Why fractional and negative indices feel harder than they are
Fractional and negative indices are one of those GCSE maths topics that look intimidating on the page but follow a small set of rules that never change. If you can evaluate $27^{-\frac{2}{3}}$ confidently, you can pick up marks on a question that a lot of students leave blank — and on the Higher paper, that is exactly the kind of question that separates a grade 6 from a grade 7.
The good news is that every fractional and negative indices question, however scary it looks, breaks down into the same three moves: deal with the minus sign, deal with the root, deal with the power. Once you have those three steps in the right order, the topic becomes one of the most reliable sources of marks on the paper.
This guide covers what negative indices actually mean, how fractional indices work, how to combine the two, and the mistakes examiners see most often — with worked examples for each.
Negative indices: flip the fraction, not the sign
A negative index does not make the answer negative. This is the single most important thing to understand about the whole topic, and it is where most dropped marks come from.
A negative index means reciprocal — you flip the base into a fraction:
$$a^{-n} = \frac{1}{a^n}$$
So $2^{-3}$ is not $-8$. It is:
$$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$
The minus sign in the power is an instruction to flip, nothing more. Once you have flipped, the power becomes positive and you evaluate as normal.
This works for fractions as bases too, and here the flip makes life easier. If the base is already a fraction, the reciprocal just turns it upside down:
$$\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^{2} = \frac{25}{4}$$
Notice what happened: flipping the fraction removed the minus sign in one move, and then we squared the top and the bottom. Try saying the rule to yourself as "minus means flip" — it is short enough to remember in an exam, and it stops you reaching for a negative answer by instinct.
One more useful case: anything to the power of zero is 1, as long as the base is not zero. So $7^0 = 1$ and $\left(\frac{3}{4}\right)^0 = 1$. It comes up as a single mark surprisingly often.
Fractional indices: the denominator is the root
A fractional index combines a root and a power in one symbol. The rule is:
$$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$$
The denominator (bottom of the fraction) tells you which root to take. The numerator (top) tells you what power to apply afterwards.
Start with the simplest case, where the numerator is 1:
$$25^{\frac{1}{2}} = \sqrt{25} = 5$$
$$64^{\frac{1}{3}} = \sqrt[3]{64} = 4$$
A power of $\frac{1}{2}$ is a square root; a power of $\frac{1}{3}$ is a cube root. That is all the notation means.
Now add a numerator bigger than 1:
$$8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$$
Here is the tactical bit: always take the root first, then the power. The maths gives the same answer either way round, but the arithmetic does not. If you square 8 first you get 64, and then you need the cube root of 64 — harder numbers, more chances for something to go wrong. Root first keeps every number small.
One more example with bigger numbers, because exam questions like to use 16, 27, 32 and 81 — bases that are perfect powers:
$$81^{\frac{3}{4}} = \left(\sqrt[4]{81}\right)^3 = 3^3 = 27$$
If you are not sure what the fourth root of 81 is, ask yourself: what number multiplied by itself four times gives 81? Knowing your powers of 2, 3 and 5 up to about $2^6$, $3^4$ and $5^3$ makes this whole topic dramatically quicker. It is worth ten minutes of practice on its own.
Putting it together: a worked example with both
The hardest version of this question combines a negative sign, a fraction, and sometimes a fractional base all at once. Here is the full method on a classic exam question.
Evaluate $27^{-\frac{2}{3}}$.
Work through the three moves in order:
Step 1 — deal with the minus sign. Minus means flip:
$$27^{-\frac{2}{3}} = \frac{1}{27^{\frac{2}{3}}}$$
Step 2 — deal with the root. The denominator of the index is 3, so take the cube root of 27:
$$\sqrt[3]{27} = 3$$
Step 3 — deal with the power. The numerator is 2, so square the result:
$$3^2 = 9$$
Putting it back together:
$$27^{-\frac{2}{3}} = \frac{1}{9}$$
Now the same method on a fractional base, which is where the top grades are decided.
Evaluate $\left(\frac{16}{25}\right)^{-\frac{3}{2}}$.
Step 1 — flip:
$$\left(\frac{16}{25}\right)^{-\frac{3}{2}} = \left(\frac{25}{16}\right)^{\frac{3}{2}}$$
Step 2 — root. The denominator of the index is 2, so square root the top and bottom separately:
$$\sqrt{\frac{25}{16}} = \frac{5}{4}$$
Step 3 — power. Cube the result:
$$\left(\frac{5}{4}\right)^3 = \frac{125}{64}$$
Done. Three steps, in the same order every time: flip, root, power. Write each step on its own line in the exam — if you make an arithmetic slip somewhere, the method marks are still yours.
Common mistakes and what examiners look for
Examiner reports on indices questions mention the same handful of misconceptions year after year. Every one of them is fixable.
Treating a negative index as a negative answer. Writing $2^{-3} = -8$ is the most common wrong answer on this topic. Before you write anything down, ask yourself: what does the minus sign in the power actually tell me to do? (It tells you to flip — the answer to $a^{-n}$ is positive whenever $a$ is positive.)
Multiplying the base by the index. Some students under pressure write $5^{\frac{1}{2}} = 2.5$. A power is not multiplication. If you catch yourself doing this, slow down and rewrite the index as a root first: $5^{\frac{1}{2}} = \sqrt{5}$.
Taking the power before the root. As covered above, $8^{\frac{2}{3}}$ evaluated as $\sqrt[3]{64}$ gives the right answer but through harder arithmetic. Examiners regularly note that students who square first are far more likely to make a numerical slip. Root first, every time.
Only flipping half the job. With something like $\left(\frac{9}{4}\right)^{-\frac{1}{2}}$, some students flip the fraction but forget to then take the root, or take the root but forget the flip. The three-step routine — flip, root, power — exists precisely so nothing gets missed. Tick each step off as you do it.
Leaving the answer in an unfinished form. If the question says "evaluate", the examiner wants a number, not $\left(\sqrt[3]{27}\right)^2$ left on the page. Check the last line of your working actually answers the question asked.
It is also worth knowing how these questions are weighted: a typical "evaluate $64^{-\frac{2}{3}}$" question is worth 2 marks — one for showing a correct first step (the flip or the root), one for the final answer. Even if the arithmetic goes wrong, a correct first line keeps you on the scoreboard.
💡 Practise now: Indices questions on Bow Tie Maths — the app builds a Topic Radar from your answers so you always know where to focus next.
Summary
Fractional and negative indices come down to three moves applied in a fixed order: a minus sign means flip the base, the denominator of a fractional power is the root, and the numerator is the power you apply last. Learn your small powers of 2, 3 and 5, take the root before the power, and show each step on its own line. Pick five evaluation questions tonight and run the flip–root–power routine on each one — by the fifth, it will feel automatic.
If you want to put this into practice, try Bow Tie Maths — it generates questions on this topic at your level and tracks your progress over time.