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

The product of same exponents with same or different bases is equal to the power of a product of them. It is called as the power of a product rule.

Two or more exponential terms with same power are often involved in multiplication in mathematics but it is impossible for multiplying them directly same as the numbers. Hence, a special product rule is required to multiply the indices with same exponent. In this case, the bases of the exponential terms can be same or different.

A property for multiplying the exponents with same power is derived in mathematical form. It expresses that the product of two or more exponents with same power is equal to the power of a product of the bases.

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

The product rule of exponents with same power is used in two different cases.

- The product of exponents with same power is written as power of a product of the bases.
- The power of an exponent with a base is written as product of exponents with different bases.

Learn how to derive the power of a product rule in mathematical form.

The product of two numbers $16$ and $81$ is equal to the $1296$.

$16 \times 81 = 1296$

Express the numbers $16$ and $81$ as exponents with same power.

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

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

Now, express the relationship between the numbers $16$, $81$ and $1296$ in exponential notation.

$16 \times 81 = 1296$

$\implies$ $2^4 \times 3^4 = 1296$

Now, write the number $1296$ in exponential notation but the exponent of term should be $4$.

$1296$ $=$ $6 \times 6 \times 6 \times 6$ $=$ $6^4$

Now, factorize the base $6$.

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

Now, check both values to understand this property.

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

Therefore, it is verified that the product of exponents with same power is equal to the exponent with product of the bases.

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