Answer
$\frac{20+5\sqrt{7}+4\sqrt{3}+\sqrt{21}}{9}$
Work Step by Step
Given \begin{equation}
\frac{5+\sqrt{3}}{4-\sqrt{7}}.
\end{equation} Rationalize the denominator and simplify.
\begin{equation}
\begin{aligned}
\frac{5+\sqrt{3}}{4-\sqrt{7}}&=\frac{(5+\sqrt{3})}{(4-\sqrt{7})}\cdot \frac{(4+\sqrt{7})}{(4+\sqrt{7})}\\
&= \frac{5(4+\sqrt{7})+\sqrt{3}(4+\sqrt{7})}{16-7}\\
&= \frac{20+5\sqrt{7}+4\sqrt{3}+\sqrt{21}}{9}.
\end{aligned}
\end{equation} Hence \begin{equation}
\frac{5+\sqrt{3}}{4-\sqrt{7}}= \frac{20+5\sqrt{7}+4\sqrt{3}+\sqrt{21}}{9}.
\end{equation}
