Pull out the largest square factor
, so . Take out the biggest perfect square in one go.
Only like surds add
, but . Simplify each surd first to spot like terms.
Rationalise to clear the denominator
For multiply top and bottom by ; for multiply by the conjugate .
Drawn from real examiner reports.
Rationalising with the wrong multiplier
Using an incorrect multiplier for a binomial denominator — e.g. instead of the correct conjugate, or the wrong sign.
Errors with the rationalising multiplier (typically $\sqrt{2}-1$) were seen on Nov 2024 Paper 1H (Q17b).
Typing the surd into a calculator
When working is requested, entering the fraction and copying the decimal display scores no marks.
On Nov 2024 Paper 1H (Q17b) this approach "scored no marks as working was requested".
Writing only the integer part
Giving just a whole number with no surd factor (and no working) is not credited.
On Nov 2024 Paper 1H (Q17a), writing only "5" earned nothing without the surd shown.
Assuming √a + √b = √(a+b)
You can only combine LIKE surds: . In general , so , not . Simplify each surd first to reveal any like terms.
(general exam technique)
Not simplifying each surd first
Before adding, reduce every surd: and , so . Skipping this hides the like terms and leads to leaving an un-simplified sum or writing an incorrect single root.
(general exam technique)
Stay in surd form; show each step
Never round to a decimal mid-question, and show the surd manipulation — examiners require the working for full marks.
Use the conjugate for a binomial denominator
For multiply top and bottom by the conjugate ; the denominator becomes by the difference of two squares. Same terms, opposite sign — using or the wrong sign is the classic error.
Remember (√q)² = q when expanding
When multiplying brackets with surds, use . So . Forgetting that the squared surd becomes a whole number is a common sign and arithmetic trap.
Split off the largest perfect-square factor: Doing it in one step (not , which isn't fully simplified) is cleaner and avoids errors.
What is (√q)²?
Simplify , giving your answer in the form .
