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Proof of Power of a Product rule

Take two quantities and write them in product form as equal number of factors ($m$) on the basis of two different quantities $b$ and $c$. The two quantities are expressed in exponential notation as $b^{\displaystyle m}$ and $c^{\displaystyle m}$ respectively.

$(1) \,\,\,$ $b^{\displaystyle m} \,=\, \underbrace{b \times b \times b \ldots b}_{\displaystyle m \, factors}$

$(2) \,\,\,$ $c^{\displaystyle m} \,=\, \underbrace{c \times c \times c \ldots c}_{\displaystyle m \, factors}$

Now, let’s learn the process of deriving the power of a product rule in mathematics.

Multiply the Same exponents with Different bases

To obtain the product of them, multiply the same exponents with different bases.

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

Factorize the Unlike terms

$b$ and $c$ are two unlike terms and they both are factored as $m$ factors in the product. If the unlike terms are multiplied each other, the total number of factors is also equal to $m$.

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

Exponential form of Terms

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.

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

Therefore, it is proved that product of same exponents with different bases is equal to the exponent with product of the bases. This product rule can also be used for more than two terms in the mathematics.

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