To multiply two algebraic terms, multiply the numbers (coefficients) together first, then deal with each letter separately by adding their indices. For example, 3x² × 4x³ = 12x⁵ because 3 × 4 = 12 and x² × x³ = x²⁺³ = x⁵. The same method works for terms with multiple variables.

What is the rule for multiplying algebraic terms?

Every algebraic term has two parts: the coefficient (the number) and the variable part (the letters with their indices).

Rule:

  1. Multiply the coefficients together.
  2. For each letter that appears, add its indices. (Remember: x means x¹.)
  3. Write any letters that appear in only one term, keeping their indices unchanged.

This works because of the multiplication law of indices: aᵐ × aⁿ = aᵐ⁺ⁿ.

How do you multiply a term by a single variable?

Worked example 1: Simplify 5n × n

  1. Coefficients: 5 × 1 = 5
  2. Variable part: n¹ × n¹ = n¹⁺¹ = n²
  3. Answer: 5n²

Worked example 2: Simplify 3x × 4y

  1. Coefficients: 3 × 4 = 12
  2. Variable parts: x and y are different letters, so write both.
  3. Answer: 12xy (by convention, letters are written alphabetically)

When two different letters appear, they cannot be combined — simply write them side by side.

How do you multiply terms that both contain the same letter?

Worked example 3: Simplify 4x³ × 7x²

  1. Coefficients: 4 × 7 = 28
  2. x parts: x³ × x² = x³⁺² = x⁵
  3. Answer: 28x⁵

Worked example 4: Simplify −2a⁴ × 5a

  1. Coefficients: −2 × 5 = −10 (negative × positive = negative)
  2. a parts: a⁴ × a¹ = a⁵
  3. Answer: −10a⁵

Remember: a term without an explicit index has index 1. So a = a¹ and x = x¹.

How do you multiply terms with two different letters?

Worked example 5: Simplify 3a²b × 5ab³

  1. Coefficients: 3 × 5 = 15
  2. a parts: a² × a¹ = a³
  3. b parts: b¹ × b³ = b⁴
  4. Answer: 15a³b⁴

Worked example 6: Simplify 4xy² × 3x²y × 2

  1. Coefficients: 4 × 3 × 2 = 24
  2. x parts: x¹ × x² = x³
  3. y parts: y² × y¹ = y³
  4. Answer: 24x³y³

Three terms can be multiplied in one go — handle coefficients first, then each letter separately.

What happens when a term is multiplied by a negative term?

The sign rules for multiplication apply exactly as with numbers:

Sign combination Result
Positive × Positive Positive
Positive × Negative Negative
Negative × Positive Negative
Negative × Negative Positive

Worked example 7: Simplify −3p²q × −4pq²

  1. Signs: − × − = +
  2. Coefficients: 3 × 4 = 12, so result is +12
  3. p parts: p² × p¹ = p³
  4. q parts: q¹ × q² = q³
  5. Answer: 12p³q³

Multiplying algebraic terms is the core skill inside expanding brackets. When you write 3x(2x + 5), you are multiplying 3x by each term inside the bracket:

  • 3x × 2x = 6x²
  • 3x × 5 = 15x

So 3x(2x + 5) = 6x² + 15x

Every term produced inside an expansion is the product of two algebraic terms — the method you have learned here is what makes bracket expansion work.

What mistakes should you avoid?

Mistake Wrong answer Correct answer
Adding coefficients instead of multiplying 3x × 4x = 7x² 3 × 4 = 12; answer is 12x²
Multiplying indices instead of adding 5x² × 3x³ = 15x⁶ x² × x³ = x⁵; answer is 15x⁵
Forgetting that x = x¹ 4x³ × x = 4x³ x = x¹; x³ × x¹ = x⁴; answer is 4x⁴
Dropping the sign −6a × 2a = 6a² −6 × 2 = −12; answer is −12a²
Treating different letters as like 3x × 4y = 12x Different letters: answer is 12xy

Frequently asked questions

What if the coefficient is 1 or −1?

Write the variable part without the 1, since 1x = x and −1x = −x. For example, 5m × m = 5m², not 5m¹m or 5 × 1m². Similarly, −m × 3m = −3m². The invisible "1" counts but is not written in the final answer.

Can I multiply more than two algebraic terms together?

Yes. Multiply any two terms first, then multiply the result by the next term. For example, 2x × 3x × 4x: first 2x × 3x = 6x², then 6x² × 4x = 24x³. Alternatively, multiply all the coefficients together and add all the indices for each letter in one step: (2 × 3 × 4)x¹⁺¹⁺¹ = 24x³.

How is this different from adding algebraic terms?

When adding, only like terms combine: 3x + 4x = 7x but 3x + 4y cannot be simplified. When multiplying, any two terms can be multiplied regardless of their letters or powers: 3x × 4y = 12xy. Multiplication always produces a result; addition only simplifies if terms match.

Multiplying algebraic terms uses the multiplication law of indices (aᵐ × aⁿ = aᵐ⁺ⁿ). This topic builds the algebra fluency you need before tackling expanding double brackets, factorising and later the full range of GCSE index laws with negative and fractional indices.

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