Index: Definition, Laws of Indices, and Solved Examples

The index of a number is also known as the power or exponent. It actually tells us how many times we have to multiply the number by itself. For example, we consider 34. It means we have to multiply 3 by itself four times. In other words, 34= 3×3×3×3

Definition of Index

The power/exponent raised to a number is called the index of that number.

Mathematically, we understand the index of a number as follows. For a real number a and for a positive integer n, we have an=a×a××a(m times) Here the number n is called the index of a and the number a is called the base. In the above example 34= 3×3×3×3, the number 4 is the index of 3, and the number 3 is the base.

Note that the plural form of index is indices. ♣

 

Laws of Indices

We now discuss the laws of indices (or the rules of indices). This will help us to solve the problems of indices.

• Zero index rule: a0=1

Thus, if the index of any non-zero number is 0, then the value will be 1. For example, 10=1,70=1. But, remember that 001. Note 00 is meaningless.

 

Negative index rule:an=1an

Thus, if the index of any number is negative, then the value will be the reciprocal of the positive index raised to that number itself. For example, 71=171=17

 

Quotient rule of indices: 

(i) am÷an =aman =amn,   (ii) (ab)n =anbn

Examples:

(i) 24÷22 =2422 =242, (ii) (42)2 =4222

 

Product rule of indices: 

(i) am.an =am+n, (ii) (am)n =amn, (iii) (ab)n =an.bn

Examples:

(i) 22.23=22+3, (ii) (23)2 =23.2, (iii) (2.3)2 =22.32

 

Fraction rule of indices: amn=amn

So if the index is a fraction, then the value can be expressed as a radical form. For example, 713 =713 =73 (cube root of 7)

 

Some Remarks on Indices

(R1) Simple proof of a0=1

Proof: a0= amm [0=mm]

=amam [axy=axay]

=1

(R2) Meaning of an when n is a negative integer: write n=m, where m is a positive integer. So an= am =1am (negative index rule)

(R3) Meaning of an when n is a fraction: we write n=pq, where p is an integer (positive or negative) and q is a positive integer. So an= apq =(ap)1q =apq (fraction rule of indices)

 

Before providing solved examples, we now summarize the fundamental laws of indices.

a0=1

an=1an

am.an=am+n

am÷an=amn

(am)n=am×n

(ab)n=an.bn

(ab)n=anbn

a1n=an

amn=amn

 

Solved Examples of Indices

Ex 1: Simplify 82/3

Solution. 

Note that 8=2×2×2=23

82/3= (23)2/3 =23×2/3 =22 =4

 

Ex 2: Calculate {(27)37}13

Solution. 

{(27)37}13

={27×37}13

=(23)13

=23×13

=21

=12

 

Ex 3: If xa=xb, then a=b?

Solution.

Case 1: x0

Now xa=xbxaxb=1xab=1

So ab=0 as x0

a=b

Case 2: x=0

Note that 02=05=0, but 25

Conclusion: xa=xb does not always imply that a=b

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