Negative and Fractional Indices: GCSE Maths

Negative and fractional indices GCSE questions apply two extra rules beyond the basic index laws: a negative power flips the base into a reciprocal, and a fractional power turns into a root. Master those two ideas and every higher-tier index question becomes a short, mechanical chain of steps to work through.

What do negative and fractional indices actually mean?

A negative index tells you to take the reciprocal of the base raised to the positive version of that power. A fractional index tells you to take a root: the denominator of the fraction is the type of root, and the numerator is a power applied before or after the root is taken.

These are not separate topics from the standard laws of indices — they are extensions of the same three rules you already know (multiplying, dividing, and raising a power to a power). Higher-tier GCSE papers expect you to combine negative and fractional powers in a single question, so understanding why each rule works matters more than memorising it blindly.

What are the index laws for negative and fractional powers?

These sit alongside the multiplication law ($a^m \times a^n = a^{m+n}$), the division law ($a^m \div a^n = a^{m-n}$), and the power law ($(a^m)^n = a^{mn}$) that you meet earlier in the course. Every negative or fractional indices GCSE question can be solved by applying one or more of these rules in sequence.

How do you simplify a negative index?

1. Rewrite the term as a fraction with the positive index on the bottom.
2. Evaluate the positive power first, then take the reciprocal.
3. Simplify the resulting fraction if the numbers cancel.

Worked example: Evaluate $4^{-3}$.

$$4^{-3} = \frac{1}{4^3} = \frac{1}{64}$$

The base (4) stays the same throughout — only the sign of the exponent changes, which flips the term from a whole number into a fraction.

How do you simplify a fractional index?

1. Identify the denominator of the fraction — this tells you which root to take.
2. Identify the numerator — this tells you what power to raise the result to.
3. Take the root first (usually smaller numbers to work with), then apply the power.

Worked example: Evaluate $27^{2/3}$.

The denominator is 3, so take the cube root of 27 first: $\sqrt[3]{27} = 3$. The numerator is 2, so square that result: $3^2 = 9$.

$$27^{2/3} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$$

Taking the root before the power keeps the numbers small — cubing 27 first would give 19,683, which is far harder to find the cube root of.

How do you combine negative and fractional indices in one question?

Higher-tier papers often stack both ideas together. Treat the negative sign and the fraction as two separate instructions applied one after the other: deal with the fraction to find the positive-power value, then take the reciprocal for the negative sign.

Worked example: Evaluate $16^{-1/2}$.

1. The fraction $\frac{1}{2}$ means "square root": $\sqrt{16} = 4$.
2. The negative sign means "reciprocal": $\dfrac{1}{4}$.

$$16^{-1/2} = \frac{1}{16^{1/2}} = \frac{1}{\sqrt{16}} = \frac{1}{4}$$

What are the steps for tackling any negative or fractional indices question?

1. Check the sign of the index. If negative, plan to take a reciprocal at the end.
2. Check for a fraction in the index. If present, identify which root (denominator) and which power (numerator) it represents.
3. Take the root first, before applying any power — this keeps the arithmetic manageable.
4. Apply the power to the result of the root.
5. Apply the reciprocal last, if the original index was negative.
6. Simplify the final fraction or number fully.

Following this order every time — root, then power, then reciprocal — removes the guesswork from even the most stacked index expressions.

Which mistakes cost the most marks?

The most common error is treating a negative index as meaning "the answer is negative" — it does not. $2^{-3}$ equals $\frac{1}{8}$, a small positive fraction, not $-8$. A second frequent slip is applying the power before the root, which is mathematically valid but leads to much larger, error-prone numbers. A third is forgetting that $a^0 = 1$ for any non-zero value of $a$, regardless of what $a$ is.

Exam questions on this topic frequently ask you to simplify expressions like $\left(\dfrac{8}{27}\right)^{-2/3}$, which combines a fraction base, a fractional index, and a negative sign all in one. Working through the same root-power-reciprocal order handles this without difficulty: cube root of $\frac{8}{27}$ is $\frac{2}{3}$, squared gives $\frac{4}{9}$, and the reciprocal gives $\frac{9}{4}$.

Frequently asked questions

Does a negative index make the answer negative?

No. A negative index changes a number into its reciprocal — it does not change its sign. $5^{-2}$ equals $\frac{1}{25}$, which is positive, because only the base and the sign of the exponent are involved in the reciprocal rule, not the sign of the final value.

Which do I do first: the root or the power, in a fractional index?

Take the root first wherever possible. For $a^{m/n}$, finding $\sqrt[n]{a}$ before raising to the power of $m$ keeps the numbers smaller and the arithmetic simpler, especially with larger values of $a$.

What does a zero index equal?

Any non-zero number raised to the power of zero equals 1. This follows from the division law of indices: $a^n \div a^n = a^{n-n} = a^0$, and any number divided by itself equals 1, so $a^0$ must also equal 1.

Do negative and fractional indices appear on the foundation tier?

Fractional indices are a higher-tier-only topic on all major exam boards. Negative indices appear on both tiers, but foundation-tier questions rarely combine them with fractions — that combination is reserved for higher-tier papers.

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