The multiplication of two or more exponents with different bases and same exponents equals to the product of different bases raised to the power of the exponent is called the power of a product rule. The property is also called as multiplying powers with same exponents product rule.

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

Two or more exponential terms which contain same exponents but having different bases, often participate in multiplication, whereas it is not possible to find the product of them directly.

Therefore, a product rule is required to find the product of them mathematically and it is called the power of a product rule.

$b$ and $c$ are two literals. Multiply the literal $b$ by same literal number $m$ times to represent their product as $b^{\displaystyle m}$. Similarly, multiply the literal $c$ by the same literal $m$ times to represent their product as $c^{\displaystyle m}$ mathematically.

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

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

01

Multiply both exponential terms to obtain the product of them.

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

02

The bases are different but the number of multiplying factors of them is same. So, each multiplying factor of a term can be multiply the each multiplying of the second term.

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

03

The total number of multiplying factors of product of different bases $b$ and $c$ is also $m$. As per exponentiation, it can be written in exponential notation as follows.

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

It is proved that the multiplication of exponents with different bases and same exponents is equal to the product of the bases raised to the power of the same exponent. The property can be extended to more two terms as well.

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

$2^4$ and $3^4$ are two exponential terms and they both have same exponent $4$ but different bases.

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

$(2) \,\,\,\,\,\,$ $3^4 = 3 \times 3 \times 3 \times 3 = 81$

Multiply both of them.

$2^4 \times 3^4 = 16 \times 81 = 1296$

Now, multiply both bases and find the value of the product of them raised to the power of $4$.

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

Now, check both values to understand this property.

$\,\,\, \therefore \,\,\,\,\,\,$ $2^4 \times 3^4 \,=\, {(2 \times 3)}^4 \,=\, 1296$

Thus, the power of a product rule is used for multiplying powers with same different bases in mathematics.