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Fractional and Negative Indices

HigherHigher tier onlyAQAEdexcelOCR

Master fractional and negative indices for GCSE Maths with structured, exam-style practice. This Higher resource covers fractional and negative indices and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. A negative index means 'one over', and a fractional index means a root.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Fractional indices represent roots. The denominator of the fraction tells you which root to take, so \(x^{\frac{1}{2}}\) is the square root and \(x^{\frac{1}{3}}\) is the cube root.

When the numerator is not \(1\), it becomes a power applied after the root. So \(x^{\frac{2}{3}}\) means the cube root of \(x\), then squared. Taking the root first keeps the numbers small, which matters on a non-calculator paper.

Combining a fractional index with a negative one is common at Higher tier. Deal with the negative first by taking the reciprocal, then handle the fraction as a root and a power. Working in that order avoids most of the errors.

Revision notes

Roots from denominators

\(x^{\frac{1}{n}} = \sqrt[n]{x}\). The denominator names the root.

So \(25^{\frac{1}{2}} = 5\), \(27^{\frac{1}{3}} = 3\) and \(16^{\frac{1}{4}} = 2\).

Powers from numerators

\(x^{\frac{m}{n}} = \left(\sqrt[n]{x}\right)^m\). Take the root first, then raise to the power.

For \(8^{\frac{2}{3}}\), the cube root of \(8\) is \(2\), and \(2^2 = 4\). Squaring first would mean finding the cube root of \(64\), which is more work for the same answer.

Negative fractional indices

Take the reciprocal first, then apply the root and the power.

So \(16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}}\). The fourth root of \(16\) is \(2\), and \(2^3 = 8\), giving \(\frac{1}{8}\).

Key points

  • \(x^{\frac{1}{n}} = \sqrt[n]{x}\).
  • The denominator gives the root; the numerator gives the power.
  • \(x^{\frac{m}{n}} = \left(\sqrt[n]{x}\right)^m\).
  • Take the root before applying the power.
  • A negative fractional index means the reciprocal as well.
  • Deal with the negative sign first, then the fraction.

Worked examples

Example 1

Work out \(49^{\frac{1}{2}}\).

Working

\[49^{\frac{1}{2}} = \sqrt{49}\]a denominator of 2 means the square root
\[= 7\]evaluate the root

Example 2

Work out \(27^{\frac{2}{3}}\).

Working

\[\sqrt[3]{27} = 3\]the denominator 3 means take the cube root first
\[3^2 = 9\]then apply the numerator as a power

Example 3

Work out \(16^{-\frac{3}{4}}\).

Working

\[\frac{1}{16^{\frac{3}{4}}}\]the negative index means take the reciprocal
\[\sqrt[4]{16} = 2 \text{, so } 2^3 = 8\]fourth root then cube
\[= \frac{1}{8}\]combine to give the answer

Common mistakes

  • Reading the numerator as the root.

    In x^(2/3) the 3 gives the root and the 2 gives the power, not the other way round.

  • Applying the power before the root.

    Both orders give the same answer, but powering first produces much larger numbers, which is harder without a calculator.

  • Making a negative fractional index give a negative answer.

    16^(−3/4) is 1/8, a positive fraction.

  • Forgetting to apply the numerator at all.

    8^(2/3) is 4, not 2. The cube root is only the first step.

Exam tips

  • Write out the root and the power as two separate steps.
  • Take the root first to keep the numbers manageable.
  • Handle a negative index by writing the reciprocal before anything else.
  • Check that your answer is positive when the base is positive.

Key terms

Fractional index
A power written as a fraction, representing a root and a power.
Cube root
The value that multiplies by itself three times to give a number.
Radical
The root symbol.
Reciprocal
One divided by a number, produced by a negative index.

Related topics

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