Power and Index | Quantitative Techniques for CLAT PDF Download

Chapter 2 Power and Index  

In pⁿ, p is called the base and n is called the power or index
In (-5)⁷, -5 is te base and 7 is the power
In  , is the base and 7 is the power
   

And
=
Remarks: In the exponential notation, the base can be any rational number and the power can be any integer.

Note: If the power of a rational number is 1 its value will be the rational number itself
i.e. (-10)¹ = -10,  and in general
 . 

Example: Find the value of
 
Solution:




Product law for exponents: If p is a non-zero rational number and m and n are two positive integers then p^m x pⁿ = p ^m+n
also p^m x pⁿ x p^r x p^s = p^m+n+r+s
also (p^m)ⁿ = p^mn

Quotient law for exponents: It p is a non- zero rational number and m and n are two positive integers
then p^m ÷   pⁿ = p^m-n   for m > n
and p^m ÷ pⁿ =       for m < n

If power is zero (o): If p is a non-zero rational number then p^o = 1
If power is (-1): If p is a non-zero rational number then p^-1 denotes the reciprocal of p and (p)^-1 =
A negative integer as power
p^-m =
other laws of exponents
p^m x q^m = (p x q)^m
 

Few examples showing the application of laws of exponents.
Example1: Simplify (a) 

Solution: a)




= 2^-2 x 3^-4+2 = 2^-2 x 3^-2

Example2: Find m if

Solution
(a) LHS=
RHS =
Equating LHS and RHS
 
Because base is same, powers must be equal
So -2m + 1 = -27
or -2m  = -27 – 1
= -28
or  m  = 14
(b) LHS =
RHS = 2^m
So 2^m = 2⁵
Or m = 5.

Example: Solve for x
a) 3^x = 81              b) (7^2x)^-2 = (2401)^-1

Solution a) RHS = 81 = 3⁴
So 3^x = 3⁴ or x = 4
Solution b) RHS= (2401)^-1 = (7⁴)^-1 = 7^-4
LHS = (7^2x)^-2 = 7^-4x
Equating LHS and RHS
7^-4X = 7^-4
Base in same, powers must be equal
-4x = -4
Or  x = 1

What is the difference between exponents

 = p^m x p^m x p^m x p^m --------- n times
 =p^m+m+m+m ------ n times
=p^{m n}
Where as =
Let us simplify it with the help of an example
Find the difference between
(2²)³ = 2² x 2² x 2² = 2⁶
  = 2^2x2x2 = 2⁸
So the diff. is very clear.