Theory:

Rationalisation of surds
Rationalisation of surds is defined as the term which is multiplied or divided with the surd to make it as a rational number.
Example:
1. The rationalising factor of \(\sqrt{19}\) is \(\sqrt{19}\). Because multiplying both the surds, we have \(\sqrt{19} \times \sqrt{19} = 19\) which is a rational number.
2. The rationalising factor of \(\sqrt[5]{4^2}\) is \(\sqrt[5]{4^3}\). Because multiplying both the surds, we have \(\sqrt[5]{4^2} \times \sqrt[5]{4^3} = \sqrt[5]{4^5} = 4\) which is a rational number.
Conjugate surds
Conjugate surds of a binomial term is defined as the term having the same identical terms with an opposite sign in the middle.
Example:
Find the conjugate of the surd \(4 - \sqrt{12}\) and simplify it.
 
Solution:
 
The given surd is \(4 - \sqrt{12}\).
 
To simplify the surd, let us find the conjugate by changing the sign in the middle.
 
Therefore, the conjugate of the surd \(4 - \sqrt{12}\) is \(4 + \sqrt{12}\).
 
Now, we shall simplify the surd \(4 - \sqrt{12}\) by multiplying it with its conjugate.
 
\((4 - \sqrt{12})(4 + \sqrt{12}) = 4^2 - (\sqrt{12})^2\) [\(a^2 - b^2 = (a + b)(a - b)\)]
 
\(= 16 - 12\)
 
\(= 4\)
 
Therefore, the solution is \(4\).