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Proof of Zero Power Rule

$b$, $m$ 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.

Find the Quotient of Exponents

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.

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

Possibility of Zero exponent

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

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

Power Rule of Zero exponent

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.

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