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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 \(4\).