RATIONALISATION OF SURDS
Rationalising factors of Binomial surds
1) R.F of \(\sqrt a + \sqrt b = \sqrt a - \sqrt b \) and R.F of \(\sqrt a - \sqrt b = \sqrt a + \sqrt b \)
\(\because \,\,\,\left( {\sqrt a + \sqrt b } \right)\left( {\sqrt a - \sqrt b } \right) = {\left( {\sqrt a } \right)^2} - {\left( {\sqrt b } \right)^2} = \frac{{a - b}}{1}\) (R.N)
Illustration 1:-
1) Rationalising factor \(\sqrt 7 + \sqrt 3 \) is \(\sqrt 7 - \sqrt 3 \)
\(\therefore \,\,\,\,\,\left( {\sqrt 7 + \sqrt 3 } \right)\left( {\sqrt 7 - \sqrt 3 } \right) = {\sqrt 7 ^2} - {\sqrt 3 ^2} = 7 - 3 = \frac{4}{1}\) (R.N)
Illustration2 :- Rationalising factor \(\sqrt[3]{a} + \sqrt[3]{b}\) is \(\sqrt[3]{{{a^2}}} - \sqrt[3]{{ab}} + \sqrt[3]{{{b^2}}}\)
explanation:-
we Know \({x^3} + {y^3} = (x + y)({x^2} - xy + {y^2})\)
Put \(x = \sqrt[3]{a} = {a^{\frac{1}{3}}}.y = \sqrt[3]{b} = {b^{\frac{1}{3}}}\)
\({\left( {{a^{\frac{1}{3}}}} \right)^3} + {\left( {{b^{\frac{1}{3}}}} \right)^3} = \left( {\sqrt[3]{a} + \sqrt[3]{b}} \right)\left( {{{\left( {\sqrt[3]{a}} \right)}^2} - \sqrt[3]{a}.\sqrt[3]{b} + {{\left( {\sqrt[3]{b}} \right)}^2}} \right)\)
\(a + b = \left( {\sqrt[3]{a} + \sqrt[3]{b}} \right)\left( {{{\left( {\sqrt[3]{a}} \right)}^2} - \sqrt[3]{a}.\sqrt[3]{b} + {{\left( {\sqrt[3]{b}} \right)}^2}} \right)\)
\(a + b = \frac{{a + b}}{1}\)=(R.N)
\(\therefore \,\,\sqrt[3]{a} + \sqrt[3]{b}\) and \(\left( {\sqrt[3]{{{a^2}}} - \sqrt[3]{{ab}} + \sqrt[3]{{{b^2}}}} \right)\) are rationalised factors of each other
Ex:- 1) Rationalising factors of \(\sqrt[3]{5} + \sqrt[3]{2}\)
\(a = \sqrt[3]{5},\,\,b = \sqrt[3]{2}\)
R.F of \(\sqrt[3]{5} + \sqrt[3]{2} = {\left( {\sqrt[3]{5}} \right)^2} - \sqrt[3]{5}.\sqrt[3]{2} + {\left( {\sqrt[3]{2}} \right)^2}\)
\(= {5^{\frac{2}{3}}} - {10^{\frac{1}{3}}} + {2^{\frac{2}{3}}}\)
Ex:2 R.F of \({\sqrt[3]{7}^2} - \sqrt[3]{7}.\sqrt[3]{7} + {\left( {\sqrt[3]{4}} \right)^2}\) is \(\left( {\sqrt[3]{7} + \sqrt[3]{4}} \right)\left[ {\because a = \sqrt[3]{7},b = \sqrt[3]{4}} \right]\)
Note :- \(\left( {\sqrt[3]{a} - \sqrt[3]{b}} \right)\left( {\sqrt[3]{{{a^2}}} + \sqrt[3]{{ab}} + \sqrt[3]{{{b^2}}}} \right)\) are R.F’s of each other