Surds
Surds are mathematical expressions containing square roots. However, it must be emphasized that the square roots are 'irrational' i.e. they do not result in a whole number, a terminating decimal or a recurring decimal.
The rules governing surds are taken from the Laws of Indices.
rule #1
examples
rule #2
examples
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Some Useful Expressions
expression #1
expression #2 - (the difference of two squares)
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Rationalising Surds - This is a way of modifying surd expressions so that the square root is in the numerator of a fraction and not in the denominator.
The method is to multiply the top and bottom of the fraction by the square root.
Rationalising expressions using the 'difference of two squares'
Remembering that : .... 
.....from 'useful expressions' above.
Example #1 - simplify 
multiplying top and bottom by 
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Example #2 - rationalise 
multiply top and bottom by 
Reduction of Surds - This is a way of making the square root smaller by examining its squared factors and removing them.
Rational and Irrational Numbers - In the test for rational and irrational numbers, if a surd has a square root in the numerator, while the denominator is '1' or some other number, then the number represented by the expression is 'irrational'.
examples of irrational surds: 
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