Exponents

Introduction to Exponents

We know multiplication is repeated addition and can be written in a short form known as Product form.

Example: 3+3+3=3×3

Similarly, repeated multiplication of a number is called Powers and can be written in a short form known as Exponential form.

Example: 3×3×3×3= 3⁴

Let us look at the above example in detail:

Here, 3 is the base and 4 is the Exponent/ Index/Power/Order.

The term exponent indicates how often the base value needs to be multiplied. In this context, the exponent 4 indicates that the base 3 has to be multiplied by itself 4 times.

Thus, a number's exponent indicates how many times it has been multiplied by itself.

Let us look into some more examples:

Number	Base	Exponent	Expanded form	Value
42	4	2	4×4	16
(-3)3	3	3	-3×-3×-3	-21
(3/5)2	3/5	2	3/5×3/5	9/25
201	20	1	20×1	20

Expressing a number in exponential form

Any number can be expressed in exponential form by the prime factorisation method.

Example 1: 16 can be expressed as 2⁴

Explanation: 16 is 2×2×2×2

Example 2: 225 can be written as 5²×3²

Explanation: 225 can be written as 5×5×3×3

Exponents of Negative Numbers

• If the power/index of a negative integer is an odd number, then the final value will be negative.
  
  Example: (-2)⁵, here the exponent 5 is an odd number; thus, the final value will be a negative number, i.e., -2×-2×-2×-2×-2=-32

• If the power/index of a negative integer is an even number, then the final value will be positive.
  
  Example: (-2)⁴; here, the exponent 4 is an even number; thus, the final value will be a positive number, i.e., (-2) × (-2) × (-2) × (-2) =+16

Rules of exponents

With the same bases:

1. p^m×pⁿ=p^(m+n)
   
   When two equal bases are multiplied, the base remains the same, and their powers/ exponents get added.
   Example: 6²×6²= 6⁽²⁺²⁾ =6⁴, 6×6×6×6=1296
   
2. p^m/pⁿ=p^(m-n)
   
   When two equal bases are divided, the base remains the same, and their powers get subtracted.
   
   Example: 2⁴/2²=2^(4-2) =2²=2×2=4
   
3. (p^m)ⁿ=p^m*n
   
   When a base is raised to a power which is raised to another power, the base remains the same, and the powers get multiplied.
   
   Example: (4²)⁴=4⁸=4×4×4×4×4×4×4×4

With different bases:

1. p^m×q^m=(p×q) ^m
   
   When two different bases with the same powers are multiplied, the power remains the same, and the bases are multiplied.
   
   Example: 2²×4²= (2×4)²=8²=8×8=64
   
2. p^m/q^m=(p/q) ^m
   
   When two different bases with the same powers are divided from each other, the power remains the same, and the bases are divided.
   
   Example: 4²/2²= (4/2)²=2²=2×2=4

Zero Exponent Rule: p⁰=1

When a base is raised to the power 0, the value is always one, irrespective of the base number.

Example: 5⁰=1

Negative Power Rule: p^-m=1/p^m

Whenever a base is raised to the negative value of an exponent, reciprocate the number to make the power positive.

Example: 4^-2= (1/4²) =1/4×4=1/16