Quotient Law of Exponents with same Exponent

Formula

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

Proof

$b$ and $c$ are two literal numbers. The literal $b$ is multiplied by itself $m$ number of times to represent a quantity. Similarly, the literal number $c$ is multiplied by the same literal $n$ number of times to represent another quantity.

They both are expressed in exponential form as follows.

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

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

Divide the term $b^{\displaystyle m}$ by $c^{\displaystyle m}$ to obtain the quotient of them.

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

The number of multiplicative factors of both exponential terms is same and it is $m$.

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

The total $\Bigg(\dfrac{b}{c}\Bigg)$ terms in the product is $m$. So, the product can be expressed in exponential notation.

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

The property of the exponents is called the quotient rule or division rule of exponents with same exponent.

Verification

$2^{\displaystyle 4}$ and $5^{\displaystyle 4}$ are two exponential terms and they both have $4$ as their exponent commonly. Divide $2^{\displaystyle 4}$ by $5^{\displaystyle 4}$ to obtain the quotient of them.

$\dfrac{2^{\displaystyle 4}}{5^{\displaystyle 4}} = \dfrac{2 \times 2 \times 2 \times 2}{5 \times 5 \times 5 \times 5} = \dfrac{16}{625}$

Now, calculate the value of product of quotient of $2$ by $5$ four number of times.

${\Bigg(\dfrac{2}{5}\Bigg)}^{\displaystyle 4} = \dfrac{2}{5} \times \dfrac{2}{5} \times \dfrac{2}{5} \times \dfrac{2}{5} $

$\implies {\Bigg(\dfrac{2}{5}\Bigg)}^{\displaystyle 4} = \dfrac{2 \times 2 \times 2 \times 2}{5 \times 5 \times 5 \times 5} = \dfrac{16}{625}$

Compare both results to understand the quotient identity of exponents.

$\therefore \,\, {\Bigg(\dfrac{2}{5}\Bigg)}^{\displaystyle 4} = \dfrac{2^{\displaystyle 4}}{5^{\displaystyle 4}} = \dfrac{16}{625}$


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