\(g\left(3\right)=\frac{3+\sqrt{5}}{\sqrt{3}+\sqrt{3+\sqrt{5}}}+\frac{3-\sqrt{5}}{\sqrt{3}-\sqrt{3-\sqrt{5}}}\)
\(=\frac{\left(3+\sqrt{5}\right)\left(\sqrt{3}+\sqrt{3+\sqrt{5}}\right)}{-\sqrt{5}}+\frac{\left(3-\sqrt{5}\right)\left(\sqrt{3}-\sqrt{3-\sqrt{5}}\right)}{\sqrt{5}}\)
\(=\frac{\left(3+\sqrt{5}\right)\left(\sqrt{6}+\sqrt{6+2\sqrt{5}}\right)}{-\sqrt{10}}+\frac{\left(3-\sqrt{5}\right)\left(\sqrt{6}-\sqrt{6-2\sqrt{5}}\right)}{\sqrt{10}}\)
\(=\frac{\left(3+\sqrt{5}\right)\left(\sqrt{6}+\sqrt{5}+1\right)}{-\sqrt{10}}+\frac{\left(3-\sqrt{5}\right)\left(\sqrt{6}-\sqrt{5}+1\right)}{\sqrt{10}}\)
\(=\frac{3\sqrt{6}-4\sqrt{5}-\sqrt{30}+8}{\sqrt{10}}-\frac{3\sqrt{6}+4\sqrt{5}+\sqrt{30}+8}{\sqrt{10}}\)
\(=\frac{-8\sqrt{5}-2\sqrt{30}}{\sqrt{10}}=\frac{-8-2\sqrt{6}}{\sqrt{2}}=-4\sqrt{2}-2\sqrt{3}\)
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