Question

B=\(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)

Answer 1

\(B^2=8+2\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\left(4-\sqrt{10+2\sqrt{5}}\right)}\\ B^2=8+2\sqrt{\left(16-\left(10+2\sqrt{5}\right)\right)}\\ B^2=8+2\sqrt{6-2\sqrt{5}}\\ B^2=8+2\sqrt{\left(\sqrt{5}-1\right)^2}\\ B^2=8+2\left(\sqrt{5}-1\right)\\ \Rightarrow B=\sqrt{\left|8+2\left(\sqrt{5}-1\right)\right|}=\sqrt{6+2\sqrt{5}}\\ B=\sqrt{\left(1+\sqrt{5}\right)^2}=1+\sqrt{5}\)