The multiplication of two or more exponents with same base equals to the base raised to the power of summation of the exponents, is called the adding exponents with the same base product rule or simply called as the adding powers product rule. It is also called as the multiplying powers with the same base product rule.

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

Two or more exponential terms which contain same base often participate in multiplication but the product of them cannot be calculated directly, whereas a product rule is essential to multiply two or more different powers with same base and the product rule is called the adding exponents with same base product rule.

Take, two quantities are expressed in exponential form on the basis of a literal $b$. The number of multiplying factors of $b$ for the both quantities are $m$ and $n$, and the quantities in exponential notation are represented as $b^m$ and $b^n$ in mathematics.

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

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

01

Now, multiply the exponential terms $b^{\displaystyle m}$ and $b^{\displaystyle n}$ to get their product.

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

02

The total number of multiplying factors of $b$ for representing term $b^{\displaystyle m}$ is $m$ and the total number of multiplicative factors of $b$ for denoting the term $b^{\displaystyle n}$ is $n$. The total number of multiplying factors should be $m+n$ if the two exponential terms are multiplied each other.

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

03

According to exponentiation, it can be expressed in exponential notation simply as follows.

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

The property of adding exponents with same base product rule can be applied to more than two terms and it can be written in general form as follows.

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

For example, $16$ and $64$ are two numbers and the product of them is $1024$. It can be expressed in mathematical form as follows.

$16 \times 64 \,\,=\,\, 1024$

Express, $16$, $64$ and $1024$ in exponential form on the basis of number $4$.

$(1)\,\,\,\,\,\,$ $16 = 4 \times 4 = 4^2$

$(2)\,\,\,\,\,\,$ $64 = 4 \times 4 \times 4 = 4^3$

$(3)\,\,\,\,\,\,$ $1024 = 4 \times 4 \times 4 \times 4 \times 4 = 4^5$

Now, replace the numbers in exponential notation.

$16 \times 64 \,\,=\,\, 1024$

$\implies 4^2 \times 4^3 \,\,=\,\, 4^5$

The expression property has proved that the product of exponential terms which contain same base is equal to the base raised to the power of summation of the exponents.

$\therefore \,\,\,\,\,\, 4^2 \times 4^3 \,\,=\,\, 4^{2\,+\,3}$