# 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 $b^m$ and $c^m$ can be written in product form as follows.

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

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

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 $b^m$ by the $c^m$ for obtaining the quotient of them.

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

### 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 $m$.

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

### Exponential form of Terms

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

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

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.

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