Algebraic notation: reading and writing expressions correctly
Algebra is shorthand for arithmetic. Once you can translate fluently between English, numbers and letters you have unlocked half of GCSE maths. This topic looks tiny but examiners punish sloppiness here ruthlessly.
The unwritten conventions
When we write 3a we mean 3 × a. The multiplication sign is dropped between a number and a letter, and between two letters: ab means a × b. Numbers always come first, so write 5x, never x5.
Powers are written as superscripts: a × a = a², a × a × a = a³. Note 3a² means 3 × a × a, not (3a)². To square the whole 3a you must write (3a)² = 9a².
Division uses a fraction bar in algebra. Write a ÷ 2 as a/2 or ½a. Coefficients in front of fractions are usually preferred: (2x)/3 rather than 2x ÷ 3.
Like terms and combining
A term is a number, a letter, or a product of numbers and letters separated from the rest of the expression by + or –. In 4x + 3y - 2x + 5 the terms are +4x, +3y, -2x, +5. Note the sign belongs to the term that follows.
Like terms share the same letters with the same powers. 5x² and -2x² are like; 5x² and 5x are not.
Worked example: simplify 7p + 3q - 2p + q - 5.
Group: (7p - 2p) + (3q + q) - 5 = 5p + 4q - 5.
Multiplication and powers
When multiplying, multiply the numbers and add the indices on each letter:
3a²b × 4ab³ = (3 × 4)(a² × a)(b × b³) = 12a³b⁴.
Dividing flips the rule — subtract indices: 12a⁵b³ ÷ 4a²b = 3a³b².
Brackets
A number outside a bracket multiplies every term inside. 3(2x - 5) = 6x - 15. Don't forget the second term — this is the most common single-mark error in F-tier algebra.
A negative outside flips every sign. -(4x - 7) = -4x + 7. Build the habit: if there's a minus in front of a bracket, picture a -1 there.
⚠Common mistakes— Common mistakes (examiner traps)
- Confusing
3a²with(3a)². The first is3 × a²; the second is9a². Brackets matter. - Combining unlike terms.
x + x² ≠ x³. You can only add like terms. - Sign errors when a minus precedes a bracket.
5 - (2x - 3) = 5 - 2x + 3 = 8 - 2x, not5 - 2x - 3. - Writing
x · x = 2x. No —x × x = x². The "doubling" only happens with addition (x + x = 2x). - Dropping coefficients of 1 inconsistently.
xand1xare identical;-xmeans-1x. Be careful when collecting like terms.
➜Try this— Quick check
Simplify 6x + 4 - 2(x - 5).
Step 1: Expand bracket: 6x + 4 - 2x + 10. Step 2: Collect: 4x + 14. Note the +10, not -10, because -2 × -5 = +10.
AI-generated · claude-opus-4-7 · v3-deep-algebra