Indices and roots
OCR J560 tests index laws across all six papers. Foundation focuses on integer indices and simple roots; Higher pushes into fractional and negative indices.
The three index laws
For any non-zero base a:
- Multiplication: a^m × a^n = a^(m+n)
- Division: a^m ÷ a^n = a^(m−n)
- Power of a power: (a^m)^n = a^(mn)
Special cases
- a^0 = 1 (any non-zero base to the power 0).
- a^1 = a.
- a^(−n) = 1/a^n (negative index = reciprocal).
Roots and fractional indices
- a^(1/2) = √a (the positive square root).
- a^(1/3) = ∛a.
- a^(1/n) = nth root of a.
- a^(m/n) = (a^(1/n))^m = (nth root of a) raised to power m.
Example: 8^(2/3) = (∛8)² = 2² = 4.
Negative fractional indices
a^(−m/n) = 1 / a^(m/n).
Example: 16^(−1/2) = 1 / √16 = 1/4.
Working with surds
A surd is a root that doesn't simplify to a rational number, e.g. √2, √7, ∛5.
Surd manipulation rules:
- √a × √b = √(ab)
- √a / √b = √(a/b)
- √(a²b) = a√b (extract square factors)
Example: √50 = √(25 × 2) = 5√2.
Rationalising: to remove a surd from a denominator, multiply top and bottom by it.
- 1/√3 = √3/3.
OCR mark scheme conventions
- B1 for using an index law correctly (often awarded for an intermediate step).
- M1 for combining indices in a multi-step calculation.
- A1 for the final answer.
- For surd questions: A1 cao for the simplified form (e.g. 5√2, not √50).
⚠Common mistakes
- Writing a^m × a^n = a^(mn) instead of a^(m+n).
- Forgetting that a^0 = 1, not 0.
- Misreading a^(2/3) as "2/3 of a" — it means cube root, then square.
- Not simplifying surds: leaving √50 instead of 5√2.
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves