Cambridge International Mathematics

142 Exponents and surds (Chapter 6)

When an expression involves division by a surd, we can write the expression with aninteger denominator

which doesnotcontain surds.

If the denominator contains a simple surd such as

p

athen we use the rule

p

a£

p

a=a:

For example:

6

p

3

can be written as

6

p

3

£

p

3

p

3

since we are really just multiplying

the original fraction by 1.

6

p

3

£

p

3

p

3

then simplifies to

6

p

3

3

or 2

p

3.

Example 27 Self Tutor

Express with integer denominator: a

7

p

3

b

10

p

5

c

10

2

p

2

a

7

p

3

=

7

p

3

£

p

3

p

3

=

7

p

3

3

b

10

p

5

=

10

p

5

£

p

5

p

5

=^105

p

5

=2

p

5

c

10

2

p

2

=

10

2

p

2

£

p

2

p

2

=

10

p

2

4

=

5

p

2

2

Example 28 Self Tutor

Express

1

3+

p

2

with integer denominator.

1

3+

p

2

=

μ

1

3+

p

2

¶μ

3 ¡

p

2

3 ¡

p

2

¶

=

3 ¡

p

2

32 ¡(

p

2)^2

fusing (a+b)(a¡b)=a^2 ¡b^2 g

=

3 ¡

p

2

7

H DIVISION BY SURDS [1.10]

We are really

multiplying by one,

which does not change

the value of the

original expression.

If the denominator has the form a+

p

b then we can remove the surd from the denominator by multiplying

both the numerator and the denominator by itsradical conjugate a¡

p

b. This produces a rational

denominator, so the process is calledrationalisationof the denominator.

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y:\HAESE\IGCSE01\IG01_06\142IGCSE01_06.CDR Friday, 10 October 2008 9:10:01 AM PETER