Rationalize the denominator: \(\frac{2}{\sqrt{5}-\sqrt{3}} \cdot \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}=\frac{2\sqrt{5}+2\sqrt{3}}{{\sqrt{5}}^{2}-{\sqrt{3}}^{2}}\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\frac{2\sqrt{5}+2\sqrt{3}}{{\sqrt{5}}^{2}-{\sqrt{3}}^{2}}+\frac{1}{2+\sqrt{5}}\end{aligned}\]
Factor out the common term \(2\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\frac{2(\sqrt{5}+\sqrt{3})}{{\sqrt{5}}^{2}-{\sqrt{3}}^{2}}+\frac{1}{2+\sqrt{5}}\end{aligned}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\frac{2(\sqrt{5}+\sqrt{3})}{5-{\sqrt{3}}^{2}}+\frac{1}{2+\sqrt{5}}\end{aligned}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\frac{2(\sqrt{5}+\sqrt{3})}{5-3}+\frac{1}{2+\sqrt{5}}\end{aligned}\]
Simplify \(5-3\) to \(2\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\frac{2(\sqrt{5}+\sqrt{3})}{2}+\frac{1}{2+\sqrt{5}}\end{aligned}\]
Cancel \(2\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\sqrt{5}+\sqrt{3}+\frac{1}{2+\sqrt{5}}\end{aligned}\]
Rationalize the denominator: \(\frac{1}{2+\sqrt{5}} \cdot \frac{2-\sqrt{5}}{2-\sqrt{5}}=\frac{2-\sqrt{5}}{{2}^{2}-{\sqrt{5}}^{2}}\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\sqrt{5}+\sqrt{3}+\frac{2-\sqrt{5}}{{2}^{2}-{\sqrt{5}}^{2}}\end{aligned}\]
Simplify \({2}^{2}\) to \(4\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\sqrt{5}+\sqrt{3}+\frac{2-\sqrt{5}}{4-{\sqrt{5}}^{2}}\end{aligned}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\sqrt{5}+\sqrt{3}+\frac{2-\sqrt{5}}{4-5}\end{aligned}\]
Simplify \(4-5\) to \(-1\).
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\sqrt{5}+\sqrt{3}+\frac{2-\sqrt{5}}{-1}\end{aligned}\]
Move the negative sign to the left.
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\sqrt{5}+\sqrt{3}-(2-\sqrt{5})\end{aligned}\]
Remove parentheses.
\[\begin{aligned}&2-4\sqrt{3}\\&\frac{1}{2+\sqrt{3}}+\sqrt{5}+\sqrt{3}-2+\sqrt{5}\end{aligned}\]
2-4*sqrt(3);1/(2+sqrt(3))+sqrt(5)+sqrt(3)-2+sqrt(5)
