Surds and Indices Cheat Sheet

Surds and Indices Cheat Sheet

Quick Revision

Indices are powers, such as a³; surds are roots that cannot be simplified to a whole number, such as √2. The same small set of laws governs both.

1. Laws of Indices

aᵐ × aⁿ = a⁽ᵐ⁺ⁿ⁾
aᵐ ÷ aⁿ = a⁽ᵐ⁻ⁿ⁾
(aᵐ)ⁿ   = a⁽ᵐ × ⁿ⁾
(a × b)ⁿ = aⁿ × bⁿ

2. Zero, Negative and Fractional Powers

a⁰   = 1          (a ≠ 0)
a⁻ⁿ  = 1 / aⁿ
a^(1/n) = ⁿ√a
a^(m/n) = ⁿ√(aᵐ)

3. Rules for Surds

√a × √b = √(a × b)
√a ÷ √b = √(a / b)

Simplify a surd by pulling out the largest square factor, e.g. √50 = √(25 × 2) = 5√2.

4. Rationalising the Denominator

Remove a surd from the denominator by multiplying by its conjugate:

1 / (√3 − 1) = (√3 + 1) / [(√3 − 1)(√3 + 1)] = (√3 + 1) / 2

5. Worked Example

2³ × 2²  = 2⁵ = 32
8^(2/3)  = (³√8)² = 2² = 4
√50      = 5√2

6. Exam Quick Tips

• You can only add or subtract like surds: 2√3 + 5√3 = 7√3, but √2 + √3 does not simplify.
• A negative power means reciprocal, not a negative number.
• Always simplify a surd before comparing or adding.

Keep revising with the rest of the Maths section:

• All Maths topics
• Logarithms Cheat Sheet
• Simplification Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.