7723

723

1473

7273

Rationalising Surds

Rationalise the denominator

55

5710

5710

752

57510

Rationalise the Denominator

7 3= 32 6= 3

7 10= 15

2 29

2 15= 21

3 6= 11

37

64

10314

294

3752

21136

Conjugate Pairs - Starter Questions

Multiply out :

= 3= 14

33

1414

)312)(312(

Conjugate Pairs.

This is a conjugate pair. The brackets are identical apart from the

sign in each bracket .Multiplying out the brackets we get :

When the brackets are multiplied out the surds ALWAYS cancel out leaving a rational expression

5 x 5 - 2 5 + 2 5- 4 = 5 - 4

= 1

(5 + 2)(5 - 2)

Conjugate Pairs - Third Rule

7 3 7 3

a b a b a b

11 5 11 5

= 7 – 3 = 4

= 11 – 5 = 6

Eg.

Rationalising Surds

Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate:

152

1515

152

4)15(2

15)15(2

2

15

Rationalising Surds

Another one ...

)23(7

)23()23(

)23(7

)23(7)23()23(7

Rationalising the Denominator

Rationalise the denominator in the expressions below :

= 3 + 6

- 5=6( 6 - 4)- 6=7( 5 - 11)

5( 7 + 2)= 3)27(5 )23(

3

1246

7115

Trial and Improvement

A method which involves making a guess and then systematically improving it until you reach the answer

Eg. x 2 + 5 = 24 What is x?Make an initial guess, maybe x = 3Try it and then keep improving the guess

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Trial and ImprovementTry Working Out

x2 + 5 Result

x = 3 32 + 5 = 14 Too small

x = 4 42 + 5 = 21 Too small

x = 5 52 + 5 = 30 Too big

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x = 4.5 4.52 + 5 = 25.25 Too big

x = 4.4 4.42 + 5 = 24.36 Too big

x = 4.3 4.32 + 5 = 23.49 Too small

Trial and Improvement

There is an answer between 4.3 and 4.4

So x= 4.36 to 2 dp

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x = 4.35 4.352 + 5 = 23.9225 Too small

x = 4.36 4.362 + 5 = 24.0096 Too big

Session Summary

SurdsSimplifying SurdsRationalising SurdsConjugate PairsTrail & Improvement

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