# Exponentiation

## Definition

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

Any number can be raised to the power of another number to represent a particular number. It expresses large or small numbers in a special simple mathematical form. This process is called the exponentiation in mathematics.

### Example

For example, $2$ and $3$ are two numbers. The number $3$ is raised to the number $2$. It is written as $2^3$ in mathematics.

$2^3 = 2 \times 2 \times 2 = 8$

Similarly, it can be written in inverse operation.

$8 = 2 \times 2 \times 2 = 2^3$

The mathematical procedure of writing $2^3$ as $8$ and also the mathematical operation of expressing $8$ as $2^3$ by the expansion is called the exponentiation. This example is a basic example for your understanding of the exponentiation and you may not understand the purpose of exponentiation from this example.

#### Purpose

Exponentiation is mainly useful in expressing large or small quantities in simple form.

For example, $9765625$ is a large number. It is difficult to remember, pronounce and write every time. Hence, it is written in a special simple form by the exponentiation.

Split the number $9765625$ as product of multiplying factors of a number, for example $5$.

$9765625$ $\,=\,$ $5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5$

$\implies 9765625$ $\,=\,$ $\underbrace{5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5}_{\displaystyle 10 \, multiplying \, factors \, of \, 5}$

The product of $10$ multiplying factors of $5$ is equal to $9765625$. So, write the number $5$ is raised to the number of multiplying factors and it is $10$.

$\implies 9765625 \,=\, 5^{10}$

The mathematical procedure is called the exponentiation. The large number $9765625$ is written as $5^{10}$ simply. It can be remembered, pronounced and also represented the number $9765625$ easily anywhere and any number of times in this form.

The same number can also be written in terms of $25$ by the procedure of exponentiation.

$\implies 9765625$ $\,=\,$ $\underbrace{25 \times 25 \times 25 \times 25 \times 25}_{\displaystyle 5 \, \, factors \, of \, 25}$

$\therefore \,\,\,\,\,\,\, 9765625 \,=\, 25^5$

In this example, the large number $9765625$ can be written as $5^{10}$ or $25^5$ simply in mathematics. Thus, the exponentiation is used to write quantities in a special simple form.

#### General form

Exponentiation can be expressed in general form by the algebra.

$b$ is a literal number and assume it is multiplied by the same number $n$ number of times. The product of them is written as $b^{\displaystyle n}$.

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

The process of expressing product of $n$ times of $b$ as $b^{\displaystyle n}$ is called the exponentiation.

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

Similarly, the expansion of $b^{\displaystyle n}$ as the product of $n$ multiplying factors of $b$ is also called as the exponentiation.

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