# Exponential form

## Definition

A mathematical notation of representing any number in terms of a particular number and number of its multiplicative factors is called the exponential notation.

On the basis of a particular number, every number can be written in its terms in product form and the product of them is written in a special mathematical form by writing the base number first and then the number of its multiplying factors as its superscript.

This type of special mathematical notation is called the exponential notation and also called as the exponential form.

### Example

$32$ is a number and write it as number of multiplying factors on the basis of number $2$.

$32 = 2 \times 2 \times 2 \times 2 \times 2$

The total number of multiplying factors is $5$ in this case. So, write the base number $2$ first and then the total number of multiplying factors as its superscript.

$32 = 2^{\displaystyle 5}$

The term $2^{\displaystyle 5}$ is called the exponential notation or exponential form of $32$ on the basis of number $2$.

#### More Examples

Observe the following examples to learn how to write any number in exponential form on the basis of a number.

$(1) \,\,\,$ $81 = 3 \times 3 \times 3 \times 3 = 3^{\displaystyle 4}$

$(2) \,\,\,$ $125 = 5 \times 5 \times 5 = 5^{\displaystyle 3}$

$(3) \,\,\,$ $81 = 9 \times 9 = 9^{\displaystyle 2}$

$(4) \,\,\,$ $16807 = 7 \times 7 \times 7 \times 7 \times 7 = 7^{\displaystyle 5}$

$(5) \,\,\,$ $1771561 = 11 \times 11 \times 11 \times 11 \times 11 \times 11 = 11^{\displaystyle 6}$

##### Terminology

The exponential notation consists of two parts and they are called by two special names.

A number which is used to expand any number in its terms.

The number of times a number is multiplied to itself to obtain another number.

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