A quantity that splits another quantity as its multiplying factors is called the base.

A number can be expressed as the product of factors of another number. It can be done by the exponentiation by expressing a number as multiplying factors on the basis of another number. Hence, the number which is considered to split a particular number as its multiplying factors, is known as the base.

$81$ is a number and it can be expressed as the product of multiplying factors of another number. For example, express the number $81$ in terms of $3$.

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

Express the relation between the number $81$ and the product of multiplicative factors of number $3$ in exponential notation.

$81 = 3^4$

On the basis of number $3$, the number $81$ is expressed as four multiplicative factors of $3$. Hence, the number $3$ is called the base of the exponential notation for the number $81$.

Observe the following examples to understand what exactly the base is in exponential notation.

$(1)\,\,\,\,\,\,\,$ $8 = 2 \times 2 \times 2 = 2^3$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $2$ is the base of the exponential form.

$(2)\,\,\,\,\,\,\,$ $16 = 4 \times 4 = 4^2$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $4$ is the base of the exponential form.

$(3)\,\,\,\,\,\,\,$ $3125 = 5 \times 5 \times 5 \times 5 \times 5 = 5^5$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $5$ is the base of the exponential form.

$(4)\,\,\,\,\,\,\,$ $343 = 7 \times 7 \times 7 = 7^3$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $7$ is the base of the exponential form.

$(5)\,\,\,\,\,\,\,$ $28561 = 13 \times 13 \times 13 \times 13 = 13^4$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $13$ is the base of the exponential form.

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

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

Therefore, the number $b$ is called the base of the exponential notation $b^n$.

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