This module is about reasoning when the answer must stay exact: surds, exact trigonometric values, expressions that simplify neatly once you notice the pattern, short proofs and the habit of checking a result a second way. One idea runs through all of it: look at the structure of the question before you start calculating.
Check the current Cambridge Additional Mathematics 0606 syllabus for calculator rules, given formulae and exact-value requirements in your exam year. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need confident factorising, expanding brackets, simplifying surds and the basic ideas of trigonometry. If quadratic work is shaky, revisit quadratic structure and discriminants, and for identities see trigonometric identities.
An orienting example
Given that x = 2 + √3, find the exact value of x + 1/x, and then of x² + 1/x².
Step 1, find 1/x: 1/(2 + √3). Multiply top and bottom by 2 − √3. The bottom becomes 4 − 3 = 1, so 1/x = 2 − √3.
Step 2, add: x + 1/x = (2 + √3) + (2 − √3) = 4. The surds cancel.
Step 3, use structure, not substitution: (x + 1/x)² = x² + 2 + 1/x², so x² + 1/x² = 4² − 2 = 14.
Check by the long way: x² = 7 + 4√3 and 1/x² = 7 − 4√3. Adding gives 14. Both methods agree.
Squaring 2 + √3 and its reciprocal would have worked, but the structure gave the answer in one line.
In which order should you study it?
- Use exact surd and trigonometric values: the exact values for 30°, 45° and 60° and rationalising, the vocabulary of the whole module.
- Choose algebra before numerical substitution: shows how a sum and product or a sum and reciprocal gives an answer without finding the unknown.
- Simplify a long expression by recognising structure: factorising and common powers turn long fractions into short ones.
- Explain a proof step that a calculator cannot supply: why testing values is not proof, and how to justify the key step.
- Check a result using a second independent method: how to catch slips before the marks are lost.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Forgetting the middle term when squaring a bracket with a surd, so (√3 + 1)² becomes 4 instead of 4 + 2√3.
- Mixing up exact values, such as giving sin 60° as 1/2.
- Cancelling terms across a sum instead of factorising first.
- Checking a pattern with three examples and calling it a proof.
- Repeating the same method as the check, which repeats the same slip.
Each lesson shows one of these in full and then corrects it.
How should you use the practice set?
Attempt each question on paper with no calculator, and write the structure you noticed in one line before you start. Then compare with the worked answer.
The non-calculator working trainer gives a second opinion on exact arithmetic, and the mistake log helps you keep track of which trap caught you. For help with this topic see Additional Mathematics tuition.