Surds

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A surd is an expression of the form √? or − √? where ? is a rational number that is not a square of another rational number.

For example, √2, √ 12 13 and −√1.5 are surds, but √100, −√ 4 9 and √ 48 243 are not surds.

1 + √3, 4√2 or √40 + 5√57 are also surds.

√a × √b = √ab

√a / √b = √a/b

√a × √a = a

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Rationalising the Denominator

The conjugate surd of ? + ?√? is ? − ?√?.

The product of a surd with its conjugate surd is not a surd!

(? + ?√?)(? − ?√?) = ?2 − ??√? + ??√? − ?2 ? = ?2 − ?2 ?

. By multiplying the numerator and denominator of an expression involving surds by the conjugate surd of the denominator, the expression transforms into one whose denominator is rational. This is called rationalising the denominator.