Surds

A surd is a square root which cannot be reduced to the whole number. If we can't simplify a number to remove a square root (or cube root) then it is a surd.

The numbers left in the square root form or cube root form etc. is called surds. The reason we leave them as surds is because in the decimal form. They would go on forever and this is a clumsy way of writing them.

Example: √3 ( square root of 3 ) can't be simplified further.

Addition and Subtraction of Surds

Adding and subtracting surds are simple- however we need the numbers being square rooted ( or cube rooted) to be the same.

5√8 + 2√8 = 10√8

6√7 - 3√7 = 3√7

However, if the number in the square root sign isn't prime, we might be able to split it up in order to simplify an expression. For example :

√12 + √27 = \(\sqrt{4×3}\) + \(\sqrt{9×3}\)

= \(\sqrt{4×3}\) + \(\sqrt{9×3}\)

= 2√3 + 3√3 = 5√3

= 5√3

Rationalising the Denominator

It is untidy to have to have the fraction which has the surd denominator. This can be tidied up by multiplying the top and bottom of the fraction by the particular expression. This is known as rationalising the denominator. For example: \(\frac{1}{√2}\) has an irrational denominator. We multiply the top and bottom by √2.

\(\frac{1}{√2}\) = \(\frac{1}{√2}\)× \(\frac{√2}{√2}\) = \(\frac{√2}{2}\)

Now the denominator has the rational number.