What Are Surds?
A surd is a square root that doesn’t give a whole number. So √3 and √50 are surds but √4 and √25 are not. If you type √3 into your calculator, you’ll get something like 1.73205… If the questions wants an answer to 1 decimal place, that would be 1.7 If the question asked for an exact answer, you’d leave it as √3.
Splitting and Combining Surds
The most useful technique with surds is splitting them into their factors, for example
√20 = √(4x5) = √4 x √5 = 2 x √5
(usually written 2√5)
This is called simplifying the surd
Choosing the right factor pair is key here. We could have said:
√20 = √(2x10) = √2 x √10
That’s not wrong but it isn’t simpler and wouldn’t have been very useful. Try to find a factor that’s a perfect square.
We can also reverse this process:
3√5 = √9√5 = √45
A common question in National 5 maths is to add/subtract surds.You do this by simplifying each of them to leave the same surd. For example:
√18 + √50 = √9√2 + √25√2 = 3√2 + 5√2 = 8√2
Sometimes you need to multiply out brackets and simplify, for example a question like this:
√3(√12 + √300)
Here you can either simplfy the surds then multiply out the brackets or do it the other way round:
√3(√12 + √300) = √3(√4√3 + √100√3)
= √3(2√3 + 10√3) = √3(12√3) = 36
or
√3(√12 + √300) = √36 + √900 = 6 + 30 = 36
Rationalising the Denominator
Sometimes you will be given a fraction with a surd on the bottom and asked to ‘rationalise the denominator’. This simply means ‘make the bottom a whole number, not a surd’. Assuming the bottom is simple, just multiply top and bottom by the surd then simplfy if possible. For example:
