untitled

If ais any real number, successive multiplication of ntimes a; that is,

auauau..... uais written in the form an, where nis a positive integer.

auauau..... ua (ntimesa) = an.

Here na ooindex base orpower

Again, conversely, an = auauau..... ua (ntimes a). Exponents may

not only be positive integer, it may also be negative integer or positive fraction or
negative fraction. That is, for aR(set of real numbers) and nQ (set of
rational numbers), an is defined. Besides, it may also be irrational exponent. But as
it is out of curriculum, it has not been discussed in this chapter.
4.2 Formulae for exponents
Let, aR;m,nN.

Formula 1. amuan amn

Formula 2.
° ̄

°

®

­

!

! z





m n

n ma

mn

n anm

m

a

a

a , when

(^1) , when , 0
Fill in the blanks of the following table :
m!n n!m
0
,
az
aman
m 5 ,n 3 m 3 ,n 5
amuan
8 5 3
(^53) ( ) ( )

u u u u u u u
u u u u u u u u
a a
a a a a a a a a
a a a a a a a a a a a^3 ua^5 =
n
m
a
a
3
5
a
a
53
1
2
1
5
3

u u u u
u u
a a
a a a a a
a a a
a
a
?amuan=amn
and
̄
®
­
!
!


m n
n m
mn
anm
n
m
a
a
a when
when
,
(^1) ,
Formula 3. (ab)n anubn
We observe, ( 5 u 2 )^3 ( 5 u 2 )u( 5 u 2 )u( 5 u 2 )[a^3 auaua;a 5 u 2 ]
53 23
( 5 5 5 ) ( 2 2 2 )
5 2 5 2 5 2
u
u u u u u
u u u u u
In general, (ab)n abuabuabu.......uab [Successive multiplication of ntimesab]