Surds – GCSE Maths Revision and Worksheets

What you need to know:

Surds are defined to be numbers that we leave in root form, \sqrt{2}, rather than decimal form, 1.414213…, so to express their exact value. You may be asked to simplify surds by various applications of the rule \sqrt{ab} = \sqrt{a}\sqrt{b}; this rule is very important for all the work we do surrounding surds, so familiarity with it is necessary.

We say that a number is in surd form when it cannot be simplified any further. For example, \sqrt{3} is in surd form, however \sqrt{27} can be simplified using the above rule as follows:

\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}

Exam questions often ask that you leave your answers in surd form. In this example, you would likely be deducted a mark if you said that the final answer was \sqrt{27} rather than 3\sqrt{3}.

You are also required to know how to rationalise the denominator of a fraction involving surds. This involves manipulating the fraction so that the bottom of it no longer contains any surds – only whole numbers. You should be able to do this with fractions where the denominator is itself a surd (multiply top and bottom by that surd) as well as with fractions where the denominator is an expression of the form a + \sqrt{b} (multiply top and bottom by a - \sqrt{b}). More than anything else this requires care, so take your time with the arithmetic, especially at the beginning.