Proof of Power of a Quotient rule

Consider two quantities and express each quantity in exponential notation but the exponent of both terms is same and it is denoted by $m$. Similarly, the bases of both terms are different and they are $b$ and $c$. The exponential terms $b^{\displaystyle m}$ and $c^{\displaystyle m}$ can be written in product form as follows.

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

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

Now, let us derive the power of a quotient rule in algebraic form in mathematics.

Divide the Same exponents with Different bases

Divide the exponential term $b^{\displaystyle m}$ by the $c^{\displaystyle m}$ for obtaining the quotient of them.

$\dfrac{b^{\displaystyle m}}{c^{\displaystyle m}}$ $\,=\,$ $\dfrac{b \times b \times b \times \ldots \times b}{c \times c \times c \times \ldots \times c}$

Factorize the Unlike terms

In this division, $b$ and $c$ are two unlike terms but the total number of factors of each term is $m$. Now, factorise the unlike terms but the total number of factors is $m$.

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

Exponential form of Terms

According to exponentiation, the product form of factors can be written in exponential form to prove this property of exponents.

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

Therefore, it is proved that the quotient for the division of same exponents with different bases is equal to the power of a quotient of them.