Algebraic Expressions and Simplifying

Algebra is the branch of mathematics that uses letters and symbols to represent numbers and quantities. This lesson covers the essential building blocks of algebra for AQA GCSE Mathematics: writing algebraic expressions, collecting like terms, substitution, and using index notation. These skills underpin almost every other algebra topic, so a confident grasp here will pay dividends throughout your course.

Key Vocabulary

Before diving in, make sure you understand the following terms:

Term	Meaning	Example
Variable	A letter that represents an unknown value	x, y, n
Term	A single number, variable, or number multiplied by a variable	5, 3x, -2y
Coefficient	The number in front of a variable	In 7x, the coefficient is 7
Constant	A term that is just a number (no variable)	4, -3, 12
Expression	A collection of terms joined by + or - signs (no equals sign)	3x + 2y - 5
Equation	A statement that two expressions are equal	3x + 2 = 14
Formula	An equation that shows the relationship between variables	A = l x w
Identity	An equation that is true for all values of the variable	2(x + 1) = 2x + 2

Exam Tip: The AQA exam may ask you to state whether a mathematical statement is an expression, equation, formula, or identity. Remember: an expression has no equals sign, an equation is true for particular values, a formula links variables, and an identity is always true.

Writing Algebraic Expressions

In algebra, we follow certain conventions:

We write 4n instead of 4 x n (no multiplication sign).
We write n/3 or n over 3 instead of n divided by 3.
We write n squared as n^2 and n cubed as n^3.
The number always goes before the letter: we write 5x, not x5.
We write 1x simply as x and -1x as -x.

Worked Example 1

Write an expression for each of the following:

(a) I think of a number n, multiply it by 4 and add 3.

Answer: 4n + 3

(b) I think of a number x, square it, then subtract 7.

Answer: x^2 - 7

(c) A rectangle has length (2a + 1) and width 3. Write an expression for the perimeter.

Answer: Perimeter = 2 x (2a + 1) + 2 x 3 = 4a + 2 + 6 = 4a + 8

Collecting Like Terms

Like terms are terms that have exactly the same variable raised to the same power. You can add or subtract like terms to simplify an expression.

Like Terms	Not Like Terms
3x and 7x	3x and 3y
5y^2 and -2y^2	5y^2 and 5y
4ab and -ab	4ab and 4a

Rules for Collecting Like Terms

Identify terms with the same variable and power.
Add or subtract their coefficients.
Write the simplified expression.

Worked Example 2

Simplify: 5x + 3y - 2x + 7y - 4

Group like terms:

x terms: 5x - 2x = 3x
y terms: 3y + 7y = 10y
constants: -4

Answer: 3x + 10y - 4

Worked Example 3

Simplify: 4a^2 + 3a - 2a^2 + 5a - 1

Group like terms:

a^2 terms: 4a^2 - 2a^2 = 2a^2
a terms: 3a + 5a = 8a
constants: -1

Answer: 2a^2 + 8a - 1

Exam Tip: A very common mistake is to combine terms like 3x and 3x^2 — these are not like terms because the powers of x are different. Always check the power before combining.

Simplifying Expressions with Multiplication and Division

When multiplying algebraic terms:

Multiply the coefficients together.
Multiply the variables together (add the powers if the base is the same).

When dividing algebraic terms:

Divide the coefficients.
Divide the variables (subtract the powers if the base is the same).

Worked Example 4

Simplify: 3a x 4b

Multiply coefficients: 3 x 4 = 12

Multiply variables: a x b = ab

Answer: 12ab

Worked Example 5

Simplify: 2x^3 x 5x^2

Multiply coefficients: 2 x 5 = 10

Multiply variables: x^3 x x^2 = x^(3+2) = x^5

Answer: 10x^5

Worked Example 6

Simplify: 12x^4 divided by 4x

Divide coefficients: 12 / 4 = 3

Divide variables: x^4 / x = x^(4-1) = x^3

Answer: 3x^3

Index Notation (Laws of Indices)

Index notation (also called powers or exponents) is a shorthand for repeated multiplication.

Rule	In Words	Example
a^m x a^n = a^(m+n)	When multiplying, add the powers	x^3 x x^4 = x^7
a^m / a^n = a^(m-n)	When dividing, subtract the powers	x^8 / x^3 = x^5
(a^m)^n = a^(m x n)	When raising a power to a power, multiply	(x^2)^3 = x^6
a^0 = 1	Anything to the power 0 is 1	5^0 = 1
a^1 = a	Anything to the power 1 is itself	x^1 = x
a^(-n) = 1/a^n	A negative power means the reciprocal [H]	x^(-2) = 1/x^2
a^(1/n) = the nth root of a	A fractional power means a root [H]	8^(1/3) = 2

Worked Example 7

Simplify: (3x^2)^3

Cube the coefficient: 3^3 = 27

Multiply the power of x: 2 x 3 = 6

Answer: 27x^6

Exam Tip: When a bracket contains a coefficient, you must raise the coefficient to the power as well as the variable. A common error is to write (3x^2)^3 = 3x^6, forgetting to cube the 3.

Substitution

Substitution means replacing variables with given numerical values and then calculating the result.

Worked Example 8

If a = 3, b = -2, and c = 5, find the value of:

(a) 2a + b

2(3) + (-2) = 6 - 2 = 4

(b) a^2 - 3b

(3)^2 - 3(-2) = 9 + 6 = 15

(c) 3c - ab

3(5) - (3)(-2) = 15 + 6 = 21

(d) b^2 + 4c

(-2)^2 + 4(5) = 4 + 20 = 24

Common Pitfalls with Substitution

Mistake	Correct Approach
Forgetting that (-2)^2 = 4, not -4	Always put negative values in brackets when squaring
Writing 3 x -2 = -5	3 x -2 = -6 (multiply, do not add)
Ignoring order of operations	Follow BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction

Using a Flow Diagram for Expressions

The following diagram shows how to build and simplify an algebraic expression:

flowchart LR
    A[Start with a word problem] --> B[Identify the unknown - assign a letter]
    B --> C[Write each operation as an algebraic term]
    C --> D[Combine into an expression]
    D --> E{Can you collect like terms?}
    E -- Yes --> F[Simplify by collecting like terms]
    E -- No --> G[Expression is already in simplest form]

Summary

An expression is a collection of terms with no equals sign.
Like terms have the same variable raised to the same power and can be combined.
When multiplying terms, multiply coefficients and add powers of the same base.
When dividing terms, divide coefficients and subtract powers of the same base.
The laws of indices govern how to handle powers during multiplication, division, and bracket operations.
Substitution replaces letters with numbers — always use BIDMAS and take extra care with negative values.

Exam Tip: Always show each step of your working when simplifying or substituting. Even if your final answer is wrong, you can still earn method marks for correct intermediate steps.