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.

What Are Negative and Fractional Indices?

The standard index laws (multiplying, dividing, and raising powers) still apply, but negative and fractional indices introduce two new ideas.

A negative index means "take the reciprocal." So x⁻¹ = 1/x, and x⁻² = 1/x². The negative sign does not make the answer negative — it flips the base to the bottom of a fraction.

A fractional index means "take a root." The denominator of the fraction tells you which root to take, and the numerator tells you the power. So x^(1/2) = √x, x^(1/3) = ∛x, and x^(2/3) = (∛x)². You can apply the root first and then the power, or vice versa — but taking the root first usually keeps the numbers smaller.

Key Formulas

Step-by-Step Method

1. If the index is negative, write the reciprocal (flip to 1 over the base raised to the positive index).
2. If the index is a fraction, identify the denominator (root) and the numerator (power).
3. Take the root first to keep numbers small, then raise to the power.
4. If combining with other index laws, apply the laws as normal (add indices when multiplying, subtract when dividing).
5. Simplify the result.

Worked Example 1 — Foundation Level

Question: Evaluate 25^(1/2).

Working:

Step 1 — The index 1/2 means "square root."

Step 2 — √25 = 5.

Answer: 5

Worked Example 2 — Higher Level

Question: Evaluate 8^(−2/3).

Working:

Step 1 — The negative index means reciprocal: 8^(−2/3) = 1 / 8^(2/3).

Step 2 — The denominator 3 means cube root: ∛8 = 2.

Step 3 — The numerator 2 means square: 2² = 4.

Step 4 — So 8^(2/3) = 4, and 8^(−2/3) = 1/4.

Answer: 1/4

Worked Example 3 — Exam Style

Question: Simplify fully (16x⁸)^(3/4). (3 marks)

Working:

Step 1 — Apply the power 3/4 to each part separately: 16^(3/4) × (x⁸)^(3/4).

Step 2 — 16^(3/4): the fourth root of 16 is 2, and 2³ = 8. So 16^(3/4) = 8.

Step 3 — (x⁸)^(3/4) = x^(8 × 3/4) = x⁶.

Answer: 8x⁶

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.

Practice Questions

Q1 (Higher): Evaluate 27^(1/3).

Q2 (Higher): Evaluate 49^(−1/2).

Q3 (Higher): Simplify (27x⁶)^(2/3).

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Related Topics

• Indices and Index Laws
• Powers and Roots
• Simplifying Surds

Summary

• A negative index means reciprocal: x⁻ⁿ = 1/xⁿ.
• A fractional index means a root: x^(1/n) = nth root of x.
• For x^(m/n), take the nth root first, then raise to the power m.
• x⁰ = 1 for any non-zero value of x.
• Take the root before the power to keep numbers manageable.
• Show each step clearly in exams — separate the root and power for method marks.