$b^{\displaystyle m} \times c^{\displaystyle m} = (b \times c)^{\displaystyle m}$

Two or more exponential terms are appeared in multiplication but in a special case, the exponents of exponential terms are same. A special procedure is used to multiply them in mathematics and it is called as product law of exponential terms having same exponent.

The product rule tells that the product of two or more exponential terms whose exponents is same, is equal to the value of same number of times of the product of both bases.

$b^{\displaystyle m}$ is an exponential term in algebraic form and it is a product of $m$ times of a literal quantity $b$. The term $c^{\displaystyle m}$ is another exponential term in algebraic form and it is a product of $m$ times of a literal quantity $c$. The exponential terms $b^{\displaystyle m}$ and $c^{\displaystyle m}$ contain same number of multiplying factors and the total multiplicative factors in each term is $m$.

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

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

Multiply both exponential terms to obtain the product of them.

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

The product of $m$ times $b$ and $m$ times $c$ is equal to the product of $m$ times product of $b$ and $c$.

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

There are $m$ multiplying factors totally in the product and each multiplying factor is $b \times c$. So, it can be expressed in exponential notation.

$b^{\displaystyle m} \times c^{\displaystyle m} = (b \times c)^{\displaystyle m}$

This formula expresses that the product of two exponential terms whose exponents are same, is equal to the value of same number of times of the product of their bases.

It is obtained on the bases of product rule between them. Hence, this rule is called as product law of exponential terms of having same exponents. The product rule is not limited to two terms and it can be extended to several number of exponential terms. Therefore, it can be written in the following form.

$b^{\displaystyle m} \times c^{\displaystyle m} \times d^{\displaystyle m} \ldots = (b \times c \times d \ldots)^{\displaystyle m}$

The numerical verification is essential to use this property in mathematics. So, take any two exponential terms, for example $2^{\displaystyle 4}$ and $3^{\displaystyle 4}$ and write their products.

$2^{\displaystyle 4} = 2 \times 2 \times 2 \times 2 = 16$ and $3^{\displaystyle 4} = 3 \times 3 \times 3 \times 3 = 81$

Now multiply both exponential terms to obtain the product of them and the product of $2^{\displaystyle 4}$ and $3^{\displaystyle 4}$ is $1296$.

$2^{\displaystyle 4} \times 3^{\displaystyle 4} = 16 \times 81 = 1296$

Now multiply bases of both terms and take their common exponent as its power.

${(2 \times 3)}^{\displaystyle 4} = 6^{\displaystyle 4} = 6 \times 6 \times 6 \times 6 = 1296$

The product of exponential terms $2^{\displaystyle 4}$ and $3^{\displaystyle 4}$ is $1296$ and also the product of four times product of $2$ and $3$ is $1296$. There is a product relation between them and it can be learned from this example.

$2^{\displaystyle 4} \times 3^{\displaystyle 4} = (2 \times 2 \times 2 \times 2) \times (3 \times 3 \times 3 \times 3)$

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

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

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

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

$\implies 2^{\displaystyle 4} \times 3^{\displaystyle 4} = 6 \times 6 \times 6 \times 6$

$\therefore \,\, 2^{\displaystyle 4} \times 3^{\displaystyle 4} = (2 \times 3)^{\displaystyle 4} = 6^{\displaystyle 4}$

On the basis of this product rule, the product law of exponential terms whose exponent is same, is derived in mathematics.

It is verified for all the values. Hence, it is called an identity in mathematics. Due to derivation of this product rule in algebraic terms, it is generally called as an algebraic identity but it is in terms of exponential terms. So, it can also be called as an exponential identity.

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