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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.


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}

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square root

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


A number containing one or more irrational square roots.


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

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