Exponents & Roots • Topic 2 of 5

Negative & Fractional Exponents

A negative exponent means "take the reciprocal": a⁻ⁿ = 1/aⁿ, so 2⁻³ = 1/8 and (3/4)⁻² = (4/3)² = 16/9. Nothing about the sign of the answer changes — a negative exponent never makes a positive base negative. A fractional exponent packs a root and a power together: a^(m/n) = the n-th root of a, raised to the m-th power. So 8^(2/3) = (∛8)² = 2² = 4, and 16^(3/4) = (⁴√16)³ = 2³ = 8. Read the denominator as the root and the numerator as the power, and always take the root first — the numbers stay small and calculator-free. A classic GRE Quantitative Comparison exploits the fact that a base between 0 and 1 behaves "backwards": (1/2)⁻¹ = 2 is larger than the base, and (1/2)³ = 1/8 is smaller than the base. Slow down whenever the base is a proper fraction or the exponent is negative.

✅ Solved examples

1. Evaluate 2⁻⁴.
Negative exponent ⇒ reciprocal: 2⁻⁴ = 1/2⁴ = 1/16.
2. Evaluate 27^(2/3).
Root first: ∛27 = 3, then square: 3² = 9.
3. Evaluate (3/5)⁻².
Flip the fraction, drop the sign: (5/3)² = 25/9.
4. Quantitative Comparison. Let 0 < x < 1. Quantity A: x⁻¹. Quantity B: x². Which is greater?
For 0 < x < 1, x⁻¹ = 1/x > 1 (bigger than 1) while x² < x < 1. So Quantity A is greater.

✏️ Practice — try these, take hints as needed

1. Evaluate 5⁻².
Reciprocal of 5².
1/5².
1/25.
1/25
2. Evaluate 16^(3/4).
Take the 4th root first.
⁴√16 = 2.
Then 2³.
8
3. Evaluate (2/3)⁻³.
Flip the fraction.
(3/2)³.
27/8.
27/8
4. Simplify 9^(−1/2).
Negative ⇒ reciprocal; 1/2 ⇒ square root.
1/√9.
1/3.
1/3
5. Evaluate 32^(2/5).
Fifth root of 32 first.
⁵√32 = 2.
Then 2².
4

📝 Topic test — 8 questions

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