Division Rule of Exponential terms having same Base

Formula

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

quotient rule of exponents with same base

The quotient of ratio between two exponential terms having same base is equal to the product of number of times (obtained from subtraction of exponents) of same base.

This fundamental property gives a quotient of two exponential terms. Hence, the rule is called as quotient rule of exponential terms having same base. The quotient from both exponential terms is obtained by division law. So, it is also called as the division law of exponents with same base.

Proof

$b^{\displaystyle m}$ and $b^{\displaystyle n}$ are two exponential terms but they both have same base. The number of multiplying factors in the terms $b^{\displaystyle m}$ and $b^{\displaystyle n}$ are $m$ and $n$ respectively.

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

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

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

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

The numerator and denominator have product of same literal quantities. So, they can be cancelled each other but remember the number of multiplying factors in numerator is not equal to the number of multiplicative factors in denominator.

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

There are some multiplicative factors left in the numerator. The numerator contains $m$ terms and denominator contains $n$ terms. The $n$ factors in denominator cancel the $n$ factors from the $m$ factors in numerator. So, the total number of factors left in the numerator is $m-n$.

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

Now, the product consists of $m-n$ multiplicative factors and it can be expressed in exponential notation.

$\dfrac{b^{\displaystyle m}}{b^{\displaystyle n}} = b^{\displaystyle (m-n)}$

Verification

The quotient property of exponential terms having same base can be verified in number system.

$3^{\displaystyle 7}$ and $3^{\displaystyle 5}$ are two exponential terms but they have same base. Now, divide the $3^{\displaystyle 7}$ by $3^{\displaystyle 5}$ to obtain the quotient.

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

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

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

The numerator and denominator contains same multiplicative factors but the number of multiplying factors in numerator is not equal to the number of multiplicative factors in denominator. So, the number of multiplicative factors in denominator can cancel same number of multiplicative factors in numerator. However, the remaining number of multiplicative factors can be obtained by subtracting the exponents.

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

Due to successful numerical verification of quotient rule of exponents having same base, it is called an identity but the identity is derived in algebraic form. Therefore, it is also called as an algebraic identity in mathematics.

The algebraic identity is expressed in terms of exponential form. For this reason, the quotient rule of exponents with same base can also call as exponential identity.

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