Week 9 - Surds 07 April 2019 07/04/2019

Contents Simplifying a Surd Rationalising a Surd Conjugate Pairs Trial & Improvement 07/04/2019

Starter Questions = 6 = 12 = 3 = 2 Use a calculator to find the values of : = 6 = 12 = 3 = 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 = 12 = 6 Surds

Adding & Subtracting Surds Note : √2 + √3 does not equal √5 Adding & Subtracting Surds To add or subtract surds such as 2, treat as a single object. Eg.

Multiplying Surds Eg List the first 10 square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Simplifying Surds = 2 3 12 = 4 x 3 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 -  45  32  72 You need to look for square numbers = 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

Simplifying Surds Simplify the following square roots : (1)  20 (2)  27 (3)  48 (4)  75 (5)  4500 (6)  3200 = 25 = 33 = 43 = 53 = 305 = 402

= ¼ = ¼ Starter Questions √20 = 2√5 √18 = 3√2 1 x 1 2 2 1 x 1 √4 √4 Simplify : √20 = 2√5 √18 = 3√2 1 x 1 2 2 = ¼ 1 x 1 √4 √4 = ¼

Second Rule Examples

Rationalising Surds 1 Numerator 2 Denominator Remember fractions – Fractions can contain surds in the numerator, denominator or both: 1 Numerator 2 Denominator

Rationalising Surds Removing the surd form numerator or denominator Remember the rules This will help us to rationalise a surd fraction

Rationalising Surds Multiply top and bottom by the square root you are trying to remove: Multiply top and bottom by √5 Remember 5 x 5 =  25 = 5 )

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

Rationalising Surds Rationalise the denominator

Rationalise the Denominator

Conjugate Pairs - Starter Questions Multiply out : = 3 = 14

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 + 2)(5 - 2) 5 x 5 - 2 5 + 2 5 - 4 = 5 - 4 = 1

Conjugate Pairs - Third Rule Eg. = 7 – 3 = 4 = 11 – 5 = 6

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

Rationalising Surds Another one ...

Rationalising the Denominator Rationalise the denominator in the expressions below :

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 = 3 Try it and then keep improving the guess 07/04/2019

Trial and Improvement Try Working Out x2 + 5 Result x = 3 32 + 5 = 14 Too small x = 4 42 + 5 = 21 x = 5 52 + 5 = 30 Too big x = 4.5 4.52 + 5 = 25.25 Too big x = 4.4 4.42 + 5 = 24.36 x = 4.3 4.32 + 5 = 23.49 Too small 07/04/2019

Trial and Improvement There is an answer between 4.3 and 4.4 So x= 4.36 to 2 dp x = 4.35 4.352 + 5 = 23.9225 Too small x = 4.36 4.362 + 5 = 24.0096 Too big 07/04/2019

Session Summary Surds Simplifying Surds Rationalising Surds Conjugate Pairs Trail & Improvement 07/04/2019