Week 9 - Surds

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Week 9 - Surds
03 January 2013
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Contents

• Simplifying a Surd
• Rationalising a Surd
• Conjugate Pairs
• Trial Improvement

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Starter Questions

• Use a calculator to find the values of

6
12
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2
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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

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Surds
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Adding Subtracting Surds
Note v2 v3 does not equal v5

• To add or subtract surds such as ?2, treat as a
  single object.
• Eg.

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Multiplying Surds

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

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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
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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
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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
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Starter Questions

• Simplify

2v5
3v2
v20
v18
¼
¼
1 x 1 2 2
1 x 1 v4 v4
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Second Rule
Examples
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Rationalising Surds

• Remember fractions
• Fractions can contain surds in the numerator,
  denominator or both

1 Numerator 2 Denominator
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Rationalising Surds

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

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Rationalising Surds

• Multiply top and bottom by the square root you
  are trying to remove

Multiply top and bottom by v5
Remember ?5 x ?5 ? 25 5 )
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Rationalising Surds

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

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Rationalising Surds

• Rationalise the denominator

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Rationalise the Denominator
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Conjugate Pairs - Starter Questions

• Multiply out

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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
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Conjugate Pairs - Third Rule

• Eg.

7 3 4
11 5 6
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Rationalising Surds

• Rationalise the denominator in the expressions
  below by multiplying top and bottom by the
  appropriate conjugate

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Rationalising Surds

• Another one ...

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Rationalising the Denominator

• Rationalise the denominator in the expressions
  below

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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

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Trial and Improvement
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Trial and Improvement

• There is an answer between 4.3 and 4.4
• So x 4.36 to 2 dp

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Session Summary

• Surds
• Simplifying Surds
• Rationalising Surds
• Conjugate Pairs
• Trail Improvement