# Proof of Zero Power Rule

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

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

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

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 $b^\, m$ by $b^\, n$ to find their quotient. It can be evaluated by using the quotient rule of exponents with same base.

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

### 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.

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

### Power Rule of Zero exponent

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

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

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

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

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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