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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 $$4$$.