Quotient Law of Exponents with same base

Formula

$\large \dfrac{b^{\displaystyle m}}{b^{\displaystyle n}}$ $\,=\,$ $\large b^{\displaystyle \,m-n}$

Proof

$b$ is a literal and it is multiplied by itself $m$ number of times to represent a quantity and it is denoted by $b^{\displaystyle m}$ in exponential notation. Similarly, $b$ is multiplied by itself $n$ number of times to represent another quantity and it is denoted by $b^{\displaystyle n}$.

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

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

01

If m > n

Take, the number of multiplying factors $m$ is greater than the number of multiplying factors $n$. Divide the term $b^{\displaystyle m}$ by another term $b^{\displaystyle n}$ to obtain the quotient of them.

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

$\implies \require{cancel} \dfrac{b^{\displaystyle m}}{b^{\displaystyle n}} \,=\, \dfrac{\cancel{b} \times \cancel{b} \times \cancel{b} \times \ldots \times b}{\cancel{b} \times \cancel{b} \times \cancel{b} \times \ldots \times \cancel{b}}$

As per the division rule, each factor in numerator is divided by the same factor in the denominator. Due to more factors in numerators than denominator, there are some non-cancellable factors left in the numerator. The non-cancellable factors can be evaluated by subtracting the number of factors in the denominator from the number of factors in the numerator. So, the total number of factors left in the numerator is $m-n$.

$\implies$ $\dfrac{b^{\displaystyle m}}{b^{\displaystyle n}} = \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle m-n \, factors}$

It can be expressed in exponential notation.

$\therefore \,\,\,\,\,\,$ $\dfrac{b^{\displaystyle m}}{b^{\displaystyle n}} = b^{\displaystyle m-n}$

02

If m < n

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

$\implies \require{cancel} \dfrac{b^{\displaystyle m}}{b^{\displaystyle n}} \,=\, \dfrac{\cancel{b} \times \cancel{b} \times \cancel{b} \times \ldots \times \cancel{b}}{\cancel{b} \times \cancel{b} \times \cancel{b} \times \ldots \times b}$

$\implies$ $\dfrac{b^{\displaystyle m}}{b^{\displaystyle n}} = \dfrac{1}{\underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle n-m \, factors}} $

$\therefore \,\,\,\,\,\,$ $\dfrac{b^{\displaystyle m}}{b^{\displaystyle n}} = \dfrac{1}{b^{\displaystyle n-m}}$

The property is actually derived by the division of the exponential terms. Hence, it is called as division rule of exponents. The division of them is known as the quotient of them. Hence, the identity is called as the quotient rule of exponents.

Verification

For example, $3^{\displaystyle 7}$ and $3^{\displaystyle 5}$ are two exponential terms and they both have same base. Now, divide the term $3^{\displaystyle 7}$ by $3^{\displaystyle 5}$ to obtain the quotient of them.

$\dfrac{3^{\displaystyle 7}}{3^{\displaystyle 5}} = \dfrac{3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3}{3 \times 3 \times 3 \times 3 \times 3}$

$\implies \require{cancel} \dfrac{3^{\displaystyle 7}}{3^{\displaystyle 5}} = \dfrac{\cancel{3} \times \cancel{3} \times \cancel{3} \times \cancel{3} \times \cancel{3} \times 3 \times 3}{\cancel{3} \times \cancel{3} \times \cancel{3} \times \cancel{3} \times \cancel{3}}$

$\implies \require{cancel} \dfrac{3^{\displaystyle 7}}{3^{\displaystyle 5}} \,=\, 3 \times 3 \,=\, 3^{\displaystyle 2}$

It can also be determined by subtracting the total number of factors in denominator from the total number of factors in numerator.

$\implies \require{cancel} \dfrac{3^{\displaystyle 7}}{3^{\displaystyle 5}} \,=\, 3^{\displaystyle 7-5} \,=\, 3^{\displaystyle 2}$


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