# The Laws Of Surds. - ARN Maths The Laws Of Surds 144 = The above roots have exact values 12 6 36 = and are called rational 2 1.41 3 212.76 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 16 23 - 7 23 =10 2 10 3 +7 3 - 4 3 =9 23 =13 3 a b ab Examples 4 6 24 4 10 40

List the first 10 square numbers 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 = 4 x 3 =2 To simplify 12 we must split 12 into factors with at least one being a square number. Now simplify the square root. Have a go !

Think square numbers 45 = 9 x 5 = 35 32 = 16 x 2 = 42 72 = 4 x 18 = 2 x 9 x 2 =2x3x 2 = 62

Simplify the following square roots: (1) 20 = 25 (2) 27 (3) 48 = 33 = 43 (4) 75 (5) 4500 (6) 3200

= 53 = 305 = 402 Simplify : 1. 20 = 25 3. 1 1 = 2 2

2. 18 = 32 1 1 4. = 4 4 a a a Examples 4 4 4

13 13 13 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 denominator Fractions can contain surds: 2 3 5 4 7 3 2 3- 5

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 Rationalising Surds To rationalise the denominator multiply the top and bottom of the fraction by the square root you are trying to remove: 3 3 5 =

5 5 5 3 5 = 5 ( 5 x 5 = 25 = 5) Rationalising Surds Lets try this one : Remember multiply top and bottom by root you are trying to remove 3 3 7 3 7 3 7 =

= = 14 2 7 2 7 7 2 7 Rationalising Surds Rationalise the denominator 10 10 5 10 5 2 5 = = = 7 5 7 5 5 7 5 7 Rationalise the denominator of the following : 7 3

4 9 2 7 3 = 3 2 2 9 4 6 2 6 = 3 14 3 10

7 10 = 15 2 5 7 3 2 15 = 21 6 3 11 2 3 6 = 11 Conjugate Pairs. Multiply out :

1. 3 3= 2. 14 14 = 3. 3 14 12 +3 12 - 3 = 12- 9 = 3

Rationalising Surds Conjugate Pairs. 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- =1 4 multiplied out the surds When the brackets are ALWAYS cancel out and we end up seeing that the expression is rational ( no root sign ) Conjugate Pairs.

a b Examples a 7 3 b a b 7

3 =73=4 11 5 11 = 11 5 = 6 5 Rationalising Surds

Conjugate Pairs. Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: 2 5-1 2( 5 +1) = ( 5 - 1)( 5 +1) 2( 5 +1) 2( 5 +1) = = ( 5 5 - 5 + 5 - 1) (5 - 1) ( 5 +1) =

2 Rationalising Surds Conjugate Pairs. 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)

Rationalise the denominator in the expressions below : 5 ( 7-2) 5( 7 +2) = 3 3 =3 + 6 ( 3 - 2) Rationalise the numerator in the expressions below : 6 +4 = - 5 5 + 11 = - 6 7( 5 - 11) 6( 6 - 4)

12 7