# Proof of Negative Power Rule

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

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

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

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

### Find the Quotient of Exponents

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

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

### Possibility of Negative exponent

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

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

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

### Power Rule of Negative exponent

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

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

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

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

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