# Power

If a quantity is divided as multiplying factors on the basis of another quantity, then the total number of multiplying factors is called the power. It is also called as exponent and also called as index.

## Introduction

On the basis of a number, any number can be expressed as the multiplying factors of that number by the exponentiation. The total number of multiplying factors is called the power or exponent or index. The power is usually written as superscript of the base of an exponential form of a number.

### Example

$32$ is a number and express this number as multiplying factors on the basis of a number, for example $2$.

$32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5$

In other words, $32 = \underbrace{2 \times 2 \times 2 \times 2 \times 2}_{\displaystyle 5 \, factors}$

The total number of multiplying factors of $2$ is $5$ and it is called the power of an exponential term $2^5$.

#### 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 form of $8$.

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

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

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

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

#### Algebraic form

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

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

The number of multiplying factors of $b$ is $n$ in this case. Therefore, $n$ is called as the power or index or exponent of an exponential form of the quantity $m$.
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