$b$, $m$ and $n$ are literals and they represent constants. The three constants form two exponential terms $b^{\displaystyle \, m}$ and $b^{\displaystyle \, n}$.

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

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

Now, let’s derive the negative exponent rule by using these basic steps.

Divide the exponential term $b^{\displaystyle \, m}$ by $b^{\displaystyle \, n}$ to evaluate their quotient. It is evaluated by using the quotient rule of exponents with same base.

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

The total number of factors in exponential term $b^{\displaystyle \, m}$ is equal to zero. It means $m = 0$

$\implies$ $\dfrac{b^0}{b^{\displaystyle \, n}} \,=\, b^{\displaystyle \, 0-n}$

$\implies$ $\dfrac{b^0}{b^{\displaystyle \, n}} \,=\, b^{\displaystyle \, -n}$

According to zero power rule, the $b$ raised to the power of zero is one.

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

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

Therefore, it is proved that value of negative exponent with a base is equal to the reciprocal of the exponent with same base.

Latest Math Topics

Mar 21, 2023

Feb 25, 2023

Feb 17, 2023

Feb 10, 2023

Latest Math Problems

Mar 03, 2023

Mar 01, 2023

Feb 27, 2023

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved