Exact working means keeping surds, fractions and π as symbols until the last line, and only rounding if the question asks for it. The reason is precision: a decimal you write early is already wrong in the later digits, and every step after it carries the error.
The fix is to change what you do with a square root: treat it like a letter, not like a number to be converted.
Why do students reach for decimals?
A calculator makes √5 look like 2.236…, and that feels like progress. On a paper where exact answers are expected, it is the opposite. The decimal hides the structure that later steps need, such as a difference of two squares.
Checking whether a calculator is allowed is the first step. Read the current 0606 syllabus on the Cambridge page and confirm the rules with your exam centre. Exact working is worth practising in both cases.
What are the core exact moves?
- Add and subtract like terms: 2√5 + 3√5 = 5√5, just as 2x + 3x = 5x.
- Multiply surds: √5 × √5 = 5.
- Expand with brackets: (3 + √5)² = 9 + 6√5 + 5 = 14 + 6√5.
- Rationalise: remove the surd from the denominator with a conjugate.
- Keep fractions exact: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8, and stay with 15/8 instead of writing 1.875 unless asked.
Worked example: rationalising the denominator
Express (3 + √5)/(3 − √5) in the form a + b√5, where a and b are rational numbers.
Step 1. The conjugate of 3 − √5 is 3 + √5. Multiply top and bottom by it.
Step 2. Numerator. (3 + √5)(3 + √5) = 9 + 3√5 + 3√5 + 5 = 14 + 6√5.
Step 3. Denominator. (3 − √5)(3 + √5) = 9 − 5 = 4, using the difference of two squares.
Step 4. The fraction is (14 + 6√5)/4. Divide each term by 2: (7 + 3√5)/2.
Step 5. In the required form, this is 7/2 + (3/2)√5, so a = 7/2 and b = 3/2.
Check. Use √5 ≈ 2.236. The original is 5.236/0.764 ≈ 6.85. The answer is (7 + 6.708)/2 ≈ 6.85. The two agree.
What is the plausible mistake?
A student multiplies top and bottom by 3 − √5 instead, the same sign as the denominator. The denominator becomes (3 − √5)² = 9 − 6√5 + 5 = 14 − 6√5, which still contains a surd. The method has made nothing simpler.
Another common slip is rounding early: replacing √5 by 2.2 in the middle of the working and giving a final answer such as 6.8. It looks close, but it is not exact, and it would be marked as approximate when the question asks for exact form.
The repair is to use the conjugate, which flips the sign between the terms, so the middle terms cancel and only a square remains.
Self-check
- Simplify 6/√3.
- Expand and simplify (√7 + 2)(√7 − 2).
- Express 1/(2 + √3) in the form a + b√3.
Show answer
- Multiply top and bottom by √3: 6√3/3 = 2√3.
- Difference of two squares: 7 − 4 = 3.
- Multiply top and bottom by 2 − √3. Numerator: 2 − √3. Denominator: 4 − 3 = 1. So the answer is 2 − √3, with a = 2 and b = −1.
How do I build exact working as a habit?
Decide at the start whether the answer is exact, then keep every step in that form. Use the non-calculator working trainer to check each fraction step, and the quadratic structure explorer to see roots that come out as surds, such as those from x² − 4x + 1 = 0.
The advanced non-calculator reasoning module covers this in order: start with exact surd and trigonometric values, then choosing algebra before numerical substitution and checking a result by a second method. Key words are in the Additional Mathematics terminology guide.
If exact working keeps costing you marks, online one-to-one Additional Mathematics tuition lets a teacher watch your pen, find the step where you reach for a decimal, and rebuild that step.