RATIONALISATION OF SURDS
Rationalation of Trinomial surds
1) \(\left( {{x^{\frac{1}{3}}} + {y^{\frac{1}{3}}} + {z^{\frac{1}{3}}}} \right)\left( {{x^{\frac{2}{3}}} + {y^{\frac{2}{3}}} + {y^{\frac{2}{3}}} - {x^{\frac{1}{3}}}.{y^{\frac{1}{3}}} - {y^{\frac{1}{3}}}.{z^{\frac{1}{3}}} - {z^{\frac{1}{3}}}.{x^{\frac{1}{3}}}} \right)\) are rationalising factors of each other.
Explanation:-
We know \((a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ca) = {a^3} + {b^3} + {c^3} - 3abc\)
put \(a = {x^{\frac{1}{3}}},b = {y^{\frac{1}{3}}},c = {z^{\frac{1}{3}}}\)
\(\left( {{x^{\frac{1}{3}}} + {y^{\frac{1}{3}}} + {z^{\frac{1}{3}}}} \right)\left( {{x^{\frac{2}{3}}} + {y^{\frac{2}{3}}} + {y^{\frac{2}{3}}} - {x^{\frac{1}{3}}}.{y^{\frac{1}{3}}} - {y^{\frac{1}{3}}}.{z^{\frac{1}{3}}} - {z^{\frac{1}{3}}}.{x^{\frac{1}{3}}}} \right)\) \(= {\left( {{x^{\frac{1}{3}}}} \right)^3} + {\left( {{y^{\frac{1}{3}}}} \right)^3} + {\left( {{z^{\frac{1}{3}}}} \right)^3} - 3.{x^{\frac{1}{3}}}.{y^{\frac{1}{3}}}.{z^{\frac{1}{3}}}\)
= \(x + y + z - 3.\sqrt[3]{{xyz}}\)
Again we find the rationalising factor of \(\left( {x + y + z} \right) - 3.\sqrt[3]{{xyz}}\) till we get rational number
2. R.F of \(\sqrt[n]{a} - \sqrt[n]{b} = \sqrt[n]{{{a^{n - 1}}}} + \sqrt[n]{{{a^{n - 2}}}}.b + \sqrt[n]{{{a^{n - 3}}.{b^2}}} + ........... + \sqrt[n]{{{b^{n - 1}}}},\rlap{--} Vn \in N\)
Illustration:-
R.F of \(\sqrt[4]{2} - \sqrt[4]{3} = \sqrt[4]{{{2^3}}} + \sqrt[4]{{{2^2} \times 3}} + \sqrt[4]{{2 \times {3^2}}} + \sqrt[4]{{{3^3}}}\)
=\(\sqrt[4]{8} + \sqrt[4]{{12}} + \sqrt[4]{{18}} + \sqrt[4]{{27}}\)
3. R.F of \(\sqrt[n]{a} + \sqrt[n]{b} = \sqrt[n]{{{b^{n - 1}}}} + \sqrt[n]{{{a^{n - 2}}.b}} + \sqrt[n]{{{a^{n - 3}}.{b^2}}} + ........... + \sqrt[n]{{{b^{n - 1}}}}\), where ‘n’ is odd number
Illustration :- Find R.F of \(\sqrt[5]{2} + \sqrt[5]{3}\)
Solution:- R.F of \(\sqrt[5]{2} + \sqrt[5]{3} = \sqrt[4]{{{2^3}}} - \sqrt[4]{{{2^2} \times 3}} + \sqrt[4]{{2 \times {3^2}}} - \sqrt[4]{{{3^3}}}\)
\(= \sqrt[4]{8} - \sqrt[4]{{12}} + \sqrt[4]{{18}} - \sqrt[4]{{27}}\)
4. R.F of \(\sqrt[n]{a} + \sqrt[n]{b} = \sqrt[n]{{{b^{n - 1}}}} - \sqrt[n]{{{a^{n - 2}} - b}} + \sqrt[n]{{{a^{n - 3}} \times {b^2}}} + ........... + {( - 1)^{n - 1}}.\sqrt[n]{{{b^{n - 1}}}}\), where n is even
Illustration :- R.F of \(\sqrt[4]{2} + \sqrt[4]{3}\)
Solution :- R.F of \(\sqrt[4]{2} + \sqrt[4]{3} = \sqrt[4]{{{2^3}}} - \sqrt[4]{{{2^2} \times 3}} + \sqrt[4]{{2 \times {3^2}}} - \sqrt[4]{{{3^3}}}\)
\(= \sqrt[4]{8} - \sqrt[4]{{12}} + \sqrt[4]{{18}} - \sqrt[4]{{27}}\)