# Quotient Rule of Exponents with same base

## Formula

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

The quotient of exponents with same base is equal to the difference of their exponents with same base. It is called as the quotient law of exponents with same base and also called as division rule of exponents with same base.

### Introduction

In mathematics, two exponents with the same base are involved in division but it is not possible to divide an exponential term by another, same as dividing the numbers. Hence, a special quotient rule is required for dividing the indices with the same base.

There is a property for dividing the powers with same base and it is used to find the quotient of any two exponents with same base by subtracting their exponents with the same base.

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

#### Use

The quotient property for exponents with same base is mainly used in two different cases.

1. The quotient of exponents with same base is simplified as the difference of exponents with same base.
2. The subtraction of exponents with a base is expanded as the quotient of exponents with the same base.

### Proof

Learn how derive the quotient rule for exponents with same base.

##### Verification

$3^7$ and $3^5$ are two exponents with same base. Now, divide an exponential term by another but express each exponential term in product form.

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

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

$\implies$ $\dfrac{3^7}{3^5} \,=\, 3 \times 3$

$\implies$ $\dfrac{3^7}{3^5} \,=\, 3^2$

Look at the exponents of the exponential terms in left-hand side of the equation, they are $7$ and $5$. Now look at the exponent of the exponential term in the right-hand side of the equation. It is $2$. It can be obtained by subtracting the power $5$ from $7$.

$\,\,\, \therefore \,\,\,\,\,\,$ $\dfrac{3^7}{3^5} \,=\, 3^{7-5}$

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