Negative & Zero Exponents

A negative exponent tells you to flip, not to go negative.

2⁻³ = 1/2³ = 1/8. The answer is positive.

Anything (except 0) to the power 0 is 1. Not 0.

A factor with a negative exponent moves across the fraction bar and the exponent turns positive. x⁻² in the numerator becomes x² in the denominator, and vice versa.

Simplest routine for an expression like 3x⁻²: only the x moves, because the exponent is attached to x alone. It becomes 3/x², not 1/(3x²).

The definitions a⁰ = 1 and a⁻ⁿ = 1/aⁿ (a ≠ 0) are forced by requiring the quotient rule to hold for all integers, as shown by aᵐ/aᵐ = a⁰ and a⁰/aⁿ = a⁻ⁿ. They are the unique consistent extension, not a convention chosen for convenience.

Consequently (a/b)⁻ⁿ = (b/a)ⁿ, which is often the fastest simplification route for a compound fraction raised to a negative power.

Two precision points. First, the exponent binds only to its immediate base: in 3x⁻² the 3 is unaffected, while in (3x)⁻² it is. Second, a⁻ⁿ is positive for positive a regardless of n; sign and exponent are independent, so (−2)⁻³ = −1/8 is negative because the *base* is negative, not because the exponent is.

Standard simplified form leaves no negative exponents, which is why the final step of most exponent problems is rewriting them as reciprocals.