Cambridge Additional Mathematics

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