Core 1: Surds
This section summarises all the key
ideas required for the surd section at ASlevel.
Contents
1)
Adding, subtracting and
multiplying simple surds
2)
Simplifying surds
3)
Rationalising surds
4)
Surd problems
Adding, subtracting and
multiplying surds

Treat like algebra: Consider: 

Consider: 

Consider: 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. 

Consider Treat surds in the same way 

Generally, write the coefficient before the surd sign 

Consider 

Notice how the surds vanish. This is an important result Generally: 
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: So, 
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 Remember: 

We do not like the surd as a denominator. We therefore multiply top and bottom by Remember: 

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: 
Simplify the surds where possible: 
So the question becomes: 
Rationalise each surd separately: Make the denominators the same. Write over the same denominator Simplify Write as two separate fractions. 