Negative and fractional exponents

Extending exponents

The exponent laws work for all exponents, not just positive whole numbers. Following the patterns leads to two new ideas: negative and fractional exponents.

Negative exponents

Look at the pattern:

Each step down divides by 2. So:

x⁻ⁿ = 1/xⁿ

A negative exponent means "take the reciprocal". It does not make the number negative.

• 5⁻² = 1/25
• (2/3)⁻² = (3/2)² = 9/4
• 1/x⁻³ = x³

Fractional (rational) exponents

What is 9^(1/2)? By the power law, (9^(1/2))² = 9¹ = 9. So 9^(1/2) is the number that squares to 9: 3.

x^(1/n) = ⁿ√x

x^(m/n) = (ⁿ√x)ᵐ = ⁿ√(xᵐ)

A fractional exponent (or rational exponent) means a root. The denominator is the root; the numerator is the power.

Tip: Take the root first; it keeps numbers small.

Combining the laws

All the usual laws still apply:

• x^(1/2) × x^(3/2) = x²
• (x⁶)^(1/3) = x²
• x³ ÷ x⁻² = x⁵
• (4x⁻²)^(1/2) = 2x⁻¹ = 2/x

Writing answers

Usually give final answers with positive exponents:

• 3a⁻²b = 3b/a²
• x^(−1/2) = 1/√x

Why it matters

Rational and negative exponents appear in:

• Compound interest and growth formulas
• Science (10⁻⁹ m = 1 nanometre)
• Scaling laws in biology and physics

Worked examples

Example 1: 4⁻³ = 1/64.

Example 2: 81^(3/4) = (⁴√81)³ = 3³ = 27.

Example 3: Simplify (x² y⁻³)⁻² = x⁻⁴ y⁶ = y⁶/x⁴.

Common mistakes

• Thinking x⁻² is negative: 2⁻² = 1/4, not −4.
• Mixing up the root and power in m/n.
• Applying the exponent to only part of a product.

Try it yourself

1. 3⁻⁴
2. 64^(1/3)
3. 9^(3/2)
4. 16^(−1/4)
5. Simplify a⁵ × a⁻⁸.

Answers: 1) 1/81 2) 4 3) 27 4) 1/2 5) a⁻³ = 1/a³

Quick recap

• Negative exponent: x⁻ⁿ = 1/xⁿ (a reciprocal).
• Fractional exponent: x^(m/n) = (ⁿ√x)ᵐ.
• The exponent laws apply to all exponents.