A long expression simplifies when you recognise its structure: a difference of squares, a common factor, a repeated power. The skill appears in fractions, indices and exact arithmetic, and it supports the rest of advanced non-calculator reasoning.
What structures should you look for?
Four patterns cover most cases:
- Common factor: 6x + 12 = 6(x + 2).
- Difference of two squares: x² − 9 = (x − 3)(x + 3).
- Quadratic that factorises: x² − x − 12 = (x − 4)(x + 3).
- Common power in indices: 2ⁿ⁺³ = 2ⁿ × 8, so 2ⁿ⁺³ − 2ⁿ⁺¹ = 2ⁿ(8 − 2).
The habit is to write the structure beside each piece before doing anything else. Division by a fraction becomes multiplication by its reciprocal, and then the cancelling is visible.
Worked example
Simplify (x² − x − 12)/(x² − 9) ÷ (x − 4)/(x + 3).
Step 1, factorise each part: x² − x − 12 = (x − 4)(x + 3), and x² − 9 = (x − 3)(x + 3).
Step 2, turn the division into multiplication: the expression becomes (x − 4)(x + 3)/((x − 3)(x + 3)) × (x + 3)/(x − 4).
Step 3, cancel matching factors: (x − 4) cancels, and one (x + 3) in the numerator cancels with the (x + 3) in the bottom. This leaves (x + 3)/(x − 3).
Step 4, note restrictions: the original expression is undefined for x = 3, x = −3 and x = 4.
Answer: (x + 3)/(x − 3).
Check with x = 5: the original is (25 − 5 − 12)/(25 − 9) ÷ (1/8) = (8/16) × 8 = 4. The answer gives 8/2 = 4. They agree.
The mistake to watch for
A common slip is to cancel terms that are added, not factors that are multiplied.
Mistaken working: (x² − x − 12)/(x² − 9) = (−x − 12)/(−9)
The student crossed out x² from both lines as if it were a factor.
The correction is to factorise first. Only brackets that multiply the whole numerator or the whole bottom can be cancelled. Testing one number, such as x = 5, shows the wrong answer gives 17/9 while the original gives 8/16, which is 1/2.
Check yourself
Try these without a calculator, then open each answer.
1. Simplify (x² − 16)/(x² + x − 12).
Show answer
x² − 16 = (x − 4)(x + 4) and x² + x − 12 = (x + 4)(x − 3). Cancel (x + 4): (x − 4)/(x − 3). Check x = 5: the original gives 9/18 = 1/2, and the answer gives 1/2 as well.
2. Simplify (2ⁿ⁺³ − 2ⁿ⁺¹)/2ⁿ.
Show answer
2ⁿ⁺³ = 8 × 2ⁿ and 2ⁿ⁺¹ = 2 × 2ⁿ. So the numerator is 2ⁿ(8 − 2) = 6 × 2ⁿ, and dividing by 2ⁿ gives 6. Check n = 1: (16 − 4)/2 = 6.
3. Find 37 × 43 + 37 × 57 without a calculator.
Show answer
Take out the common factor 37: 37 × (43 + 57) = 37 × 100 = 3700. Check: 1591 + 2109 = 3700.
Where this leads next
Once structure comes quickly, move to explaining a proof step that a calculator cannot supply. The quadratic structure explorer shows how a quadratic factorises and why, and the non-calculator working trainer lets you practise exact steps.
Some students can factorise in isolation but do not see the factors when an expression is long. That first-glance scan is what our teachers coach in online one-to-one Additional Mathematics tuition.