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

Exponential terms with same base are often involved in multiplication. The product of them can be obtained by evaluating the value of an exponential term having same base but contains summation of exponents of exponential terms as its power.

This fundamental product rule is called as the product law of exponents with same base.

$b$ is a letter which represents a number. Assume, it is multiplied by itself $m$ times and the product of them is written as $b^{\displaystyle m}$. Similarly, the letter $b$ is also multiplied to itself $n$ times and the product of them is written in exponential notation as $b^{\displaystyle n}$.

The expansions of exponential terms $b^{\displaystyle m}$ and $b^{\displaystyle n}$ are written in following mathematical form.

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

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

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

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

The total multiplicative factors of $b$ in the term $b^{\displaystyle m}$ is $m$ and the total multiplying factors of $b$ in the term $b^{\displaystyle n}$ is $n$.

So, the total multiplicative factors in the product of $b^{\displaystyle m}$ and $b^{\displaystyle n}$ is equal to the summation of the total multiplicative terms of $b^{\displaystyle m}$ and the total multiplicative factors of $b^{\displaystyle n}$. Therefore, the total multiplicative factors of product of $b^{\displaystyle m}$ and $b^{\displaystyle n}$ is $m+n$.

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

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

This formula is called the product rule of indices with same base. The product law of indices is not limited to two terms and can be used for numerous exponential terms. So, the product law can be written in the following form.

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

The product rule reveals that the product of two or more exponential terms having same base is equal to the value of an exponential term whose base is same but contains summation of all the exponents of all the exponential terms as its index.

The product rule of exponents having same base can be verified in numerical system by testing the formula for all the values.

$4^{\displaystyle 2}$ and $4^{\displaystyle 3}$ are two exponential terms and they both have $4$ as their common base. Expand both terms to obtain their values.

$4^{\displaystyle 2} = 4 \times 4 = 16$

$4^{\displaystyle 3} = 4 \times 4 \times 4 = 64$

Multiply both exponential terms.

$4^{\displaystyle 2} \times 4^{\displaystyle 3} = 16 \times 64$

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

The product of exponential terms $4^{\displaystyle 2}$ and $4^{\displaystyle 3}$ is $1024$. Now, form another exponential term whose base is $4$ but it contains the summation of exponents of two terms as its power. So, the term is $4^{\displaystyle (2+3)}$ and find its value.

$4^{\displaystyle 5} = 4 \times 4 \times 4 \times 4 \times 4 = 1024$

The product of the exponential terms $4^{\displaystyle 2}$ and $4^{\displaystyle 3}$ is $1024$ and also the value of $4^{\displaystyle 5}$ is $1024$. It is possible due to the product relation between them.

$4^{\displaystyle 5} = (4 \times 4) \times (4 \times 4 \times 4) = 1024$

$4^{\displaystyle 5} = 4^{\displaystyle 2} \times 4^{\displaystyle 3} = 1024$

On the basis of this product relation, the product rule $b^{\displaystyle m} \times b^{\displaystyle n} = b^{\displaystyle m+n}$ is derived in mathematics. It is verified for all the values. Hence, it is called as an identity but it is derived in algebraic form. Therefore, it is generally called as an algebraic identity. This algebraic identity is in exponential terms form. So, it can also be called as an exponential identity.

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