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

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

3. Starter Questions = 6 = 12 = 3 = 2
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. 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.

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

7. 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

8. 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

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
= 53
= 305
= 402

10. = 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
= ¼
1 x 1
√4 √4
= ¼

11. Second Rule
Examples

12. Rationalising Surds 1 Numerator 2 Denominator Remember fractions –
Fractions can contain surds in the numerator, denominator or both:
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 = What is x?
Make an initial guess, maybe x = 3
Try it and then keep improving the guess
07/04/2019

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

26. Trial and Improvement There is an answer between 4.3 and 4.4
So x= 4.36 to 2 dp
x = 4.35 =
Too small
x = 4.36 =
Too big
07/04/2019

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