# Power of a Quotient Rule

## Formula

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

The quotient of division of same exponents with different bases is equal to the exponent with the quotient of their bases. It is called the power of a quotient rule, also called as the quotient or division rule of same exponents.

### Introduction

In some cases, two exponential terms are involved in division with same powers and different bases but it is not possible to evaluate the quotient of them directly like product of numbers. So, a mathematical identity is essential to find the quotient of same exponents with different bases.

There is a property for dividing the same indices with different bases and it is called power of a quotient rule. It states that the quotient of any two same exponents with different bases is equal to the exponent with quotient of their bases.

$b$, $c$ and $m$ are literals and they are constants. They form two exponential terms $b^{\displaystyle m}$ and $c^{\displaystyle m}$.

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

#### Use

The quotient property for the same exponent with different bases is used as a formula in two different cases.

1. The quotient of same exponents with different bases is simplified as the exponent with quotient of their bases.
2. The power of a quotient is expanded as the quotient of same exponents with different bases.

### Proof

Learn how to derive the power of a quotient rule in algebraic form in mathematics.

#### Verification

$2^4$ and $5^4$ are two exponential terms in which the exponents are same but their bases are different. Now, Divide $2^4$ by $5^4$ for obtaining the quotient of them.

$\dfrac{2^4}{5^4}$

Now, express each exponential term in product form.

$\implies$ $\dfrac{2^4}{5^4}$ $\,=\,$ $\dfrac{2 \times 2 \times 2 \times 2}{5 \times 5 \times 5 \times 5}$

The fraction can be written as the product of the some factors.

$\implies$ $\dfrac{2^4}{5^4}$ $\,=\,$ $\dfrac{2}{5} \times \dfrac{2}{5} \times \dfrac{2}{5} \times \dfrac{2}{5}$

Now, express the product of factors in exponential notation.

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

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

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