# Exponentiation

The mathematical operation of raising a quantity to the power of another quantity is called the exponentiation.

## Introduction

Mathematically, a number can be raised to the power of another number to represent a particular quantity in simple form. It is useful to express small or large numbers easily in simple form. It is also useful to express and pronoun such numbers quickly.

### Example

Factor method is the fundamental method, which is used to perform the exponentiation in mathematics. For example, $16$ is a number and it can be expressed as the product of some same numbers on the basis of a number.

$(1)\,\,\,\,\,\,$ $16 = 2 \times 2 \times 2 \times 2 = 2^4$

In this case, the number $16$ is split as factors on the basis of number $2$. The total number of multiplying factors is $4$. Hence, the number $16$ is expressed as $2$ is raised to the power of $4$. The mathematical approach of expressing the number $16$ as $2$ raised to the power of $4$ is called the exponentiation.

$(2)\,\,\,\,\,\,$ $16 = 4 \times 4 = 4^2$

On the basis of another number, the same number can also be expressed in another form. It is possible to split the number $16$ as the factors of $4$. In this case, the product of two times $4$ is equal to the number $16$. So, it can be expressed as $4$ raised to the power of $2$. The process is also known as exponentiation.

#### Purpose

The above two examples are the basic examples for the exponentiation but the following example understands you the importance and advantage of exponents in mathematics.

$19487171$ is a number. Actually, it is a large number. It is not easy to write it everywhere, hard to remember it and also hard to pronounce it every time.

$19487171$ $\,=\,$ $\underbrace{11 \times 11 \times 11 \times 11 \times 11 \times 11 \times 11}_{\displaystyle 7 \, factors \, of \, 11}$ $\,=\,$ $11^7$

Now, it is easy to remember. The number $19487171$ can be simply written as $11^7$ everywhere and any number of times. Similarly, it is also easy to pronounce. Exponentiation solved all the problems.

#### General form

The process of exponentiation can be written in standard form algebraically.

Assume $m$ is a literal and it represents a quantity. Take, the quantity is expressed as factors on the basis of another literal $b$ and the number of multiplying factors is $n$.

$\large m \,=\,$ $\large \underbrace{b \times b \times b \times \cdots \times b}_{\displaystyle n \, factors}$ $\,=\,$ $\large b^{\displaystyle n}$

It is simply written as $\large m \,=\, b^{\displaystyle n}$