The Laws Of Surds

2

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 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

4 6 24

a b ab

4 10 40

List the first 10 square numbers

Examples

1, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100

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

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

4 4 4

a a a

13 13 13

Examples

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

a a a

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

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

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.

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 )

7 3 7 3

a b a b a b

11 5 11 5

Examples

Conjugate Pairs.

= 7 – 3 = 4

= 11 – 5 = 6

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

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