Question 1: \((vi)\) \(\quad\) \(\large{\frac{4\sqrt{3}}{\sqrt{7}+\sqrt{5}}} \)
Solution:
\[\begin{align*}
\frac{4\sqrt{3}}{\sqrt{7}+\sqrt{5}}&=\frac{4\sqrt{3}}{\sqrt{7}+\sqrt{5}} \times \frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}-\sqrt{5}}\\
&=\frac{4\sqrt{3}\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}\right)^2-\left(\sqrt{5}\right)^2}\\
&=\frac{4\sqrt{3}\left(\sqrt{7}-\sqrt{5}\right)}{7-5}\\
&=\frac{4\sqrt{3}\left(\sqrt{7}-\sqrt{5}\right)}{2}\\
&=2\sqrt{3}\left(\sqrt{7}-\sqrt{5}\right)\\
\end{align*}\]