$b$, $m$ 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.
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