Notation, vocabulary and manipulation: simplify and manipulate algebraic expressions by: •• collecting like terms •• multiplying a single term over a bracket •• taking out common factors •• simplifying expressions involving sums, products and powers, including the laws of indices

The idea

This skill set covers four linked algebraic manipulations. Collecting like terms means combining terms with identical variable parts, for example 5x + 3x - 2x = 6x. Multiplying a single term over a bracket uses the distributive law: 4(2a - 3) = 8a - 12. Taking out common factors reverses that process, so 6x² + 9x = 3x(2x + 3). Simplifying sums, products and powers applies the laws of indices to algebraic terms: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. For example, 3x² × 4x³ = 12x⁵ and (2y³)² = 4y⁶. Each step must preserve equivalence, so check by substituting a value such as x = 2 into the original and simplified forms.

What earns the marks

• Collect like terms by adding or subtracting coefficients of terms with identical variable parts, for example 7a + 4a - 3a = 8a.
• Multiply a single term over a bracket using the distributive law, for example 5(3x - 2) = 15x - 10.
• Take out the highest common factor from every term, for example 12x³ + 8x² = 4x²(3x + 2).
• Apply the laws of indices to products and powers of algebraic terms, for example a⁴ × a³ = a⁷ and (b²)⁵ = b¹⁰.
• Apply the laws of indices to quotients of algebraic terms, for example c⁶ ÷ c² = c⁴.
• Simplify expressions that combine sums, products and powers, for example 2x(3x² + 4x) = 6x³ + 8x².

Common mistakes and how to put them right

• Adding unlike terms such as 3x + 4y to get 7xy.

  Put it right: Only terms with identical variable parts can be collected, so 3x + 4y stays as 3x + 4y.

• Multiplying only the first term inside a bracket, writing 3(x + 5) as 3x + 5.

  Put it right: Multiply every term, giving 3x + 15.

• Multiplying indices instead of adding them when multiplying algebraic terms, writing x² × x³ as x⁶.

  Put it right: The laws of indices require adding the powers, giving x⁵.

• Failing to apply a power to the coefficient inside a bracket, writing (3x²)³ as 3x⁶.

  Put it right: Cube the coefficient as well, giving 27x⁶.

Exam tips

• Check a simplification by substituting a simple value such as x = 2 into both the original and simplified expressions; the results must match.
• When taking out a common factor, divide every term by the factor and confirm the bracket multiplies back to the original expression.
• Write powers with clear raised indices and keep bases distinct, so a² × b³ is not combined into a single power.
• For multiple-choice questions, test each option with a numerical substitution rather than choosing by appearance.