Math Doubts

Proof of Power of a Quotient rule

Consider two quantities and express each quantity in exponential notation but the exponent of both terms is same and it is denoted by $m$. Similarly, the bases of both terms are different and they are $b$ and $c$. The exponential terms $b^{\displaystyle m}$ and $c^{\displaystyle m}$ can be written in product form as follows.

$(1) \,\,\,$ $b^{\displaystyle m}$ $\,=\,$ $\underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle m \, factors}$

$(2) \,\,\,$ $c^{\displaystyle m}$ $\,=\,$ $\underbrace{c \times c \times c \times \ldots \times c}_{\displaystyle m \, factors}$

Now, let us derive the power of a quotient rule in algebraic form in mathematics.

Divide the Same exponents with Different bases

Divide the exponential term $b^{\displaystyle m}$ by the $c^{\displaystyle m}$ for obtaining the quotient of them.

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

Factorize the Unlike terms

In this division, $b$ and $c$ are two unlike terms but the total number of factors of each term is $m$. Now, factorise the unlike terms but the total number of factors is $m$.

$\implies$ $\dfrac{b^{\displaystyle m}}{c^{\displaystyle m}}$ $\,=\,$ $\underbrace{\Bigg(\dfrac{b}{c}\Bigg) \times \Bigg(\dfrac{b}{c}\Bigg) \times \Bigg(\dfrac{b}{c}\Bigg) \times \ldots \times \Bigg(\dfrac{b}{c}\Bigg)}_{\displaystyle m \, factors}$

Exponential form of Terms

According to exponentiation, the product form of factors can be written in exponential form to prove this property of exponents.

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

Therefore, it is proved that the quotient for the division of same exponents with different bases is equal to the power of a quotient of them.

Math Doubts
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more