Algebra : Surds

[ rules ][ expressions ][ simple rationalising ]

[ diff. 2 sqrs. ][ reduction ][ rational & irrational nos. ]






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


surds #1



surds  #3


rule #2


surds #2



surds #4


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Some Useful Expressions

expression #1

surds #5

expression #2 - (the difference of two squares)

surds #6


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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.

surds #7


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Rationalising expressions using 'difference of two squares'

Remembering that : .... surds8 .....from 'useful expressions' above.

Example #1 - simplify surds #9

multiplying top and bottom by surds #10

surds #11

Example #2 - rationalise surds #12

multiply top and bottom by surds #13



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Reduction of Surds - This is a way of making the square root smaller by examining its squared factors and removing them.

surds #15


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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: surds #16



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