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Rationalisation of Surds

Rationalisation of surds involves the multiplication of a surd by its conjugate to get a rational number. This process requires us to not leave the denominator in the surd form, but as a rational number.

Let’s consider 3/sqrt{3}. As per the definition of rationalisation of surds, we should have a rational number in the denominator, and not have a surd there. So we’ll multiply both the numerator and denominator by sqrt{3}.

3/sqrt{3}  =  3/sqrt{3}sqrt{3}/sqrt{3}

=  {3sqrt{3}}/3

=  sqrt{3}

 

Examples of rationalisation of surds

Example 1:  {2~+~sqrt{3}}/sqrt{5}

=  {2~+~sqrt{3}}/sqrt{5}  x  sqrt{5}/sqrt{5}

=  {2sqrt{5}~+~sqrt{15}}/5

 

Example 2:  2/{2~-~sqrt{3}}

=  2/{2~-~sqrt{3}}  x  {2~+~sqrt{3}}/{2~+~sqrt{3}}

=  {2(2~+~sqrt{3})}/{2^2~-~(sqrt{3})^2}

=  {2(2~+~sqrt{3})}/{4~-~3}

=  2(2~+~sqrt{3})

 

Example 3:  {2~+~sqrt{3}}/{2~-~sqrt{3}}

=  {2~+~sqrt{3}}/{2~-~sqrt{3}}  x  {2~+~sqrt{3}}/{2~+~sqrt{3}}

=  {(2~+~sqrt{3})(2~+~sqrt{3})}/{4~-~3}

=  {(2~+~sqrt{3})^2}/1

=  2^2~+~2*2*sqrt{3}~+~(sqrt{3})^2

=  4 + 4sqrt{3} + 3

=  7 + 4sqrt{3}

 

Example 4:  {2~+~sqrt{5}}/{-6~-~sqrt{5}}

=  {2~+~sqrt{5}}/{-6~-~sqrt{5}}  x  {-6~+~sqrt{5}}/{-6~+~sqrt{5}}

=  {(2~+~sqrt{5})(-6~-~sqrt{5})}/{(-6~+~sqrt{5})(-6~+~sqrt{5})}

=  {2*-6~+~2sqrt{5}~-~6sqrt{5}~+~(sqrt{5})^2}/{(-6)^2~-~(sqrt{5})^2}

=  {-12~+~2sqrt{5}~-~6sqrt{5}~+~5}/{36~-~5}

=  {-7~-~4sqrt{5}}/31