Math Doubts

Negative Power Rule


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

The negative power of a quantity is equal to the reciprocal of the power of the same quantity. It is called negative power rule of the exponents.


$b$ and $n$ are two literals and represent two constants. Assume, they formed two exponential terms $b^{\displaystyle \, n}$ and $b^{\displaystyle -n}$.

The quantity of the positive exponential term $b^{\displaystyle \, n}$ is equal to the reciprocal of the negative exponential term $b^{\displaystyle -n}$.

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

This property is called as negative exponent rule or negative power rule.


Learn how to derive the negative power rule of exponents in algebraic form.


$3^5$ is an exponential term and express its reciprocal in mathematical form.


According to power zero rule, the number $1$ in the numerator can be written as the $3$ is raised to the power of zero.

$\implies$ $\dfrac{1}{3^5} \,=\, \dfrac{3^0}{3^5}$

In the right-hand side of the equation, the bases are same. Therefore, the quotient of the exponents with same base is the equal to the difference of the exponents with same base as per quotient rule of exponents with same base.

$\implies$ $\dfrac{1}{3^5} \,=\, 3^{\,0-5}$

$\implies$ $\dfrac{1}{3^5} \,=\, 3^{-5}$

$\,\,\, \therefore \,\,\,\,\,\,$ $3^{-5} \,=\, \dfrac{1}{3^5}$

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