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Laws of Indices

FoundationChallengeAQAEdexcelOCR

Master laws of indices for GCSE Maths with structured, exam-style practice. This Foundation resource covers applying the laws of indices and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. When multiplying powers you add the indices; when dividing you subtract them.

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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.

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Topic overview

The laws of indices tell you how to combine powers. Multiplying powers of the same base adds the indices, dividing subtracts them, and raising a power to a power multiplies them.

These rules only apply when the bases match. You can simplify \(x^3 \times x^4\) to \(x^7\), but \(x^3 \times y^4\) cannot be combined into a single power.

Two results follow from the division law. Anything to the power zero is \(1\), and a negative index means the reciprocal. Both look odd until you see them as the pattern continuing downwards from the positive powers.

Revision notes

The three laws

Multiplying adds indices: \(a^m \times a^n = a^{m+n}\). Dividing subtracts them: \(a^m \div a^n = a^{m-n}\). A power of a power multiplies them: \((a^m)^n = a^{mn}\).

So \(x^2 \times x^5 = x^7\), \(x^8 \div x^3 = x^5\), and \((x^3)^4 = x^{12}\).

Coefficients

Numbers in front are handled separately from the powers.

For \(4x^3 \times 5x^2\): multiply \(4 \times 5 = 20\) and add the indices to get \(x^5\), giving \(20x^5\).

Zero and negative indices

Since \(a^3 \div a^3 = a^0\) and anything divided by itself is \(1\), it follows that \(a^0 = 1\).

Continuing downwards gives \(a^{-n} = \frac{1}{a^n}\), so \(x^{-2} = \frac{1}{x^2}\).

Key points

  • \(a^m \times a^n = a^{m+n}\).
  • \(a^m \div a^n = a^{m-n}\).
  • \((a^m)^n = a^{mn}\).
  • The bases must match for the laws to apply.
  • \(a^0 = 1\).
  • \(a^{-n} = \frac{1}{a^n}\).

Worked examples

Example 1

Simplify \(y^6 \times y^3\).

Working

\[y^{6+3}\]multiplying adds the indices
\[= y^9\]add them

Example 2

Simplify \(\dfrac{18x^7}{6x^4}\).

Working

\[18 \div 6 = 3\]divide the coefficients
\[x^{7-4} = x^3\]subtract the indices
\[3x^3\]combine the results

Example 3

Simplify \((2y^4)^3\).

Working

\[2^3 = 8\]cube the coefficient as well
\[y^{4 \times 3} = y^{12}\]multiply the indices
\[8y^{12}\]combine the results

Common mistakes

  • Multiplying the indices when multiplying powers.

    x² × x⁵ is x⁷, not x¹⁰. Multiplication adds indices.

  • Applying the laws to different bases.

    x³ × y⁴ cannot be written as a single power.

  • Forgetting to raise the coefficient.

    (2y⁴)³ is 8y¹², not 2y¹².

  • Thinking a negative index gives a negative answer.

    x⁻² is 1/x², which is positive when x is positive.

Exam tips

  • Handle coefficients and powers as separate steps.
  • Check the bases are the same before combining.
  • Write invisible indices of 1 explicitly.
  • Convert negative indices to fractions in the final answer.

Key terms

Index
A power showing repeated multiplication.
Base
The number or letter being raised to a power.
Coefficient
The number multiplying a term.
Reciprocal
One divided by a value, produced by a negative index.

Related topics

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