Starter:On your sheet the surds on the left are equal to the right hand

side.

Look at all 6 of them and make sure that you understand why.

Then use the small pieces in the plastic wallets to make

mathematical sentences.e.g. = 2

= x)

Operations with Surds

AS MathsCore 1

4

16

144

8

64

Which is the odd one out?

Try out this one

75

What square number goes into 75?

32575 325 35

Simplifying Surds

Have a go...Example

Example

Solutions

a b

c d

49 121

9

1

25

4

Example

7 11

9

1

3

1

25

4

5

2

SolutionsExample

a b

c d

77 554

233 82

e f2

32

12

3

49 7 54 20

3333 2739

16 4

2

32

416 4

1

12

3

2

1

4

1

Tarsia Time!

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

Worksheet A

Pick out the questions that you feel will benefit you.

Try the more challenging ones!

Tarsia Time!

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

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?

Rationalising Denominators

Aim: To be able to rationalise denominators of the form √a ;

(1 +/- √a) or (√a +/- √b)

Rule: Your final answer cannot contain a surd in the denominator!

18

25

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.

Rationalising Surds

18

25

29

5

23

5

Given the surd Conjugate looks like Hint

What’s a conjugate?

aa

ba ba

ba ba

aaaa 2

Rules for surds

The conjugate is the same as the original

Only the sign in between the

expression changes

We were at this step…

Now, let’s look back at that last example & rationalise the denominatorExample (Cont’d…)

We must multiply the TOP and BOTTOM by the

conjugate to rationalise the denominator!

(get the surd out of the bottom)

18

25

23

5

2

2

43

25

23

25

6

25

Express in the form , where c

and d are integers. Solution:

Rationalising SurdsExample

22

8

2dc

22

8

22

22

Multiply the top & bottom by the

conjugate!

2222224

2816

24

2816

2

2816

248

SUMMARY

To rationalise the denominator of a fraction of the form

. . . multiply the numerator and denominator by

To rationalise the denominator of a fraction of the form

. . . multiply the numerator and denominator by

qp

ba

qp

a

a

Exercises: Simplify the following by rationalising the denominators:

1. 3

13

3

3

3

3

1

Exercises: Simplify the following by rationalising the denominators:

1. 3

1

2. 18

2

3

3

3

3

3

1

29

2

23

2

2

2

23

2

6

221

3 3

2

Exercises: Simplify the following by rationalising the denominators:

1. 3

1

2. 18

2

3. 23

2

3

3

3

3

3

1

29

2

23

2

2

2

23

2

6

221

3 3

2

23

23

23

2

29

)23(2

7

)23(2

Have a go:Domino Trail

Moodle and Edpuzzle

then

What Goes In The Box ?

Rationalise the denominator of the following expressions:

3

7)1(

6

4)2(

103

14)3(

29

4)4(

37

52)5(

211

36)6(

3

37

3

62 15

107

9

22

21

152 11

63

Time's up!

Difference of 2 squares.

)63)(63( 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:

)63)(63( = 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.

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

35

2)1(

)35)(35(

)35(2

)(

)(

9-53+53-5×5

3+52=

)95(

)35(2

2

)35(

)23(

7)2(

)23)(23(

)23(7

)23(

)23(7

)23(7

In both of the above examples the surds have been removed from the denominator as required.

Have a go…

Hexagon Puzzle

What Goes In The Box ?Rationalise the denominator in the expressions below :

)27(

5)1(

)23(

3)2(

)452(

7)3(

633

)27(5

2

)25(7

End

Extension• Rationalise

• Hence write a process for rationalising a denominator with three surds.

532

1

Harder Surds

We met surds when solving quadratic equations.e.g. Find the roots of the equation

0122 xx

a

acbbx

2

42

Solution:

Using the formula for :02 cbxax

21x

Simplifying the surd: 22248

2

222x

)1(2

)1)(1(4)2(2 2 x

2

82 x

Harder Surds

We can also surds which are in the denominators of fractions.

2

1e.g.1 Write the expression in the form

p

p

Solution: Multiply the numerator and the denominator by : 2

2

1

2

1

2

2

22

2

2

2

A fraction is simplified if there are no surds in the denominator.

Harder Surds

203

2e.g.2 Simplify the expression

Solution: We first simplify the surd.

203

2

543

2

523

2

5

5

53

1

Multiply the numerator and the denominator by

5

1

1

15

5

Harder Surds

32

1

e.g.3 Write the expression in the form qp

))(( baba

Method: We know that 22 ba

So, )32)(32( 22 )3(2 34

1By multiplying the expression by the surd has disappeared.

)32( )32(

However, if we multiply the denominator by we must multiply the numerator by the same amount.

)32(

Harder Surds

32

1

32

1Solution:

34

32

32

3232

The process of removing surds from the denominator is called rationalising.

Harder Surds

SUMMARY

To rationalise the denominator of a fraction of the form

qp

ba

. . . multiply the numerator and denominator byqp