Cambridge O Level Mathematics (Syllabus D) · Syllabus 4024 · Number
Surds
What is Surds?
A surd is a root that is irrational and is therefore kept in exact root form rather than written as a decimal. Surds are simplified by extracting the largest perfect-square factor, combined only when they contain the same root, and removed from a denominator by multiplying numerator and denominator by the root itself or by the denominator's conjugate.
This definition is part of the Number chapter in Cambridge O Level Mathematics (Syllabus D).
Surds in context
Three rules cover the whole subtopic. Simplify by pulling out the largest square factor. Combine only like surds — \(3\sqrt2+5\sqrt2=8\sqrt2\), but \(\sqrt2+\sqrt3\) does not combine at all. Rationalise a denominator by multiplying top and bottom by \(\sqrt a\) for a one-term denominator, or by the conjugate for a two-term one.
Common mistakes with Surds
- 20. “\(\sqrt{288}=16.97\) is a complete answer.” Why wrongWhen a question asks for an exact value, a decimal is an approximation of it, not equal to it. Correct\(\sqrt{288}=\sqrt{144\times2}=12\sqrt2\). Keep fractions, surds and multiples of \(\pi\) exact whenever exactness is asked for or the data is exact. Say this“The largest square factor of 288 is 144, so \(\sqrt{288}=12\sqrt2\).” CheckSimplify \(\sqrt{98}\). (\(7\sqrt2\).)