What you'll learn
Simplifying and manipulating algebraic expressions is a fundamental skill that underpins all of algebra. In this guide you will learn how to collect like terms, multiply and divide algebraic terms, simplify expressions with indices, expand and simplify, and write expressions in their simplest form. These techniques are used in equations, formulae and problem solving throughout GCSE Maths.
Key terms and definitions
Expression — a collection of terms with numbers and letters, but no equals sign.
Term — a single number, variable, or product, separated by + or − signs.
Like terms — terms with exactly the same variable part (e.g. 3x and 5x).
Coefficient — the number multiplying a variable.
Simplify — write an expression in its shortest equivalent form.
Core concepts
Collecting like terms
Like terms have the same letters raised to the same powers. To simplify, add or subtract their coefficients. For example, 4x + 3x = 7x, and 5a + 2b − 3a = 2a + 2b. You cannot combine unlike terms such as x and x² or x and y — they stay separate.
Multiplying terms
To multiply algebraic terms, multiply the numbers and multiply the letters, adding indices for the same letter. For example, 3x × 4y = 12xy, and 2a × 5a = 10a². Remember 2a × 5a means the a's combine to a².
Dividing terms
To divide, divide the numbers and subtract the indices of matching letters. For example, 12x⁵ ÷ 3x² = 4x³. Algebraic fractions can be simplified by cancelling common factors in the numerator and denominator.
Using index laws
When simplifying, apply the laws of indices: multiply → add indices, divide → subtract indices, power of a power → multiply indices. For example, (2x³)² = 4x⁶. Keep coefficients and letters organised.
Expanding and simplifying
Often you expand brackets first, then collect like terms. For example, 3(x + 2) + 2(x − 1) = 3x + 6 + 2x − 2 = 5x + 4. Always finish by simplifying fully.
Worked examples
Example 1: Collecting like terms
Simplify 6x + 4y − 2x + 3y.
Combine x terms (6x − 2x = 4x) and y terms (4y + 3y = 7y): 4x + 7y.
Example 2: Multiplying
Simplify 4a × 3a².
Numbers: 4 × 3 = 12; letters: a × a² = a³. So 12a³.
Example 3: Expand and simplify
Simplify 5(2x + 1) − 3(x − 4).
10x + 5 − 3x + 12 = 7x + 17.
Common mistakes and how to avoid them
Combining unlike terms. Only add/subtract terms with identical variable parts.
Forgetting indices when multiplying. a × a = a², not a.
Sign errors when subtracting brackets. −3(x − 4) = −3x + 12; multiply the sign through.
Leaving an expression unsimplified. Always collect like terms at the end.
Mixing up multiply and add rules for indices. Add indices when multiplying like bases.
Exam technique for Simplifying Expressions
Group like terms before combining.
Multiply numbers and letters separately, using index laws.
Expand brackets carefully, watching signs.
Simplify fully — no further like terms should remain.
Check by substituting a value into the original and simplified forms.
Quick revision summary
To simplify algebraic expressions, first collect like terms — terms with identical variable parts — by adding or subtracting their coefficients (4x + 3x = 7x), keeping unlike terms separate. To multiply terms, multiply the numbers and combine the letters using the index laws (3x × 4y = 12xy; 2a × 5a = 10a²); to divide, divide the numbers and subtract indices (12x⁵ ÷ 3x² = 4x³). Apply index rules consistently: add indices when multiplying like bases, subtract when dividing, multiply for a power of a power. When brackets are involved, expand first (watching signs, especially with a leading minus) then collect like terms (5(2x + 1) − 3(x − 4) = 7x + 17). The common errors are combining unlike terms, dropping indices, sign slips, and leaving the answer unsimplified. Group like terms, handle numbers and letters separately, expand carefully, and simplify fully — these habits make all later algebra smoother.