# What is an Exponent?

A quantity that represents the number of times a quantity gets multiplied by itself is called an exponent.

## Introduction

In mathematics, a quantity is multiplied by itself once or more than once for some mathematical operations. All the factors in the product are same. In this case, the total number of factors is called an exponent.

The word “exponent” is also expressed by the following two words alternatively.

1. Index
2. Power

So, do not get confused with these three names. Now, let’s understand the concept of exponents from understandable arithmetic examples.

$2 \times 2 \times 2 \times 2 \times 2$

Two is a number and it is multiplied by itself in this example. Count the total number of times the number two is multiplied by itself.

$\underbrace{2 \times 2 \times 2 \times 2 \times 2}_{\displaystyle 5 \, factors}$

It is cleared that the number of times the number $2$ gets multiplied by itself is $5$. The mathematical relation between the numbers $2$ and $5$ is expressed in the following ways possibly.

1. The number $5$ is called an exponent of the number $2$.
2. The number $5$ is called an index of the number $2$.
3. The number $5$ is called a power of the number $2$.

### Representation

The total number of factors is appended to a quantity at superscript (right-top) position to write an exponent in mathematical form. Therefore, the exponent $5$ is displayed at superscript position of the number $2$ in this example.

$2 \times 2 \times 2 \times 2 \times 2$ $\,=\,$ $\underbrace{2 \times 2 \times 2 \times 2 \times 2}_{\displaystyle 5 \, factors}$ $\,=\,$ $2^{\displaystyle 5}$

#### Examples

Now, let us understand the concept of indices from some more arithmetic examples.

$(1).\,\,$ $4 \times 4 \times 4$ $\,=\,$ $4^3$
The number $3$ is an exponent of the number $4$.

$(2).\,\,$ $3$ $\,=\,$ $3^1$
The number $1$ is a power of the number $3$.

$(3).\,\,$ $5 \times 5 \times 5 \times 5 \times 5$ $\,=\,$ $5^5$
The number $5$ is an index of the number $5$.

$(4).\,\,$ $7 \times 7$ $\,=\,$ $7^2$
The number $2$ is a power of the number $7$.

$(5).\,\,$ $6 \times 6 \times 6 \times 6$ $\,=\,$ $6^4$
The number $4$ is an exponent of the number $6$.

You have learned the concept of exponents. It is time to express powers in mathematical form.

#### Algebraic form

Let $a$ and $n$ represent two literals. Totally $n$ times, the literal $a$ is multiplied by itself.

$\underbrace{a \times a \times a \times … \times a}_{\displaystyle n factors}$ $\,=\,$ $a^{\displaystyle n}$

Now, the literal $n$ is called an exponent or an index or a power of the literal $a$.

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