Cambridge Additional Mathematics

(singke) #1
106 Surds, indices, and exponentials (Chapter 4)

2 Write with integer denominator:

a
5
p
5

b
15
p
5

c
¡ 3
p
5

d
200
p
5

e
1
3

p
5

f
7
p
7

g
21
p
7

h
2
p
11

i
26
p
13

j
1
(

p
3)^3
3 Rationalise the denominator:

a
1
3+

p
2

b
2
3 ¡

p
2

c
1
2+

p
5

d

p
2
2 ¡

p
2

e
10
p
6 ¡ 1
f

p
3
p
7+2
g
1+

p
2
1 ¡
p
2
h

p
3
4 ¡
p
3

i
¡ 2

p
2
1 ¡
p
2
j
1+

p
5
2 ¡
p
5
k

p
3+2
p
3 ¡ 1
l

p
10 ¡ 7
p
10 + 4

Example 8 Self Tutor


Write
1
5+
p
2
in the form a+b

p
2 where a,b 2 Q.

1
5+

p
2

=

μ
1
5+

p
2


£

μ
5 ¡
p
2
5 ¡

p
2


=
5 ¡

p
2
25 ¡ 2

=
5 ¡

p
2
23
= 235 ¡ 231

p
2

4 Write in the form a+b

p
2 where a,b 2 Q:

a
3
p
2 ¡ 3

b
4
2+

p
2

c

p
2
p
2 ¡ 5

d
¡ 2

p
2
p
2+1
5 Write in the form a+b

p
3 where a,b 2 Q:

a
4
1 ¡

p
3

b
6
p
3+2

c

p
3
2 ¡

p
3

d
1+2

p
3
3+

p
3

6aSupposea,b, andcare integers, c> 0. Show that (a+b

p
c)(a¡b

p
c) is also an integer.
b Write with an integer denominator:
i
1
1+2

p
3

ii

p
2
3

p
2 ¡ 5

iii

p
2 ¡ 1
3 ¡ 2

p
2

7aSupposeaandbare positive integers. Show that (

p
a+

p
b)(

p

p
b) is also an integer.
b Write with an integer denominator:
i
1
p
2+

p
3

ii

p
3
p
3 ¡

p
5

iii

p
11 ¡

p
14
p
11 +

p
14

8 Solve the equation 2 x¡ 3

p
3=1¡x

p
3. Give your solution in the form x=a+b

p
3 , where
aandbare integers.

9 Find the positive solution of the equation (9 +

p
5)x^2 +(5¡ 2

p
5)x¡5=0. Give your answer in
the form a+b

p
5 , where a,b 2 Q.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_04\106CamAdd_04.cdr Tuesday, 14 January 2014 2:28:06 PM BRIAN

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