Cambridge Additional Mathematics

Surds, indices, and exponentials (Chapter 4) 105

Example 6 Self Tutor

Write with an integer denominator:

a

6

p

5

b

35

p

7

a

6

p

5

=

6

p

5

£

p

5

p

5

=

6

p

5

5

b

35

p

7

=

35

p

7

£

p

7

p

7

=

35

p

7

7

=5

p

7

For any fraction of the form

c

a+

p

b

, we can remove the surd

from the denominator by multiplying by

a¡

p

b

a¡

p

b

.

Expressions such as a+

p

b and a¡

p

b are known asradical conjugates. They are identical except for

the sign in the middle.

The product of radical conjugates is rational, since we have the difference between two squares. Multiplying

by

a¡

p

b

a¡

p

b

therefore produces a rational denominator, so it is sometimes calledrationalising the

denominator.

Example 7 Self Tutor

Write

5

3 ¡

p

2

with an integer denominator.

5

3 ¡

p

2

=

μ

5

3 ¡

p

2

¶μ

3+

p

2

3+

p

2

¶

=

5(3 +

p

2)

32 ¡(

p

2)^2

=

15 + 5

p

2

7

EXERCISE 4A.3

1 Write with integer denominator:

a

1

p

3

b

3

p

3

c

9

p

3

d

11

p

3

e

p

2

3

p

3

f

2

p

2

g

6

p

2

h

12

p

2

i

p

3

p

2

j

1

4

p

2

The radical conjugate of

3 - ~`2 is 3 + ~`2.

does not change its value.

Multiplying theoriginal

numberby or

p

p^7

7

p

5

p

5

1

5

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_04\105CamAdd_04.cdr Tuesday, 14 January 2014 2:28:03 PM BRIAN