A surd is a number written using the square root symbol, , or the cube root symbol, , or the fourth root symbol, , or the fifth root……..

The square numbers have square roots which are rational numbers.

Egs.

But all of the integers (whole numbers) between the square numbers have square roots that are irrational, that is numbers which have an unending decimal part that has no pattern.

Eg. correct to 31 decimal places.

So for the square root of 2, it is quicker and more accurate to write than

Because the square root of 2 is , then

If the number under the square root symbol can be factorised, then the factors can be square rooted separately.

Egs

So:

Also, in a fraction, the top and the bottom of the fraction can be square rooted separately.

Eg.

Putting these techniques together allows some expressions to be simplified.

Eg.

It is not usual to leave an answer with surds on the bottom of a fraction. So the fraction would be rationalised by multiplying top and bottom of the fraction (which is just the reverse of cancelling down) by any surds on the bottom of the fraction.

Eg.

Practise with these questions.

The method to rationalise the denominators can be extended to examples where the denominator has two, rather than one term, eg.

Here it is necessary to multiply top and bottom of the fraction by whatever is on the bottom, but with the opposite sign.

Although the numerator is now irrational, the denominator is rational and so the expression is rationalised.

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