Cambridge Additional Mathematics

Surds, indices, and exponentials (Chapter 4) 111

11 Write as powers of 2 , 3 and/or 5 :

a^19 b 161 c 1251 d^35

e 274 f

2 c

8 £ 9

g

9 k

10

h

6 p

75

12 Read about Nicomachus’ pattern on page 108 and find the series of odd numbers for:

a 53 b 73 c 123

The index laws used previously can also be applied torational indices, or indices which are written as a

fraction.

The notation an is defined to mean “amultiplied togetherntimes”. Since we cannot multiplyatogether

“half a time”, the notation a

1

(^2) is an extension of the meaning of this notation. The goal is to extend the
meaning of an so that the fundamental law
anam=an+m
remains true. If we assume that a> 0 then this law holds for rational indices.
Since x^3 =¡ 8 has x=¡ 2 as a solution, we would like to write
x=x
3
(^3) =(x^3 )
1
(^3) =(¡8)
1
(^3) =(¡ 23 )
1
(^3) =(¡2)
3
(^3) =¡ 2.
Under some circumstances it is therefore possible to extend the meaning ofanwhennis rational anda 60.
However, this is not generally so easy, and so for this course we restrict ourselves to cases where a> 0.
For a> 0 , notice that a
1
(^2) £a
1
(^2) =a
1
2 +
1
(^2) =a^1 =a findex lawsg
and
p
a£
p
a=a also.
So, a
1
(^2) =pa fby direct comparisong
Likewise a
1
(^3) £a
1
(^3) £a
1
(^3) =a^1 =a
and^3
p
a£^3
p
a£^3
p
a=a
suggests a
1
(^3) =p^3 a
In general, a
1
n =pna where pna reads ‘thenth root ofa’, for n 2 Z+.
We can now determine that n
p
am
=(am)
1
n
=a
m
n
) a
m
n =pnam for a> 0 , n 2 Z+, m 2 Z

D Rational indices

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_04\111CamAdd_04.cdr Tuesday, 14 January 2014 2:28:21 PM BRIAN