RATIONALISATION OF SURDS
Rationalising factor:-
If the product of two surds is a rational number, then each of them is called a rationalising factor of each other
Note: 1) Rationalising factor is represented by (R.F)
2) Rationalising factor of \(n\sqrt{a}\) is given by \({a^{1 - \frac{1}{n}}}\)
Illustration I:- Consider \(\sqrt 3 ,3\sqrt 3 \) are the rationalising factors of each other
\(\therefore \,\,\,\sqrt 3 \times 3\sqrt 3 = 3 \times \left( {\sqrt 3 \times \sqrt 3 } \right)\)
\(= 3 \times 3 = 9 = \frac{9}{1}\) (R.N)
Illustration 2:-
Rationalising factors of \(\sqrt {125} = \sqrt {{5^3}} \) is \({5^{\frac{3}{2} - 1}} = {5^{\frac{1}{2}}} = \sqrt 5 \left[ {\sqrt[3]{{{x^2}}} = {{({x^2})}^{\frac{1}{3}}} = {x^{\frac{1}{3}}}} \right]\)
We can see that
\(\sqrt {125} \times \sqrt 5 = \sqrt {125 \times 5} = \sqrt {625} \)
\(\sqrt {{{(25)}^2}} = {({25^2})^{\frac{1}{2}}}25 = \frac{{25}}{1}\)= (R.N)
Note :- The R.F of a given surd is not unique. A surd has infinite number of R.F’s
Illustration 3 :- \(2\sqrt 3 ,3\sqrt 3 ,4\sqrt 3 ..........\) are R.F’s of \(5\sqrt{3}\) if one R.F of a surd is known then the product of this number by a non zero rational number is also a R.F of the given surd .
Illustration 4 :- If R.F of \(\sqrt{50}\) is \(\sqrt{2}\) (\(\because \sqrt {50} = 5\sqrt 2 \)) then \(2\sqrt 2 ,3\sqrt 2 , - \frac{5}{3}\sqrt 2 \) etc also R.F’s of \(\sqrt{50}\)
Here, \(\sqrt{2}\) is called the simplest R.F’s \(\sqrt{50}\)