Surds

SurdsSurds

Simplifying a Surd

Rationalising a Surd

Conjugate Pairs

5. 2

Starter QuestionsStarter Questions

Use a calculator to find the values of :

1. 36 = 6

= 12

= 3

= 2

2. 144

33. 8 44. 16

1.41 2.7636. 21

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1. To explain what a surd is and to investigate the rules for surds.

1.1. Learn rules for surds.Learn rules for surds.

The Laws Of Surds

1.1. Use rules to simplify surds.Use rules to simplify surds.

2

What is a SurdWhat is a Surd

36 = 6

= 12

144

1.41 2.763 21

The above roots have exact values

and are called rational

These roots do NOT have exact values

and are called irrational OR Surds

Adding & Subtracting Surds

Adding and subtracting a surd such as 2. It can

be treated in the same way as an “x” variable in algebra. The following examples will illustrate this point.

4 2 + 6 2

=10 2

16 23 - 7 23

=9 23

10 3 + 7 3 - 4 3 =13 3

Note :

√2 + √3 does not equal √5

First Rule

4 6 24

a b ab

4 10 40

List the first 10 square numbers

Examples

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Simplifying Square Roots

Some square roots can be broken down into a mixture of integer values and surds. The following examples will illustrate this idea:

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

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

Have a go !Think square numbers

What Goes In The Box ?

Simplify the following square roots:

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

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

= 25

= 33

= 43

= 53

= 305 = 402


Simplify :

1. 20 = 2√5

= 3√2

= ¼

2. 18

1 13.

2 2

1 14.

4 4 =

¼


1. To explain how to rationalise a fractional surd.

1.1. Know that √a x √a = a.Know that √a x √a = a.

The Laws Of Surds

2.2. To be able to rationalise To be able to rationalise the numerator or the numerator or denominator of a denominator of a fractional surd.fractional surd.

Second Rule

4 4 4

a a a

13 13 13

Examples

Rationalising Surds

You may recall from your fraction work that the top line of a fraction is the numerator and the bottom line the denominator.

2 numerator =

3 denominatorFractions can contain surds:

23

5

4 7

3 2

3 - 5

Rationalising Surds

If by using certain maths techniques we remove the surd from either the top or bottom of the fraction then we say we are “rationalising the numerator” or “rationalising the denominator”.

Remember the rule a a a

This will help us to rationalise a surd fraction

To rationalise the denominator multiply the top and bottom of the fraction by the square root you are

trying to remove:

3

53 5

=5 5

( 5 x 5 = 25 = 5 )

3 5=

5

Rationalising Surds

Let’s try this one :

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

3

2 73 7

=2 7 7

3 7=

2 73 7

=14

Rationalising Surds

10

7 510 5

=7 5 5

10 5=

7 52 5

=7

Rationalising Surds

Rationalise the denominator

What Goes In The Box ?

Rationalise the denominator of the following :

7

34

6

14

3 10

4

9 22 5

7 36 3

11 2

7 3=

32 6

=3

7 10=

15

2 29

2 15

=21

3 6=

11

3. 12 + 3 12 - 3


Multiply out :

1. 3 3 = 3

= 14

2. 14 14

= 12- 9 = 3

Conjugate Pairs.


1. To explain how to use the conjugate pair to rationalise a complex fractional surd.

1.1. Know that Know that (√a + √b)(√a + √b)(√a - √b) (√a - √b) = a - b= a - b

The Laws Of Surds

2.2. To be able to use the To be able to use the conjugate pair to conjugate pair to rationalise complex rationalise complex fractional surd.fractional surd.

Conjugate Pairs.

Conjugate Pairs.

Rationalising Surds

Look at the expression : ( 5 2)( 5 2) This is a conjugate pair. The brackets are identical

apart from the sign in each bracket .

Multiplying out the brackets we get :

( 5 2)( 5 2) = 5 5 - 2 5 + 2 5 - 4

= 5 - 4

= 1When the brackets are multiplied out the surds ALWAYS cancel out and we end up seeing that the expression is rational ( no root sign )

Looks something like the difference of two squares

Rationalise the denominator in the expressions below by multiplying top and bottom by the

appropriate conjugate:

2

5 - 12( 5 + 1)

=( 5 - 1)( 5 + 1)

2( 5 + 1)=

( 5 5 - 5 + 5 - 1)2( 5 + 1)

=(5 - 1)

( 5 + 1)=

2

Conjugate Pairs.

Rationalising Surds

Rationalise the denominator in the expressions below by multiplying top and bottom by the

appropriate conjugate:

7

( 3 - 2)7( 3 + 2)

=( 3 - 2)( 3 + 2)

7( 3 + 2)=

(3 - 2)=7( 3 + 2)

Conjugate Pairs.

Rationalising Surds

What Goes In The Box

Rationalise the denominator in the expressions below :

5

( 7-2)3

( 3 - 2)

Rationalise the numerator in the expressions below :

6 + 412

5 + 117

= 3 + 6

- 5=6( 6 - 4)

- 6=7( 5 - 11)

5( 7 + 2)=

3

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Surds