$b$, $m$ and $n$ are literals and they represent constants. The three constants form two exponential terms $b^{\displaystyle \, m}$ and $b^{\displaystyle \, n}$.
$(1) \,\,\,\,\,\,$ $b^{\displaystyle \, m}$ $\,=\,$ $\underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle m \, factors}$
$(2) \,\,\,\,\,\,$ $b^{\displaystyle \, n}$ $\,=\,$ $\underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle n \, factors}$
Now, let’s derive the negative exponent rule by using these basic steps.
Divide the exponential term $b^{\displaystyle \, m}$ by $b^{\displaystyle \, n}$ to evaluate their quotient. It is evaluated by using the quotient rule of exponents with same base.
$\dfrac{b^{\displaystyle \, m}}{b^{\displaystyle \, n}} \,=\, b^{\displaystyle \, m-n}$
The total number of factors in exponential term $b^{\displaystyle \, m}$ is equal to zero. It means $m = 0$
$\implies$ $\dfrac{b^0}{b^{\displaystyle \, n}} \,=\, b^{\displaystyle \, 0-n}$
$\implies$ $\dfrac{b^0}{b^{\displaystyle \, n}} \,=\, b^{\displaystyle \, -n}$
According to zero power rule, the $b$ raised to the power of zero is one.
$\implies$ $\dfrac{1}{b^{\displaystyle \, n}} \,=\, b^{\displaystyle \, -n}$
$\,\,\, \therefore \,\,\,\,\,\,$ $b^{\displaystyle \, -n} \,=\, \dfrac{1}{b^{\displaystyle \, n}}$
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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