IGCSE Maths Indices & Surds Notes
Indices are powers; surds are irrational roots left in exact form. Both rely on a small set of laws that must be applied precisely.
Key concepts
- Index laws
- Add powers to multiply, subtract to divide, multiply for a power of a power.
- Negative & fractional powers
- x⁻ⁿ = 1/xⁿ and x^(1/n) = the nth root of x.
- Simplifying surds
- Split the root over a square factor, e.g. √50 = √25·√2 = 5√2.
Key formulas
Multiplying powers
xᵃ × xᵇ = xᵃ⁺ᵇ
Dividing powers
xᵃ ÷ xᵇ = xᵃ⁻ᵇ
Power of a power
(xᵃ)ᵇ = xᵃᵇ
Worked example
Simplify √72.
- 1Find the largest square factor of 72: 36.
- 2Split: √72 = √36 × √2.
- 3Simplify √36 = 6.
Answer: 6√2
Common mistakes
✗ Adding bases as well as powers.
✓ Keep the base the same; only combine the indices.
✗ Thinking x⁰ = 0.
✓ Any non-zero value to the power 0 is 1.
✗ Not simplifying a surd fully.
✓ Take out the largest square factor from under the root.
Quick check
Try these, then reveal the answers.
Simplify x⁵ × x².Reveal answer
x⁷
Evaluate 2⁻³.Reveal answer
1/8
Simplify √18.Reveal answer
3√2
Exam tips
- Rewrite roots as fractional powers to combine with index laws.
- Rationalise a denominator by multiplying top and bottom by the surd.
- Keep surds exact until the final line unless a decimal is requested.
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Frequently asked questions
What is a surd?+
A surd is an irrational root, such as √2, left in exact form rather than as a rounded decimal.
How do I rationalise a denominator?+
Multiply the numerator and denominator by the surd in the denominator so the bottom becomes a whole number.
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