GCSE Indices — Fractional and Negative Powers

The laws of indices are seven rules that govern how powers behave. Once you know them, questions involving x⁵, 2⁻³, or 16^(3/4) reduce to a few mechanical steps. Get the laws wrong and you'll be guessing on every Higher non-calculator paper. This guide walks through all seven, plus the meaning of fractional and negative powers, with examples and the mistakes I see most often.

The seven laws

For any base a (and b where it appears), and any powers m and n:

a^m × a^n = a^(m+n) — multiplying same base, add the powers
a^m ÷ a^n = a^(m−n) — dividing same base, subtract the powers
(a^m)^n = a^(mn) — power of a power, multiply the powers
a^0 = 1 — anything (except 0) to the power of zero is 1
a^(−n) = 1 / a^n — a negative power means reciprocal
a^(1/n) = ⁿ√a — a fractional power 1/n means nth root
a^(m/n) = (ⁿ√a)^m — combine: take the root, then the power

Fractional powers explained

The fractional power rule confuses most students because it looks unrelated to the others. It isn't. The trick is to read the denominator as "what root" and the numerator as "what power".

Worked example 1

Evaluate 8^(2/3).

Denominator 3 means cube root; numerator 2 means square it.

8^(2/3) = (∛8)² = 2² = 4

Worked example 2

Evaluate 16^(3/4).

Denominator 4 means fourth root; numerator 3 means cube it.

16^(3/4) = (⁴√16)³ = 2³ = 8

Examiner's note: Always take the root first, then the power. 16^(3/4) means (⁴√16)³, not (16³)^(1/4). Doing it the wrong way gives the same final answer but creates much harder arithmetic — for 16^(3/4) you'd be working with 4096 instead of 2.

Negative powers explained

A negative power means "take the reciprocal". So 2⁻³ = 1/2³ = 1/8.

Worked example 3

Evaluate (2/3)⁻².

Negative power → flip the fraction. Squared → square top and bottom.

(2/3)⁻² = (3/2)² = 9/4

Combining the laws

Worked example 4

Simplify (x³ × x⁵) / x².

x³ × x⁵ = x^(3+5) = x⁸

x⁸ / x² = x^(8−2) = x⁶

Worked example 5 — Higher tier

Simplify 27^(2/3) × 3⁻².

27^(2/3) = (∛27)² = 3² = 9. And 3⁻² = 1/9.

9 × 1/9 = 1

The mistakes that cost the most marks

Mistake 1 — Adding powers when bases are different. The "add the powers" rule only applies when the base is the same. 2³ × 3² stays as 8 × 9 = 72. You can't combine the powers.

Mistake 2 — Forgetting that anything to the 0 is 1. Including weird-looking things like (5x)⁰ = 1 and (−3)⁰ = 1. Only 0⁰ is undefined.

Mistake 3 — Taking a negative power as a negative number. 2⁻³ = 1/8, not −8. Negative power, not negative value.

Mistake 4 — Computing the power before the root in fractional indices. 16^(3/4) is much easier as (⁴√16)³ = 2³ = 8 than as (16³)^(1/4) = ⁴√4096. Same answer, way harder arithmetic.

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Indices — Fractional and Negative Powers