Negative Exponents – Powers That Create Fractions

A negative exponent doesn't mean the result is negative – it means “take the reciprocal.” x⁻ⁿ = 1/xⁿ. So 2⁻³ isn't -8; it's 1/2³ = 1/8, a small positive fraction. This rule extends the quotient law smoothly below zero: continuing the pattern x³, x², x¹, x⁰ (dividing by x each time), the next steps naturally become 1/x, 1/x², 1/x³ – exactly what the negative exponent rule predicts.

The Negative Exponent Rule

x⁻ⁿ = 1/xⁿ, for any non-zero base x.

Write 5⁻² as a fraction.

5⁻² = 1/5² = 1/25.

Evaluating a Negative Exponent

Evaluate 3⁻⁴.

3⁻⁴ = 1/(3⁴) = 1/81 = 1/81.

Real-Life Application

Scientific notation: very small numbers use negative powers of 10 (like 10⁻³ = 0.001).
Physics: units like m⁻² describe rates involving inverse area.
Algebra: simplifying expressions after applying the quotient law with a larger denominator exponent.

Key Takeaways

A negative exponent means “take the reciprocal”: x⁻ⁿ = 1/xⁿ.
The result of a negative exponent on a positive base is always a positive fraction, never a negative number.
Fractional exponents, covered next, extend powers to represent roots.

Practice: Negative Exponents

Negative Exponents as Fractions