5. AS Maths Core 1

6. Which is the odd one out?

7. Try out this one What square number goes into 75?

8. Example

9. a b c d Example

10. a b c d e f

11. Using the skills from this lesson and last, complete the domino trail in groups.

14. Worksheet A Pick out the questions that you feel will benefit you. Try the more challenging ones!

15. Using the skills from this lesson and last, complete the hexagon puzzle in groups.

16. Starting You Thinking Answer the following giving answers a simplified surds. Find the distance between the coordinates (2,5) and (-2,3) In an isosceles triangle two sides are 20 and the other is 10. What is the height? What is its area?

17. Rationalising Denominators Aim: To be able to rationalise denominators of the form √a ; (1 + / - √a) or (√a + / - √b)

18. Let’s take a look at an example where we will be left with a radical in the denominator… Example Since there is a radical in the denominator (bottom) of the fraction, we must multiply by the conjugate in order to rationalise the denominator.

19. Given the surdConjugate looks likeHint Rules for surds The conjugate is the same as the original Only the sign in between the expression changes

20.  We were at this step… We must multiply the TOP and BOTTOM by the conjugate to rationalise the denominator! (get the surd out of the bottom)

21.  Express in the form, where c and d are integers.  Solution: Multiply the top & bottom by the conjugate!

22. SUMMARY

23. Exercises: Simplify the following by rationalising the denominators: 1.

24. Exercises: Simplify the following by rationalising the denominators: 1. 2.

25. Exercises: Simplify the following by rationalising the denominators: 1. 2. 3.

26. Have a go: Domino Trail Moodle and Edpuzzle then

27. What Goes In The Box ? Rationalise the denominator of the following expressions: Time's up!

28. Difference of 2 squares. This is a conjugate pair. The brackets are identical apart from the sign in each bracket. Now observe what happens when the brackets are multiplied out: =  3 X  3 - 6  3 + 6  3 - 36 = 3 - 36 = -33 When the brackets are multiplied out the surds cancel out and we end up seeing that the expression is rational. This result is used throughout the following slide.

29. Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: In both of the above examples the surds have been removed from the denominator as required.

30. Have a go… Hexagon Puzzle

31. What Goes In The Box ? Rationalise the denominator in the expressions below : End Extension Rationalise Hence write a process for rationalising a denominator with three surds.

32. Harder Surds We met surds when solving quadratic equations. e.g. Find the roots of the equationSolution: Using the formula for : Simplifying the surd:

33. Harder Surds We can also surds which are in the denominators of fractions. e.g.1 Write the expression in the form Solution: Multiply the numerator and the denominator by : A fraction is simplified if there are no surds in the denominator.

34. Harder Surds e.g.2 Simplify the expression Solution: We first simplify the surd. Multiply the numerator and the denominator by

35. Harder Surds e.g.3 Write the expression in the form Method: We know that So, By multiplying the expression by the surd has disappeared. However, if we multiply the denominator by we must multiply the numerator by the same amount.

36. Harder Surds Solution: The process of removing surds from the denominator is called rationalising.

37. Harder Surds SUMMARY  To rationalise the denominator of a fraction of the form... multiply the numerator and denominator by