Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.33

2.5 LAWS OF INDICES

The word indices is a plural part of the word index (Power).
When we write

a ,^5 a is called the base and 5 is called the index of the base.
When we write.

a xam n then m and n are called indices.
Now a^2 =a a a a a a a a a× ,a^3 = × × ,a^4 = × × ×
i.e., the index of the base indicates the number of times the base should be multiplied .However ,if index is
a fraction it indicates the root,e.g.,

a^12 = a,a^13 =^3 a,a^23 =^3 a^2

The Radical sign with the number indicates the root,e.g.^3 indicates cube root and^4 indicates
fourth root. The Radical sign without number indicate square root.

Quantities like am n/ written as namare called radicals. The term under the radical sign is called the radicand

and the number with the radical sign (nin an m) is called the index of the radical.
It is not possible to determine exactly every root of every positive number.

For example, 2, 3, 5^2 etc. can not be determined exactly, and such quantities are called Surds or
irrational quantities.
Surds cannot be expressed as the ratio of two integers and their values are usually calculated with the help
of logarithms,
(i) a a am× n= m n+
(ii) a a am÷ n= m n−
(iii) ( )a am n= mn
where m and n are positive or negative, integral or fractional and a is a non zero real number.
Law I If m and n are positive integars then

a xam n= a x a x a x a ..... m factors

×a a a n× × ........ factors

=a a a a m n× × × ........( + ) factors am n+

Law II = −

m m n
n
d d
d
Law III ( )a a a a a nfactorsm n= m m m m× × × ×.....

=a(m m m nterms+ + +...... )=amn
We have seen that
(1) a a a a mfactorsm= × × ×....... for any non zero real number a and any positive integar m
(2) a^0 = 1 for any non-zero real number a.