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 $\large m \,=\, b^{\displaystyle n}$