\(R=\dfrac{2}{\sqrt{3-\sqrt{5}-\left(\sqrt[4]{5}-1\right)^3}}=\dfrac{2}{\sqrt{\dfrac{\left(\sqrt{5}-1\right)^2}{2}-\left(\sqrt[4]{5}-1\right)^3}}\)
\(=\dfrac{2}{\sqrt{\dfrac{\left(\sqrt[4]{5}-1\right)^2\left(\sqrt[4]{5}+1\right)^2}{2}-\left(\sqrt[4]{5}-1\right)^3}}\)
\(=\dfrac{2\sqrt{2}}{\sqrt{\left(\sqrt[4]{5}-1\right)^2\left[\left(\sqrt[4]{5}+1\right)^2-2\left(\sqrt[4]{5}-1\right)\right]}}\)
\(=\dfrac{2\sqrt{2}}{\left(\sqrt[4]{5}-1\right)\sqrt{\sqrt{5}+2\sqrt[4]{5}+1-2\sqrt[4]{5}+2}}\)
\(=\dfrac{2\sqrt{2}}{\left(\sqrt[4]{5}-1\right)\sqrt{3+\sqrt{5}}}=\dfrac{2\sqrt{2}}{\left(\sqrt[4]{5}-1\right)\sqrt{\dfrac{\left(\sqrt{5}+1\right)^2}{2}}}\)
\(=\dfrac{4}{\left(\sqrt[4]{5}-1\right)\left(\sqrt{5}+1\right)}=\dfrac{4\left(\sqrt[4]{5}+1\right)}{\left(\sqrt[4]{5}+1\right)\left(\sqrt[4]{5}-1\right)\left(\sqrt{5}+1\right)}\)
\(\)\(=\dfrac{4\left(\sqrt[4]{5}+1\right)}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}=\dfrac{4\left(\sqrt[4]{5}+1\right)}{4}\)
\(=\sqrt[4]{5}+1\)
