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 $4$.