$b$, $m$ and $n$ are three constants. They formed two exponential terms $b^{\displaystyle \, m}$ and $b^{\displaystyle \, n}$.

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

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

On the basis of this information, let us start deriving the zero exponent rule in algebraic form.

Now, divide the exponential term $b^{\displaystyle \, m}$ by $b^{\displaystyle \, n}$ to find their quotient. It can be evaluated by using the quotient rule of exponents with same base.

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

Take $m-n = 0$, then $m = n$. It expresses that the exponents must be equal to obtain the zero power in the exponential equation.

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

The quantities in both numerator and denominator are equal. So, they can be cancelled each other.

$\implies$ $\require{cancel} \dfrac{\cancel{b^{\displaystyle \, m}}}{\cancel{b^{\displaystyle \, m}}} \,=\, b^0$

$\implies$ $1 \,=\, b^0$

$\,\,\, \therefore \,\,\,\,\,\,$ $b^0 \,=\, 1$

It is called zero power rule and it states that the value of zero exponent with any base is equal to one.

Latest Math Topics

Apr 18, 2022

Apr 14, 2022

Apr 05, 2022

Mar 18, 2022

Mar 05, 2022

Latest Math Problems

Apr 06, 2022

Mar 22, 2022

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 maths problems in different methods with understandable steps.

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved