Surds Simplifying a Surd Rationalising a Surd Conjugate Pairs S5 Int2 Simplifying a Surd Rationalising a Surd www.mathsrevision.com Conjugate Pairs www.mathsrevision.com

Starter Questions = 6 = 12 = 2 = 2 S5 Int2 Use a calculator to find the values of : = 6 = 12 = 2 = 2 www.mathsrevision.com

The Laws Of Surds www.mathsrevision.com Learning Intention S5 Int2 Learning Intention Success Criteria To explain what a surd is and to investigate the rules for surds. Learn rules for surds. Use rules to simplify surds. www.mathsrevision.com www.mathsrevision.com

What is a Surd = 12 = 6 Surds The above roots have exact values S5 Int2 = 12 = 6 The above roots have exact values and are called rational These roots do NOT have exact values and are called irrational OR Surds www.mathsrevision.com

Adding & Subtracting Surds Note : √2 + √3 does not equal √5 Adding & Subtracting Surds S5 Int2 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. www.mathsrevision.com

First Rule List the first 10 square numbers S5 Int2 Examples List the first 10 square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 www.mathsrevision.com

Simplifying Square Roots S5 Int2 Some square roots can be broken down into a mixture of integer values and surds. The following examples will illustrate this idea: To simplify 12 we must split 12 into factors with at least one being a square number. 12 = 4 x 3 Now simplify the square root. = 2 3 www.mathsrevision.com

Have a go !  45  32  72 = 9 x 5 = 16 x 2 = 4 x 18 = 35 = 42 Think square numbers S5 Int2  45  32  72 = 9 x 5 = 16 x 2 = 4 x 18 = 35 = 42 = 2 x 9 x 2 = 2 x 3 x 2 = 62 www.mathsrevision.com

What Goes In The Box ? Simplify the following square roots: (2)  27 (3)  48 (1)  20 = 25 = 33 = 43 (6)  3200 (4)  75 (5)  4500 = 53 = 305 = 402 www.mathsrevision.com

= ¼ = ¼ Starter Questions = 2√5 = 3√2 Simplify : S5 Int2 www.mathsrevision.com

The Laws Of Surds www.mathsrevision.com Learning Intention S5 Int2 Learning Intention Success Criteria To explain how to rationalise a fractional surd. Know that √a x √a = a. 2. To be able to rationalise the numerator or denominator of a fractional surd. www.mathsrevision.com www.mathsrevision.com

Second Rule S5 Int2 Examples www.mathsrevision.com

Rationalising Surds S5 Int2 You may recall from your fraction work that the top line of a fraction is the numerator and the bottom line the denominator. Fractions can contain surds: www.mathsrevision.com

Rationalising Surds This will help us to rationalise a surd fraction S5 Int2 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 www.mathsrevision.com

Rationalising Surds S5 Int2 To rationalise the denominator multiply the top and bottom of the fraction by the square root you are trying to remove: ( 5 x 5 =  25 = 5 ) www.mathsrevision.com

Rationalising Surds Let’s try this one : S5 Int2 Let’s try this one : Remember multiply top and bottom by root you are trying to remove www.mathsrevision.com

Rationalising Surds Rationalise the denominator S5 Int2 www.mathsrevision.com

What Goes In The Box ? Rationalise the denominator of the following : www.mathsrevision.com

Starter Questions = 3 = 14 = 12- 9 = 3 Conjugate Pairs. Multiply out : S5 Int2 Multiply out : = 3 = 14 = 12- 9 = 3 www.mathsrevision.com

The Laws Of Surds www.mathsrevision.com Conjugate Pairs. S5 Int2 Learning Intention Success Criteria To explain how to use the conjugate pair to rationalise a complex fractional surd. Know that (√a + √b)(√a - √b) = a - b 2. To be able to use the conjugate pair to rationalise complex fractional surd. www.mathsrevision.com www.mathsrevision.com

Looks something like the difference of two squares Rationalising Surds Conjugate Pairs. S5 Int2 Look at the expression : This is a conjugate pair. The brackets are identical apart from the sign in each bracket . Multiplying out the brackets we get : = 5 x 5 - 2 5 + 2 5 - 4 = 5 - 4 = 1 When the brackets are multiplied out the surds ALWAYS cancel out and we end up seeing that the expression is rational ( no root sign ) www.mathsrevision.com

Third Rule Examples = 7 – 3 = 4 = 11 – 5 = 6 Conjugate Pairs. S5 Int2 www.mathsrevision.com

Rationalising Surds Conjugate Pairs. S5 Int2 Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: www.mathsrevision.com

Rationalising Surds Conjugate Pairs. S5 Int2 Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: www.mathsrevision.com

What Goes In The Box S5 Int2 Rationalise the denominator in the expressions below : Rationalise the numerator in the expressions below : www.mathsrevision.com