Roots and indices (laws and fractional indices)

Edexcel 1MA1 tests integer index laws on Foundation, fractional and negative indices on Higher. Higher Paper 1H frequently uses these in non-calculator settings.

Index laws (integer)

For a > 0:

• Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ.
• Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
• Power of a power: (aᵐ)ⁿ = aᵐⁿ.
• Zero index: a⁰ = 1.
• Negative index: a⁻ⁿ = 1/aⁿ.

Fractional indices (Higher)

• a^(1/2) = √a (the positive square root).
• a^(1/3) = ∛a (the cube root).
• a^(1/n) = ⁿ√a.
• a^(m/n) = (ⁿ√a)ᵐ = (aᵐ)^(1/n).

Practical tip: take the root first (smaller numbers to manipulate), then raise to the power.

Example: 8^(2/3) = (∛8)² = 2² = 4.

Negative fractional indices

a^(−m/n) = 1 / a^(m/n).

Example: 16^(−3/4) = 1 / 16^(3/4) = 1 / (⁴√16)³ = 1 / 2³ = 1/8.

Worked Higher example

Simplify 27^(2/3) × 27^(−1/3).

Method 1 — index law: 27^(2/3 − 1/3) = 27^(1/3) = 3. Method 2 — direct: 27^(2/3) = (∛27)² = 9. 27^(−1/3) = 1/3. Product = 9 × 1/3 = 3.

Combining with surds (touches A23/N8)

Surds are roots that cannot be simplified to integers. Index laws still apply:

• (√a)² = a.
• √a × √a = a.
• √(ab) = √a × √b.

Common Edexcel exam tip

On Paper 1H, "Find the value of 16^(3/4)" — show the root first, then the power: ⁴√16 = 2; 2³ = 8. Each step scores B1 in the mark scheme.