Cambridge IGCSE Mathematics · Syllabus 0580 · 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 IGCSE Mathematics.
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.” Extended only 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 \(\sqrt{98}\). (\(7\sqrt2\).)