O-Level A-Math and SEC G3 Additional Mathematics K341
A3: Surds
Simplify exact forms, rationalise denominators, and check for extraneous roots after squaring.
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Core notes
Surds preserve exact irrational values. Work with them by exposing like radicals, rationalising denominators, and checking any equation after squaring because squaring can introduce an extraneous root.
Operations and rationalising denominators
Use for suitable non-negative values and extract perfect-square factors. Add or subtract only like surds. To rationalise a single surd denominator, multiply by that surd; for a binomial denominator, multiply by its conjugate so the denominator becomes a difference of squares.
Equations involving surds
Isolate one surd before squaring. Simplify, solve the resulting equation, and substitute every candidate into the original relation. A candidate that satisfies the squared equation may fail the original sign condition.
Formulae and methods
| Relationship | Use |
|---|---|
| Rationalise a binomial surd denominator. | |
| Extract a square factor without losing the absolute-value condition. |
Worked examples
Example 1: Simplify .
- Multiply numerator and denominator by the conjugate .
- The denominator becomes .
- Expand the numerator.
Answer: .
Example 2: Solve .
- The right side must be non-negative, so .
- Square: , giving .
- The candidates are and ; only satisfies the original equation.
Answer: .
Chapter checkpoint
- Simplify every radical before deciding which terms are like surds.
- Choose a rationalising factor that removes all surds from the denominator.
- Check candidate solutions in the original equation after isolating and squaring a surd.
Exam traps and retrieval check
Avoid these traps
- Combining unlike surds before simplifying them.
- Multiplying by the same binomial instead of the conjugate.
- Accepting every root produced after squaring.
Check from memory
When are two surd terms like terms?
After simplification, they have the same radical part.
What is the conjugate of ?
.
Why can squaring create an extraneous solution?
Opposite-signed expressions have equal squares even though they are not equal in the original equation.
Official K341 coverage
This chapter maps to 2 official outcomes: A3.1, A3.2. Check the official 2027 K341 syllabus for exact assessable wording and paper details.