Proof of Product Rule of Exponents with same base

Take two quantities and they are expressed in product form as $m$ and $n$ factors respectively on the basis of a quantity $b$. Similarly, the two quantities are written in exponential notation as $b^{\displaystyle m}$ and $b^{\displaystyle n}$ respectively.

$(1) \,\,\,$ $b^{\displaystyle m}$ $\,=\,$ $\underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle m \, factors}$

$(2) \,\,\,$ $b^{\displaystyle n}$ $\,=\,$ $\underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle n \, factors}$

Now, let’s start deriving the product law of powers with same base in mathematics.

Multiply the Exponents with same base

Now, multiply the exponents with same base for obtaining their product.

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

Mix the Product form of both terms

There are $m$ factors of $b$ in the term $b^{\displaystyle m}$ and $n$ factors of $b$ in the term $b^{\displaystyle n}$. If they both are multiplied, then the total number of factors in their product is equal to $m+n$.

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

Exponential form of Terms

According to exponentiation, the right-hand side of the equation can be expressed in exponential form.

$\,\,\, \therefore \,\,\,\,\,\,$ $b^{\displaystyle m} \times b^{\displaystyle n} \,\,=\,\, b^{\displaystyle m+n}$

It is proved mathematically that the product of exponents with same base is equal to the sum of the exponents with same base. This property can also be extended to more than two terms as well.

$b^{\displaystyle m} \times b^{\displaystyle n} \times b^{\displaystyle o} \ldots$ $\,\,=\,\,$ $b^{\displaystyle m+n+o \cdots}$