what goes in the box ? rationalise the denominator of the following expressions: time's up!
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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!
Rationalising Denominators
Aim: To be able to rationalise denominators of the form √a ;
(1 +/- √a) or (√a +/- √b)
Answer exam questions involving surds and rationalising
denominators
Know the square numbers up to 152
Know the cube numbers up to 63
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.
Quick fire square numbers and cube numbers MWB
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
Homework
Video&
Quiz
Length and Midpoint of Lines
Have a go
Make mathematical sentences out of the card sets.
See how many you can do?
See how complicated you can make them, but must be correct!
Mathsnet activities
moodle
Surds Activities:
True or false
Exercise Level 2
Make more sentences
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
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