Rationalisation of Surds


 
 
Concept Explanation
 

Rationalisation of Surds

Rationalisation of Surds

 If the product of two surds is rational, then each of the surds is called a rationalising factor of the other. In general, if the surd is of type a+sqrt{b}, then its rationalising surd is a asqrt{b} .

The rationalising factor of a surd is of a surd is not unique but it is always convenient to use simplest of all rationalising factors.

   e.g.,    frac{1}{2-sqrt{3}}

can be rationalised as follows:

            frac{1}{2-sqrt3}= frac{1}{2-sqrt3} times   frac{2+sqrt{3}}{2+sqrt{3}}

                             =frac{2+sqrt{3}}{(2)^2-(sqrt{3})^2}=frac{2+sqrt{3}}{4-3}=2+sqrt{3}

Illustration . The expression 2+sqrt{2}+frac{1}{2+sqrt{2}};+;frac{1}{2-sqrt{2}}  equals to.

A.   4+sqrt{2}                B.   2sqrt{2}                  C.    4-sqrt{2}              D.    2+sqrt{2}

Solution:

2+sqrt{2}+frac{1}{2+sqrt{2}}+frac{1}{2-sqrt{2}}

=2+sqrt{2}+frac{1}{2+sqrt{2}}times frac{2-sqrt{2}}{2-sqrt{2}}+frac{1}{2-sqrt{2}}times frac{2+sqrt{2}}{2+sqrt{2}}

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