Index laws — integer and fractional indices

Index laws condense pages of repeated multiplication into a few simple rules. Master them and surds, standard form and exponential equations all become easier.

The three core laws (same base)

Let a be a non-zero base and m, n any integers (or rationals).

1. Multiplying: aᵐ × aⁿ = a^(m+n) — add the indices.
2. Dividing: aᵐ ÷ aⁿ = a^(m−n) — subtract the indices.
3. Power of a power: (aᵐ)ⁿ = a^(mn) — multiply the indices.

These ONLY work when the bases are the same. 2³ × 5² ≠ 10⁵.

Two derived rules

• a⁰ = 1 (any non-zero base; falls out of aⁿ ÷ aⁿ = a^(n−n) = a⁰).
• a⁻ⁿ = 1/aⁿ (negative index = reciprocal).

So 5⁻² = 1/25 and (2/3)⁻¹ = 3/2.

Power of a product / quotient

• (ab)ⁿ = aⁿbⁿ.
• (a/b)ⁿ = aⁿ/bⁿ.

Useful: (2x)³ = 8x³, not 2x³.

Fractional indices [Higher tier]

A fractional index represents a root.

• a^(1/n) = ⁿ√a. So 9^(1/2) = √9 = 3; 8^(1/3) = ³√8 = 2.
• a^(m/n) = (ⁿ√a)ᵐ (or equivalently ⁿ√(aᵐ)).

When evaluating a^(m/n) numerically, take the root first, then raise to the power. The numbers stay smaller and more manageable.

Worked example: 64^(2/3) = (³√64)² = 4² = 16. (vs trying 64² = 4096 first — same answer, much harder arithmetic.)

Combining rules — typical exam algebra

Worked example: simplify (8x⁶)^(2/3).

• Apply the power to each factor: 8^(2/3) × (x⁶)^(2/3).
• 8^(2/3) = (³√8)² = 2² = 4.
• (x⁶)^(2/3) = x⁴.
• Result: 4x⁴.

Worked example: simplify (2a²b)³ × (3a⁻¹b²)².

• (2a²b)³ = 8a⁶b³.
• (3a⁻¹b²)² = 9a⁻²b⁴.
• Multiply: 8 × 9 × a^(6−2) × b^(3+4) = 72a⁴b⁷.

Negative AND fractional indices together

a^(−m/n) = 1 / a^(m/n) = 1 / (ⁿ√a)ᵐ.

Worked example: 16^(−3/4).

• Reciprocal first: 1 / 16^(3/4).
• 16^(3/4) = (⁴√16)³ = 2³ = 8.
• Result: 1/8.