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

The product of exponential terms having same base is equal to the base number raised to the power of sum of the exponents. It is called the product identity of the indices or exponents.

$b$ is a literal number. Two quantities are expressed in exponential form on the basis of $b$ and the number of multiplying factors of $b$ for the both quantities are $m$ and $n$. Therefore, the quantities in exponential notation are represented as $b^{\displaystyle m}$ and $b^{\displaystyle n}$.

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

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

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

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

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.

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

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

The formula is known as the product rule of exponents and it is not limited to exponential terms and it can be extended to countless multiplying exponential terms.

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

The product law of indices is mainly used to express the product of two or more exponential terms having same base as the base number raised to the power of sum of the exponents. It is also used in the reverse operation.

For example, $16$ and $64$ are two numbers and the product of them is $1024$.

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

Now, write both numbers in exponential notation on the basis of number $4$.

$16 = 4 \times 4 = 4^2$

$64 = 4 \times 4 \times 4 = 4^3$

Similarly, write the number $1024$ in exponential form on the basis of $4$.

$1024 = 4 \times 4 \times 4 \times 4 \times 4 = 4^5$

$\implies$ $1024 = (4 \times 4) \times (4 \times 4 \times 4) = 4^5$

$\implies$ $1024 = 4^2 \times 4^3 = 4^5$

$\implies$ $1024 = 4^2 \times 4^3 = 4^{2+3}$

$\therefore \,\,\,\,\,\,$ $4^2 \times 4^3 = 4^{2+3} = 1024$

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