Division Rule of Exponential terms having same Exponent

Formula

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

quotient rule of exponents with same exponent

The quotient of ratio between two exponential terms having same exponent is equal to the product of same number of times of quotient of the ratio between the bases.

This property is used to obtain the quotient. So, the exponential rule is called the quotient law of exponential terms having same exponent. Actually, the quotient is calculated by the division and it causes to also call this rule as division law of exponential terms with same exponent.

Proof

$b^{\displaystyle m}$ and $c^{\displaystyle m}$ are two exponential terms and they are products of $m$ times of $b$ and $m$ times of $c$ respectively.

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

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

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 in $b^{\displaystyle m}$ is $m$ and the number of multiplying factors in term $c^{\displaystyle m}$ is also $m$. So, the ratio of $m$ times of $b$ to $m$ times of $c$ can be expressed as the product of $m$ times of ratio of $b$ to $c$.

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

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

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

Verification

This quotient property of exponents can be verified in numerical system.

$2^{\displaystyle 4}$ 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 four times quotient of ratio of $2$ to $5$.

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

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

The quotient of division of $2^{\displaystyle 4}$ by $5^{\displaystyle 4}$ is $\dfrac{16}{625}$ and the quotient of four times of division of $2$ by $5$ is also $\dfrac{16}{625}$. The product in numerator and product in denominator contains same number of multiplying factors, each multiplicative factor in numerator and denominator can be written as a fraction. Thus, they both get same quotient by the division.

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

Thus, the quotient rule of exponential terms having same exponent is derived in mathematics. It is verified for all the values. Hence, the quotient law of exponential terms with same exponent is called as an identity.

However, it is derived in algebraic form. Thus, it is also called as an algebraic identity in mathematics but the algebraic identity is in exponential terms. Therefore, the division law of exponents with same exponent can also be called as the exponential identity.

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