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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