The Negative Exponent Rule states that a nonzero base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent. Mathematically, it is expressed as:

$$a^{-n} = \frac{1}{a^n}, \quad \text{where } a \neq 0$$

Explanation

• The negative exponent tells us to take the reciprocal (flip the base to the denominator).
• The exponent itself remains positive in the denominator.
• If the base is already in the denominator, moving it to the numerator makes the exponent positive.

Examples

1. ( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} )
2. ( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} )
3. ( \left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2 = \frac{16}{9} )
4. ( \frac{1}{7^{-2}} = 7^2 = 49 )

This rule is useful in simplifying expressions and solving equations involving exponents.

Latex

a^{-n} = \frac{1}{a^n}, \quad \text{where } a \neq 0