GCSE Maths Indices & Surds Notes
Indices are powers; surds are exact roots. Knowing the index laws and how to simplify surds unlocks marks across number and algebra.
Key concepts
- Index (power)
- Tells you how many times to multiply the base, e.g. 5³ = 5 × 5 × 5 = 125.
- Negative index
- Means a reciprocal: a⁻ⁿ = 1/aⁿ, so 2⁻² = 1/4.
- Fractional index
- A root: a^(1/2) = √a, and a^(m/n) = (ⁿ√a)ᵐ.
- Surd
- An irrational root left in exact form, e.g. √5, kept exact for accuracy.
Key formulas
Index laws
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1
Simplifying surds
√(ab) = √a × √b
e.g. √12 = √4 × √3 = 2√3.
Rationalising
1/√a = √a / a
Worked example
Simplify √50 + √8.
- 1Simplify each surd: √50 = √25 × √2 = 5√2.
- 2√8 = √4 × √2 = 2√2.
- 3Add like surds: 5√2 + 2√2 = 7√2.
Answer: 7√2
Common mistakes
✗ Thinking √a + √b = √(a + b).
✓ You cannot add surds inside one root — simplify each first, then add like terms.
✗ Writing a⁻² as −a².
✓ A negative power is a reciprocal: a⁻² = 1/a², not a negative number.
✗ Adding indices when the bases differ.
✓ The index laws only apply when the bases are the same.
Quick check
Try these, then reveal the answers.
Simplify 3⁴ ÷ 3².Reveal answer
3² = 9
Write √18 in the form a√b.Reveal answer
3√2
Evaluate 16^(1/2).Reveal answer
4
Exam tips
- Look for a square-factor (4, 9, 16, 25 …) when simplifying surds.
- Leave surd and π answers exact on the non-calculator paper.
- To rationalise, multiply top and bottom by the surd in the denominator.
Related practice
Practise this topic
Exam-style questions with instant Study Coach feedback.
Flashcards
Quick-fire recall to lock in the key facts.
Worksheets
Printable practice you can work through offline.
Past papers
Full papers with examiner-grade marking.
Study Coach
Ask for a step-by-step explanation any time.
All Mathematics
Every Mathematics topic across every exam board.
Related topics
Key terms explained
Short, exam-ready definitions with a formula, a worked example and an examiner tip.
Frequently asked questions
What does a fractional power mean?+
The denominator is a root and the numerator is a power, so 8^(2/3) = (³√8)² = 2² = 4.
Why leave answers as surds?+
Surds are exact — rounding them early loses accuracy and can cost the final mark.
Create a free account to practise indices & surds with Study Coach feedback
Try 5 questions now, get instant examiner-grade marking and ask the Study Coach to explain anything. Free during our early launch period — no credit card required.
Related on MathsGradeUp
Keep studying
- Mathematics courses & specifications
Every board and level we cover for mathematics.
- Formula sheets
Every GCSE, IGCSE and A Level formula, searchable and explained.
- Maths & science glossary
Plain-English definitions with a formula, worked example and examiner tip.
- Free maths calculators & solvers
Step-by-step solvers for the calculations that come up in every paper.
- Past paper guides
Paper structure, timings and topic weightings board by board.
- Grade boundaries
Historic boundaries and what each grade typically needs.
- Browse subjects
Specification-mapped courses for every board and level.
Practise it
- Predicted papers
GCSE and A Level predicted papers with full mark schemes.
- Question bank
Filter thousands of exam-board questions by topic and difficulty.
- Past papers workspace
Write on real papers and get every line marked.