Proof of Negative Power Rule

$b$, $m$ 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.

Find the Quotient of Exponents

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.

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

Possibility of Negative exponent

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

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

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

Power Rule of Negative exponent

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.