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
a¡
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