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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 Power and Index | Quantitative Techniques for CLAT ,Power and Index | Quantitative Techniques for CLAT is the base and 7 is the power
Power and Index | Quantitative Techniques for CLAT   
Power and Index | Quantitative Techniques for CLAT
And Power and Index | Quantitative Techniques for CLAT
= Power and Index | Quantitative Techniques for CLAT
Remarks: In the exponential notation, the base can be any rational number and the power can be any integer.
Power and Index | Quantitative Techniques for CLAT
Note: If the power of a rational number is 1 its value will be the rational number itself
i.e. (-10)1 = -10, Power and Index | Quantitative Techniques for CLAT and in general
Power and Index | Quantitative Techniques for CLAT . 

Example: Find the value of
 Power and Index | Quantitative Techniques for CLAT
Solution: Power and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLATPower and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLAT
Product law for exponents: If p is a non-zero rational number and m and n are two positive integers then pm x pn = p m+n
also pm x pn x pr x ps = pm+n+r+s
also (pm)n = pmn

Quotient law for exponents: It p is a non- zero rational number and m and n are two positive integers
then pm ÷   pn = pm-n   for m > n
and pm ÷ pn =  Power and Index | Quantitative Techniques for CLAT     for m < n

If power is zero (o): If p is a non-zero rational number then po = 1
If power is (-1): If p is a non-zero rational number then p-1 denotes the reciprocal of p and (p)-1 = Power and Index | Quantitative Techniques for CLAT
A negative integer as power
p-m = Power and Index | Quantitative Techniques for CLAT
other laws of exponents
pm x qm = (p x q)m
 Power and Index | Quantitative Techniques for CLAT

Few examples showing the application of laws of exponents.
Example1: Simplify (a) Power and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLAT
Solution: a) Power and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLATPower and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLAT
= 2-2 x 3-4+2 = 2-2 x 3-2
Power and Index | Quantitative Techniques for CLAT

Example2: Find m if
Power and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLAT

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

Example: Solve for x
a) 3x = 81              b) (72x)-2 = (2401)
-1

Solution a) RHS = 81 = 34
So 3x = 34 or x = 4
Solution b) RHS= (2401)-1 = (74)-1 = 7-4
LHS = (72x)-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
Power and Index | Quantitative Techniques for CLAT
Power and Index | Quantitative Techniques for CLAT = pm x pm x pm x pm --------- n times
 =pm+m+m+m ------ n times
=pm n
Where as Power and Index | Quantitative Techniques for CLAT= Power and Index | Quantitative Techniques for CLAT
Let us simplify it with the help of an example
Find the difference between Power and Index | Quantitative Techniques for CLAT
(22)3 = 22 x 22 x 22 = 26
Power and Index | Quantitative Techniques for CLAT  = 22x2x2 = 28
So the diff. is very clear.

The document Power and Index | Quantitative Techniques for CLAT is a part of the CLAT Course Quantitative Techniques for CLAT.
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FAQs on Power and Index - Quantitative Techniques for CLAT

1. What is power in mathematics?
Ans. In mathematics, power refers to the operation of multiplying a number by itself a certain number of times. It is represented by an exponent, which indicates the number of times the base number is multiplied by itself.
2. How do you calculate the value of a power?
Ans. The value of a power can be calculated by multiplying the base number by itself the number of times indicated by the exponent. For example, if the base number is 2 and the exponent is 3, the value of the power would be 2 x 2 x 2 = 8.
3. What is an index or exponent in mathematics?
Ans. In mathematics, an index or exponent is a small number written above and to the right of a base number. It indicates the number of times the base number should be multiplied by itself. For example, in the expression 5^3, 3 is the exponent and 5 is the base number.
4. How does the concept of power relate to real-life situations?
Ans. The concept of power is used in various real-life situations, such as calculating the area or volume of objects, determining the growth or decay of populations, analyzing financial investments, and understanding exponential growth or decay in natural phenomena.
5. What are some common mistakes to avoid when working with powers and exponents?
Ans. When working with powers and exponents, it is important to avoid common mistakes such as forgetting to multiply the base number by itself the correct number of times, misinterpreting negative exponents, incorrectly applying the order of operations, and confusing exponentiation with multiplication or addition. Double-checking calculations and understanding the properties and rules of powers can help avoid these mistakes.
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