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Week 9 - Surds

Week 9 - Surds. 15 March 2014. Contents. Simplifying a Surd Rationalising a Surd Conjugate Pairs Trial & Improvement. Starter Questions. 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

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Week 9 - Surds

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  1. Week 9 - Surds 15 March 2014

  2. Contents • Simplifying a Surd • Rationalising a Surd • Conjugate Pairs • Trial & Improvement

  3. StarterQuestions • Use a calculator to find the values of : = 6 = 12 = 3 = 2

  4. 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

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

  6. Multiplying Surds • Eg • List the first 10 square numbers • 1, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100

  7. Simplifying Surds • 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

  8. Have a go - • You need to look for square numbers  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

  9. 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 = 305 = 402 = 53

  10. StarterQuestions • Simplify : = 2√5 = 3√2 √20 √18 1 x 1 2 2 = ¼ 1 x 1 √4 √4 = ¼

  11. Second Rule Examples

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

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

  14. 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 )

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

  16. Rationalising Surds • Rationalise the denominator

  17. Rationalise the Denominator

  18. Conjugate Pairs - Starter Questions • Multiply out : = 3 = 14

  19. 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

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

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

  22. Rationalising Surds • Another one ...

  23. Rationalising the Denominator • Rationalise the denominator in the expressions below :

  24. 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

  25. Trial and Improvement

  26. Trial and Improvement • There is an answer between 4.3 and 4.4 • So x= 4.36 to 2 dp

  27. Session Summary • Surds • Simplifying Surds • Rationalising Surds • Conjugate Pairs • Trail & Improvement

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