Negative Power Rule of Indices

Formula

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

negative power law of exponents

The reciprocal of an exponential term having a positive power is equal to the exponential term having negative power and vice-versa.

This rule of exponents is used to convert positive power of exponential terms as negative power of exponential terms and vice-versa. Hence, the power rule of indices can be called as power changing law of exponents. This power rule is used to switch the powers. Therefore, it can also be called as power swapping law of exponents.

Proof

The exponential term $b^{\displaystyle n}$ is a representation of the product of $n$ times of literal quantity $b$.

$b^{\displaystyle n} = \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle n \, factors}$

Take its reciprocal.

$\dfrac{1}{b^{\displaystyle n}} = \dfrac{1}{b \times b \times b \times \ldots \times b}$

The product of $n$ times of literal quantity $b$ is denoted by $b^{\displaystyle n}$ but it is in reciprocal form and it is denoted by its opposite power. So, the reciprocal of product of $n$ times of literal quantity is denoted by $b^{\displaystyle -n}$.

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

Replace the product of $n$ times of literal quantity $b$ by its exponential notation.

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

This formula represents how to swap the powers of exponential terms mathematically. Similarly, it can also be written as follows on the basis of same power rule.

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

The power changing rule is true for all the values but it is derived in algebraic form. So, the formula is called as an algebraic identity. However, it is also called as exponential identity due to the involvement of exponential terms.

Example

$4^{\displaystyle -5}$ is an exponential term having negative power. Use negative power rule of indices to transform the term as positive power exponential term.

$\implies 4^{\displaystyle -5} = \dfrac{1}{4^{\displaystyle 5}}$

Now, expand the term to write the term in product form.

$\implies 4^{\displaystyle -5} = \dfrac{1}{4 \times 4 \times 4 \times 4 \times 4}$

Multiply the like factors to obtain the product of them.

$\implies 4^{\displaystyle -5} = \dfrac{1}{1024}$


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