Sheet № 81 · Higher only · AQA · Edexcel · OCR
Negative and Fractional Indices –
Negative and fractional indices extend the index laws you already know into Higher tier territory. These questions appear frequently on AQA, Edexcel, and OCR papers and are often worth 2-3 marks each. Once you understand what negative and fractional powers mean, they become very manageable.
§Key definitions
Question:
Evaluate 25^(1/2).
Q1 (Higher):
Evaluate 27^(1/3).
Q2 (Higher):
Evaluate 49^(−1/2).
Q3 (Higher):
Simplify (27x⁶)^(2/3).
§Formulas to memorise
x⁻ⁿ = 1/xⁿ — a negative index means reciprocal
x^(1/n) = the nth root of x — a unit fraction index means a root
x^(m/n) = (the nth root of x)^m — root first, then power
x⁰ = 1 — any non-zero number to the power zero equals 1
Worked example
Evaluate 25^(1/2).
Working:
⚠ Common mistakes
- ✗Thinking a negative index makes the answer negative. x⁻² = 1/x², not −x². The negative sign indicates a reciprocal, not a negative number.
- ✗Applying the power before the root. While mathematically valid, computing 8² = 64 then ∛64 is harder than doing ∛8 = 2 then 2² = 4. Take the root first.
- ✗Confusing x^(1/2) with x/2. A fractional index is not the same as dividing by the denominator. x^(1/2) = √x, which is very different from x ÷ 2.
✦ Exam tips
- →When you see a fractional index, write out what the root and power are separately before calculating. This earns method marks.
- →Memorise the key roots: ∛8 = 2, ∛27 = 3, ∛64 = 4, ∛125 = 5, as well as fourth roots of 16 and 81.
- →If the question involves algebra, use the power law (xᵃ)ᵇ = x^(ab) to simplify the expression step by step.