Proof of Power Rule of Exponents

The power rule of exponents reveals that the power of an exponential term is equal to the product of the powers with same base. This property can be proved in algebraic form for using it as a formula in mathematics.

Exponential term in Product form

$b$ is a literal number. Assume, it is multiplied by itself $m$ times. The product of them is represented by $b^{\displaystyle m}$ in exponential form.

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

Power of An Exponential term

Multiply the term $b^{\displaystyle m}$ by the same term $n$ times. The product of them is written in exponential notation as ${(b^{\displaystyle m})}^{\displaystyle n}$.

${(b^{\displaystyle m})}^{\displaystyle n}$ $\,=\,$ $\underbrace{b^{\displaystyle m} \times b^{\displaystyle m} \times b^{\displaystyle m} \times \ldots \times b^{\displaystyle m}}_{\displaystyle n \, factors}$

Evaluate the Product of the factors

$b$ is a factor in the term $b^{\displaystyle m}$ and the total number of factors in each term is $m$.

$\implies$ ${(b^{\displaystyle m})}^{\displaystyle n} = \underbrace{\underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors} \times \underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors} \times \ldots \times \underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors}}_{\displaystyle n \, factors}$

There are $m$ factors in each term but total $n$ terms are involved in this product. Therefore, the total number of factors in the product of the exponential term ${(b^{\displaystyle m})}^{\displaystyle n}$ is $m \times n$.

$\implies$ ${(b^{\displaystyle m})}^{\displaystyle n}$ $\,=\,$ $\underbrace{b \times b \times b \times b \times \ldots \times b}_{\displaystyle m \times n \, factors}$

$\implies$ ${(b^{\displaystyle m})}^{\displaystyle n}$ $\,=\,$ $\underbrace{b \times b \times b \times b \times \ldots \times b}_{\displaystyle mn \, factors}$

Now, express the product in exponential notation.

$\,\,\, \therefore \,\,\,\,\,\,$ ${(b^{\displaystyle m})}^{\displaystyle n} \,=\, b^{\displaystyle mn}$

Therefore, it is proved that the power of an exponential term is equal to the product of the indices with same base. It is called as the power rule of exponents and used as a formula in mathematics.