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