Surds are square roots (or cubed roots etc.) which can’t be reduced to rational numbers. Some can be simplified using various rules or by rationalising the denominator.

Surds have a decimal which goes on forever without repeating, and are irrational numbers.

In fact “Surd” used to be another name for “Irrational” but now it is used for a root that is irrational.

When it is a root and irrational, it is a surd. But not all roots are surds.

Rationalising the denominator

Rationalising an expression means getting rid of any surds from the bottom (denominator) of fractions.

Usually when you are asked to simplify an expression it means you should also rationalise it.

Sometimes the denominator might be more complicated and include other numbers as well as the surd.

If this is the case you need to multiply the fraction by a number that will cancel out the surd. Remember to multiply the numerator by the same number or you will change the value of the fraction.

The conjugate is where we change the sign in the middle of two terms. It can be useful because when we multiply something by its conjugate we get squares like this.

(a+b) (a+b) = a2 – b2

Rules of Surds

There are three simple rules of surds that can help us simply things.