AS Maths Core 1 Which is the odd one out?
36
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)
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Transcript of AS Maths Core 1 Which is the odd one out?
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
