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A surd is a number written in a form that includes a square root. For example \sqrt 3 , 1+\sqrt 2 and 7\sqrt 5 are all examples of surds. A surd is a useful way of giving a value in a precise, or exact, form where there is no need to be concerned with accuracy or decimal places, and therefore a calculator is not needed.
Examination papers often use square numbers in questions on surds, so look out for these numbers \qquad \qquad \qquad 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256...
and simple multiples of them.
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Summary/Background

The expression \sqrt{x} means the positive square root of x and is called a surd.
Surds such as \sqrt{2} can be evaluated on a calculator, for example \sqrt{2} = 1.414... \, \,, however this immediately introduces the issue of accuracy. Instead of evaluating, we use some algebraic properties of surds in order to simplify them, for example factors that are square numbers themselves.
Remember the all-important rules:
  • \sqrt{ab} = \sqrt{a}\sqrt{b}
  • \displaystyle \sqrt{\frac{a}{b} } =\frac{ \sqrt{a} }{\sqrt{b} }
Be aware also of these common mistakes when a and b are both positive:
  • \sqrt{a + b} \ne \sqrt{a} + \sqrt{b}
  • \sqrt{a - b} \ne \sqrt{a} - \sqrt{b}

Software/Applets used on this page

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Glossary

square root

of a number n, that value that when squared equals n

surd

A number containing one or more irrational square roots.

union

The union of two sets A and B is the set containing all the elements of A and B.

Full Glossary List

This question appears in the following syllabi:

SyllabusModuleSectionTopic
AP Calculus AB (USA)1Algebra and FunctionsSurds
AP Calculus BC (USA)1Algebra and FunctionsSurds
AQA A-Level (UK - Pre-2017)C1Algebra and FunctionsSurds
AQA AS Maths 2017Pure MathsAlgebraSurds
AQA AS/A2 Maths 2017Pure MathsAlgebraSurds
CCEA A-Level (NI)C1Algebra and FunctionsSurds
B Algebra and FunctionsB2 SurdsSurds
Edexcel A-Level (UK - Pre-2017)C1Algebra and FunctionsSurds
Edexcel AS Maths 2017Pure MathsAlgebraic ExpressionsSurds
Edexcel AS/A2 Maths 2017Pure MathsAlgebraic ExpressionsSurds
I.B. Higher Level2Algebra and FunctionsSurds
I.B. Standard Level1Algebra and FunctionsSurds
Methods (UK)M1Algebra and FunctionsSurds
OCR A-Level (UK - Pre-2017)C1Algebra and FunctionsSurds
OCR AS Maths 2017Pure MathsIndices and SurdsSurds
OCR MEI AS Maths 2017Pure MathsSurds and IndicesSurds
OCR-MEI A-Level (UK - Pre-2017)C1Algebra and FunctionsSurds
Pre-U A-Level (UK)1Algebra and FunctionsSurds
Universal (all site questions)AAlgebra and FunctionsSurds
WJEC A-Level (Wales)C1Algebra and FunctionsSurds