Exponents • Topic 2 of 4

Negative Exponents

A negative exponent means a reciprocal: x⁻ⁿ = 1/xⁿ. So 2⁻³ = 1/2³ = 1/8, and 1/x⁻² = x². Moving a factor across the fraction bar flips the sign of its exponent. All the usual exponent laws still apply with negative exponents, so x⁵·x⁻² = x³. To simplify, first rewrite negatives as reciprocals or combine using the laws, then make all exponents positive in the final answer. Negative exponents appear throughout SAT scientific-notation and rational-expression problems, so reading them as "one over" is the key habit.

✅ Solved examples

1. Evaluate 2⁻³.
2⁻³ = 1/2³ = 1/8.
2. Rewrite x⁻⁴ with a positive exponent.
x⁻⁴ = 1/x⁴.
3. Simplify x⁵ · x⁻²·
Add exponents: x⁵⁺(⁻²) = x³.
4. Evaluate 5⁻¹.
5⁻¹ = 1/5.

✏️ Practice — try these, take hints as needed

1. Evaluate 3⁻².
Negative exponent → reciprocal.
1/3².
—
1/9.
2. Rewrite x⁻⁶ with a positive exponent.
x⁻ⁿ = 1/xⁿ.
—
—
1/x⁶.
3. Simplify x⁷ · x⁻³.
Add the exponents.
7 + (−3).
—
x⁴.
4. Evaluate 10⁻²·
1/10².
1/100.
—
1/100 (0.01).
5. Rewrite 1/x⁻³ with a positive exponent.
Flipping crosses the bar and flips the sign.
1/x⁻³ = x³.
—
x³.

📝 Topic test — 8 questions

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