Cambridge International Mathematics

Challenge #endboxedheading

146 Exponents and surds (Chapter 6)

Discovery Continued square roots#endboxedheading

X=

s

2+

r

2+

q

2+

p

2+

p

2+:::::: is an example of a

continued square root.

Some continued square roots have actual values which are integers.

1 Use your calculator to show that

p

2 ¼ 1 : 41421

p

2+

p

2 ¼ 1 : 84776

q

2+

p

2+

p

2 ¼ 1 : 96157 :

2 Find the values, correct to 6 decimal places, of:

a

r

2+

q

2+

p

2+

p

2 b

s

2+

r

2+

q

2+

p

2+

p

2

3 Continue the process and hence predict the actual value ofX.

4 Use algebra to find the exact value ofX.

Hint: FindX^2 in terms ofX, and solve by inspection.

5 Can you find a continued square root whose actual value is 3?

1 Find

s

3+2

p

2

3 ¡ 2

p

2

giving your answer in the form a+b

p

2 where a,b 2 Q.

2 If x=

p

5 ¡

p

3 , findx^2 andx^4. Hence find the value of x^4 ¡ 16 x^2.

Copy and complete: x=

p

5 ¡

p

3 is one of the solutions of the equation x^4 ¡ 16 x^2 =0

3aWe know that in general,

p

a+b 6 =

p

a+

p

b

Deduce that if

p

a+b=

p

a+

p

b then at least one ofaorbis 0.

b What can be deduced aboutaandbif

p

a¡b=

p

a¡

p

b?

4aFind the value of

μ

1+

p

5

2

¶n

¡

μ

1 ¡

p

5

2

¶n

for n=1, 2 , 3 and 4.

b What do you suspect about

μ

1+

p

5

2

¶n

¡

μ

1 ¡

p

5

2

¶n

for all n 2 Z+?

What to do:

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Y:\HAESE\IGCSE01\IG01_06\146IGCSE01_06.cdr Friday, 31 October 2008 9:51:20 AM PETER