Đáp án:

 `R=2\sqrt{2}`

Giải thích các bước giải:

 `R=\frac{3+\sqrt{5}}{\sqrt{2}+\sqrt{3+\sqrt{5}}}+\frac{3-\sqrt{5}}{\sqrt{2}-\sqrt{3-\sqrt{5}}}`

`->R:\sqrt{2}=\frac{3+\sqrt{5}}{\sqrt{2}.(\sqrt{2}+\sqrt{3+\sqrt{5}})}+\frac{3-\sqrt{5}}{\sqrt{2}.(\sqrt{2}-\sqrt{3-\sqrt{5}})}`

`->R:\sqrt{2}=\frac{3+\sqrt{5}}{2+\sqrt{6+2\sqrt{5}}}+\frac{3-\sqrt{5}}{2-\sqrt{6-2\sqrt{5}}}`

`=\frac{3+\sqrt{5}}{2+\sqrt{(\sqrt{5})^{2}+2.\sqrt{5}.1+1^{2}}}+\frac{3-\sqrt{5}}{2-\sqrt{(\sqrt{5})^{2}-2.\sqrt{5}.1+1^{2}}}`

`=\frac{3+\sqrt{5}}{2+\sqrt{(\sqrt{5}+1)^{2}}}+\frac{3-\sqrt{5}}{2-\sqrt{(\sqrt{5}-1)^{2}}}`

`=\frac{3+\sqrt{5}}{2+|\sqrt{5}+1|}+\frac{3-\sqrt{5}}{2-|\sqrt{5}-1|}`

`=\frac{3+\sqrt{5}}{2+\sqrt{5}+1}+\frac{3-\sqrt{5}}{2-(\sqrt{5}-1)}`

`=\frac{3+\sqrt{5}}{3+\sqrt{5}}+\frac{3-\sqrt{5}}{2-\sqrt{5}+1}`

`=1+\frac{3-\sqrt{5}}{3-\sqrt{5}}`

`=1+1`

`=2`

`->R:\sqrt{2}=2`

`->R=2\sqrt{2}`

Vậy `R=2\sqrt{2}.`