7. Rationalisation of surds and conjugate surds

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.

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 \(4\).