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Multiplying and Dividing Terms

FoundationChallengeAQAEdexcelOCR

Master multiplying and dividing terms for GCSE Maths with structured, exam-style practice. This Foundation resource covers multiplying and dividing algebraic terms and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Multiply the numbers, then add the indices of matching letters.

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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

Multiplying and dividing algebraic terms follows the index laws. Multiply the numbers, then combine the letters by adding indices when multiplying and subtracting them when dividing.

So \(3a \times 4a = 12a^2\): the numbers multiply to give \(12\), and \(a \times a\) gives \(a^2\). The number and the letter parts are handled separately.

Unlike collecting like terms, multiplication works across different letters. You cannot add \(3a\) and \(4b\), but you can multiply them, giving \(12ab\). Knowing which operations combine unlike terms and which do not is the key distinction.

Revision notes

Multiplying terms

Multiply the coefficients, then combine the letters using the index laws.

For \(5x \times 3x^2\): \(5 \times 3 = 15\) and \(x^1 \times x^2 = x^3\), giving \(15x^3\). Different letters simply sit alongside each other, so \(2a \times 3b = 6ab\).

Dividing terms

Divide the coefficients and subtract the indices.

For \(\frac{12x^5}{4x^2}\): \(12 \div 4 = 3\) and \(x^{5-2} = x^3\), giving \(3x^3\). Letters that cancel completely disappear.

Powers of terms

When a whole term is raised to a power, everything inside is raised to that power.

So \((3x^2)^3 = 3^3 \times x^6 = 27x^6\). Forgetting to raise the coefficient is a very common slip.

Key points

  • Multiply the coefficients and combine the letters.
  • Add indices when multiplying.
  • Subtract indices when dividing.
  • Different letters can be multiplied but not added.
  • Raise the coefficient too when a term is raised to a power.
  • A letter with no visible index has index 1.

Worked examples

Example 1

Simplify \(4a \times 3a^2\).

Working

\[4 \times 3 = 12\]multiply the coefficients
\[a^1 \times a^2 = a^3\]add the indices
\[12a^3\]combine the results

Example 2

Simplify \(\dfrac{20y^6}{5y^2}\).

Working

\[20 \div 5 = 4\]divide the coefficients
\[y^{6-2} = y^4\]subtract the indices
\[4y^4\]combine the results

Example 3

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

Working

\[2^4 = 16\]raise the coefficient to the power
\[x^{3 \times 4} = x^{12}\]multiply the indices
\[16x^{12}\]combine the results

Common mistakes

  • Adding the indices when dividing.

    Division subtracts indices, so x⁵ ÷ x² is x³, not x⁷.

  • Forgetting to raise the coefficient to the power.

    (3x²)³ is 27x⁶, not 3x⁶. The 3 is cubed too.

  • Trying to add unlike terms.

    3a and 4b cannot be added, but they can be multiplied to give 12ab.

  • Multiplying the indices when multiplying terms.

    x² × x³ is x⁵, not x⁶. Indices are added, not multiplied.

Exam tips

  • Handle the numbers and the letters as two separate steps.
  • Write out any invisible index of 1 to avoid confusion.
  • Remember to apply an outside power to the coefficient.
  • Check by substituting a small value such as x = 2.

Key terms

Index law
A rule for combining powers of the same base.
Coefficient
The number in front of a term.
Simplify
To write in the shortest equivalent form.
Base
The letter or number being raised to a power.

Related topics

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