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Negative Indices

FoundationChallengeAQAEdexcelOCR

Practise negative indices with this free Foundation GCSE Maths worksheet from Virtus Academy. You'll work through using negative indices, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. x⁻ⁿ is the same as 1 ÷ xⁿ.

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Topic overview

A negative index means the reciprocal of the positive power. So \(x^{-2}\) is \(\frac{1}{x^2}\), not a negative number.

The rule follows from the division law. Since \(2^3 \div 2^5 = 2^{-2}\), and working the division out directly gives \(\frac{8}{32} = \frac{1}{4}\), it must be that \(2^{-2} = \frac{1}{4}\).

The practical method is simple: flip the base to the other side of the fraction line and make the index positive. A term with a negative index in the denominator moves up to the numerator, which surprises students the first time they meet it but follows the same rule.

Revision notes

The basic rule

\(a^{-n} = \frac{1}{a^n}\). Move the base across the fraction line and change the sign of the index.

So \(3^{-2} = \frac{1}{9}\) and \(x^{-4} = \frac{1}{x^4}\). The answer is always positive when the base is positive.

Negative indices in a denominator

A negative index below the line moves up. Since \(\frac{1}{a^{-n}} = a^n\), the term simply relocates and the index turns positive.

So \(\frac{1}{2^{-3}} = 2^3 = 8\).

Fractions raised to negative powers

Raising a fraction to a negative power turns it upside down. \(\left(\frac{2}{3}\right)^{-1} = \frac{3}{2}\), and \(\left(\frac{2}{3}\right)^{-2} = \frac{9}{4}\).

Flip first, then apply the positive power — it keeps the arithmetic simple.

Key points

  • \(a^{-n} = \frac{1}{a^n}\).
  • A negative index does not make the answer negative.
  • Move the base across the fraction line and flip the sign of the index.
  • A negative index in the denominator moves to the numerator.
  • A fraction to a negative power is inverted.
  • \(a^{-1}\) is the reciprocal of \(a\).

Worked examples

Example 1

Work out \(4^{-2}\).

Working

\[4^{-2} = \frac{1}{4^2}\]a negative index means the reciprocal
\[= \frac{1}{16}\]evaluate the positive power

Example 2

Simplify \(\frac{1}{3^{-2}}\).

Working

\[\frac{1}{3^{-2}} = 3^2\]a negative index in the denominator moves up and turns positive
\[= 9\]evaluate the power

Example 3

Work out \(\left(\frac{3}{4}\right)^{-2}\).

Working

\[\left(\frac{4}{3}\right)^{2}\]a negative power inverts the fraction
\[= \frac{16}{9}\]square the numerator and the denominator

Common mistakes

  • Writing \(2^{-3}\) as \(-8\).

    The index is negative, not the answer. 2⁻³ is 1/8.

  • Leaving the index negative in a final answer.

    Questions almost always want a positive index or a fraction, so convert before writing the answer.

  • Forgetting to apply the power after flipping a fraction.

    (3/4)⁻² becomes (4/3)², so both numbers must still be squared.

  • Moving only the base and not changing the index sign.

    Both happen together: the base crosses the line and the index becomes positive.

Exam tips

  • Rewrite every negative index as a fraction before evaluating.
  • Check your answer is positive when the base is positive.
  • For fractions, invert first and then apply the power.
  • Give final answers with positive indices unless told otherwise.

Key terms

Negative index
A power indicating the reciprocal of the positive power.
Reciprocal
One divided by a number.
Base
The number being raised to a power.
Denominator
The bottom of a fraction.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.