Rationalize the denominator: \(\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\).
\[\frac{1}{\sqrt{5}}+\frac{\sqrt{2}}{\sqrt{3}}+\frac{1}{\sqrt{5}-}-\sqrt{\frac{2\sqrt{3}}{3}}\]
Simplify \(\sqrt{\frac{2\sqrt{3}}{3}}\) to \(\frac{\sqrt{2\sqrt{3}}}{\sqrt{3}}\).
\[\frac{1}{\sqrt{5}}+\frac{\sqrt{2}}{\sqrt{3}}+\frac{1}{\sqrt{5}-}-\frac{\sqrt{2\sqrt{3}}}{\sqrt{3}}\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\frac{1}{\sqrt{5}}+\frac{\sqrt{2}}{\sqrt{3}}+\frac{1}{\sqrt{5}-}-\frac{\sqrt{2}\sqrt{\sqrt{3}}}{\sqrt{3}}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{1}{\sqrt{5}}+\frac{\sqrt{2}}{\sqrt{3}}+\frac{1}{\sqrt{5}-}-\frac{\sqrt{2}\times {3}^{\frac{1\times 1}{2\times 2}}}{\sqrt{3}}\]
Simplify \(1\times 1\) to \(1\).
\[\frac{1}{\sqrt{5}}+\frac{\sqrt{2}}{\sqrt{3}}+\frac{1}{\sqrt{5}-}-\frac{\sqrt{2}\sqrt[2\times 2]{3}}{\sqrt{3}}\]
Simplify \(2\times 2\) to \(4\).
\[\frac{1}{\sqrt{5}}+\frac{\sqrt{2}}{\sqrt{3}}+\frac{1}{\sqrt{5}-}-\frac{\sqrt{2}\sqrt[4]{3}}{\sqrt{3}}\]
Regroup terms.
\[\frac{1}{\sqrt{5}}+\frac{\sqrt{2}}{\sqrt{3}}+\frac{1}{\sqrt{5}-}-\frac{\sqrt[4]{3}\sqrt{2}}{\sqrt{3}}\]
Rationalize the denominator: \(\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{5}}{5}\).
\[\frac{\sqrt{5}}{5}+\frac{\sqrt{2}}{\sqrt{3}}+\frac{1}{\sqrt{5}-}-\frac{\sqrt[4]{3}\sqrt{2}}{\sqrt{3}}\]
Rationalize the denominator: \(\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{2}\sqrt{3}}{3}\).
\[\frac{\sqrt{5}}{5}+\frac{\sqrt{2}\sqrt{3}}{3}+\frac{1}{\sqrt{5}-}-\frac{\sqrt[4]{3}\sqrt{2}}{\sqrt{3}}\]
Simplify \(\sqrt{2}\sqrt{3}\) to \(\sqrt{6}\).
\[\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{3}+\frac{1}{\sqrt{5}-}-\frac{\sqrt[4]{3}\sqrt{2}}{\sqrt{3}}\]
Rationalize the denominator: \(\frac{1}{\sqrt{5}-} \cdot \frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{5}}{\sqrt{5}-\sqrt{5}}\).
\[\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{3}+\frac{\sqrt{5}}{\sqrt{5}-\sqrt{5}}-\frac{\sqrt[4]{3}\sqrt{2}}{\sqrt{3}}\]
Rationalize the denominator: \(\frac{\sqrt{5}}{\sqrt{5}-\sqrt{5}} \cdot \frac{\sqrt{5}+\sqrt{5}}{\sqrt{5}+\sqrt{5}}=\frac{5+5}{{\sqrt{5}}^{2}-{\sqrt{5}}^{2}}\).
\[\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{3}+\frac{5+5}{{\sqrt{5}}^{2}-{\sqrt{5}}^{2}}-\frac{\sqrt[4]{3}\sqrt{2}}{\sqrt{3}}\]
Simplify \(5+5\) to \(10\).
\[\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{3}+\frac{10}{{\sqrt{5}}^{2}-{\sqrt{5}}^{2}}-\frac{\sqrt[4]{3}\sqrt{2}}{\sqrt{3}}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{3}+\frac{10}{5-\times 5}-\frac{\sqrt[4]{3}\sqrt{2}}{\sqrt{3}}\]
Rationalize the denominator: \(\frac{\sqrt[4]{3}\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt[4]{3}\sqrt{2}\sqrt{3}}{3}\).
\[\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{3}+\frac{10}{5-\times 5}-\frac{\sqrt[4]{3}\sqrt{2}\sqrt{3}}{3}\]
Simplify \(\sqrt[4]{3}\sqrt{2}\sqrt{3}\) to \(\sqrt[4]{3}\sqrt{6}\).
\[\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{3}+\frac{10}{5-\times 5}-\frac{\sqrt[4]{3}\sqrt{6}}{3}\]
sqrt(5)/5+sqrt(6)/3+10/(5-*5)-(3^(1/4)*sqrt(6))/3
