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Core 1: Surds

This section summarises all the key ideas required for the surd section at AS-level.



1)           Adding, subtracting and multiplying simple surds

2)          Simplifying surds

3)          Rationalising surds

4)          Surd problems




Adding, subtracting and multiplying surds

Treat like algebra:







Treat each surd like a different algebraic term

When multiplying or dividing surds, the numbers can be placed under one square root sign

Sometimes the result can be easily worked out

Multiply the coefficients and surds separately.



Treat surds in the same way

Generally, write the coefficient before the surd sign



Notice how the surds vanish.

This is an important result


Simplifying surds

This is a very important section. Take a while to read and understand the examples


Method 1:


Look for the largest square factor of the number under the square root.

Then break it up into two surds multiplied together.


Method 2:

Write the number as the product of its prime factors.

Every number has a unique set of prime factors:


Method 1:


36 is the highest square factor of 72

Method 2:

To speed up the next part, you can use the fact that:


Write 72 as the unique product of its prime factors



Rationalising surds:

Learn these two important facts.

Rationalising effectively means ‘move’ any surd from the denominator of a fraction to the numerator.

To do this we use ideas of equivalent fractions




We do not like the surd as a denominator.

We therefore multiply top and bottom by





We do not like the surd as a denominator.

We therefore multiply top and bottom by


When the denominator is of the form  multiply top and bottom by  and vice versa.

We will encounter more of these in detail next...

Surd problems






Simplify the surds where possible:



So the question becomes:



Rationalise each surd separately:







Make the denominators the same.



Write over the same denominator





Write as two separate fractions.