# Power

## Definition

The number of multiplicative factors when a number splits on the basis of another number to express the number in exponential notation is called an exponent.

In exponentiation, a number is split as number of multiplicative factors on the basis of another number and write it in exponential notation. The number of multiplicative factors is known as exponent.

The exponent is written as superscript to the base of the exponential notion. Hence, the exponent is also called as power of the exponential form. The exponent indexes all the multiplicative factors in a single term, hence the exponent is also called as index.

### Example

$81$ is a number and express this number as multiplicative factors on the basis of any number, for example $3$.

$81 = 3 \times 3 \times 3 \times 3$

Express the expansion in exponential form.

$81 = 3^{\displaystyle \, 4}$

On the basis of number $3$, the number $81$ is expressed as four multiplicative factors of number $3$. Hence, the number of multiplicative factors $4$ is called exponent. The number $4$ is displayed in superscript position. Due to this reason, the exponent $4$ is called as power. The exponent $4$ represents the index of all four multiplicative factors. Hence, the power is also called as index.

Observe the following two demonstrating examples for clear understanding.

$7776 = 6 \times 6 \times 6 \times 6 \times 6$
$\implies 7776 = 6^{\displaystyle \, 5}$

The number $5$ is called the exponent, or power, or index.

$262144 = 8 \times 8 \times 8 \times 8 \times 8 \times 8$
$262144 = 8^{\displaystyle \, 6}$

The number $6$ is called an exponent, or index or power.

#### Algebraic form

Assume, $m$ is a number and split the number into $n$ number of multiplicative factors on the basis of another number $b$.

$m = b \times b \times b \times … \times b (nth factor)$

Express the relation between three of them in exponential form.

$m = b^{\displaystyle n}$

The number $m$ is split into $n$ number of multiplicative factors on the basis of number $b$. Therefore, the number of multiplicative factors $n$ is called exponent, or power, or index

Save (or) Share