# Power

The total number of factors when a quantity is divided as factors on the basis of another quantity, is called an exponent. It is called as index alternatively and also called as the power.

## Introduction

On the basis of a number, any number can be expressed in product form as factors as per the exponentiation. The total number of factors is called the power or exponent or index. The power is usually written as superscript of the base of the exponential term.

### Example

$32$ is a number and express it in product form on the basis of another number $2$ by factorization.

$32$ $\,=\,$ $2 \times 2 \times 2 \times 2 \times 2$

As per exponentiation, write the expression of product form in exponential notation.

$\implies$ $32$ $\,=\,$ $\underbrace{2 \times 2 \times 2 \times 2 \times 2}_{\displaystyle 5 \, factors}$

The total number of factors of $2$ is $5$ and it is called the power of an exponential term. The exponent $5$ is written as superscript to the base $2$.

$\implies$ $32$ $\,=\,$ $2^5$

#### More Examples

The following examples demonstrate you what exactly an exponent is in an exponential term.

$(1)\,\,\,\,\,\,\,$ $8 = 2 \times 2 \times 2 = 2^3$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $3$ is the exponent of the exponential term.

$(2)\,\,\,\,\,\,\,$ $16 = 4 \times 4 = 4^2$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $2$ is the index of the exponential term.

$(3)\,\,\,\,\,\,\,$ $3125 = 5 \times 5 \times 5 \times 5 \times 5 = 5^5$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $5$ is the power of the exponential term.

$(4)\,\,\,\,\,\,\,$ $343 = 7 \times 7 \times 7 = 7^3$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $3$ is the power of the exponential term.

$(5)\,\,\,\,\,\,\,$ $28561 = 13 \times 13 \times 13 \times 13 = 13^4$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $4$ is the index of the exponential term.

#### Algebraic form

$m$ is literal and represents a quantity. Assume, the quantity $m$ is divided as factors on the basis of another quantity $b$ and the total number of factors of $b$ is $n$.

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

The number of factors of $b$ is $n$ in this case. Therefore, $n$ is called as the power or index or exponent of an exponential term $b^n$.

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