(i) As there is √5 in the denominator and we know that √5 x √5 = 5
So, multiply numerator and denominator by √5,
\(\frac{3}{\sqrt{5}}\) = \(\frac{3\times \sqrt{5}}{\sqrt{5}\times \sqrt{5}}\) = \(\frac{3\sqrt{5}}{5}\) = \(\frac{3}{5}{\sqrt{5}}\)
(ii) \(\frac{3}{2\sqrt{5}}\) x \(\frac{2\sqrt5}{2\sqrt{5}}\) = \(\frac{3\times 2\sqrt5}{(2\sqrt{5})^2}\) = \(\frac{6\sqrt{5}}{20}\) = \(\frac{3}{10}{\sqrt{5}}\)
(iii) \(\frac{1}{\sqrt{12}}\) = \(\frac{1}{\sqrt{12}} \times \frac{\sqrt{12}}{\sqrt{12}}\)
\(\frac{1}{\sqrt{12}} \times \frac{\sqrt{12}}{\sqrt{12}}\)
= \(\frac{\sqrt3\sqrt{4}}{12}\)
= \(\frac{2\sqrt{3}}{12}\)
= \(\frac{\sqrt{3}}{6}\)
(iv) \(\frac{\sqrt{2}}{\sqrt{5}}\) x \(\frac{\sqrt{5}}{\sqrt{5}}\) = \(\frac{\sqrt2\times \sqrt5}{(\sqrt{5})^2}\) = \(\frac{1}{5}{\sqrt{10}}\)
(v) \(\frac{\sqrt{3}+1}{\sqrt{2}}\) x \(\frac{\sqrt{2}}{\sqrt{2}}\) = \(\frac{\sqrt6+\sqrt{2}}{2}\)
(vi) \(\frac{\sqrt2+\sqrt5}{\sqrt{3}}\) x \(\frac{\sqrt{3}}{\sqrt{3}}\) = \(\frac{\sqrt6+\sqrt{5}}{3}\)
(vii) \(\frac{3\sqrt2}{\sqrt{5}}\) x \(\frac{\sqrt{5}}{\sqrt{5}}\) = \(\frac{3\sqrt{10}}{5}\)