Đề là : Rút gọn ạ. Em chỉ cần câu k thôi, cảm on nhiều ạ!
Đặt \(A=\sqrt{5+2\sqrt{1+\sqrt{5}}}-\sqrt{5-2\sqrt{1+\sqrt{5}}}>0\)
\(\Rightarrow A^2=10-2\sqrt{25-4\left(1+\sqrt{5}\right)}=10-2\sqrt{21-4\sqrt{5}}\)
\(=10-2\sqrt{\left(2\sqrt{5}-1\right)^2}=10-2\left(2\sqrt{5}-1\right)=12-4\sqrt{5}\)
\(=\left(\sqrt{10}-\sqrt{2}\right)^2\)
\(\Rightarrow A=\sqrt{10}-\sqrt{2}\)
\(\Rightarrow k=\sqrt{10}-\sqrt{2}+\sqrt{\left(\sqrt{3}+1\right)\left(2-\sqrt{3}\right)\left(\sqrt{4+2\sqrt{3}}\right)}\)
\(=\sqrt{10}-\sqrt{2}+\sqrt{\left(\sqrt{3}+1\right)\left(2-\sqrt{3}\right)\sqrt{\left(\sqrt{3}+1\right)^2}}\)
\(=\sqrt{10}-\sqrt{2}+\sqrt{\left(\sqrt{3}+1\right)^2\left(2-\sqrt{3}\right)}\)
\(=\sqrt{10}-\sqrt{2}+\sqrt{\left(4+2\sqrt{3}\right)\left(2-\sqrt{3}\right)}\)
\(=\sqrt{10}-\sqrt{2}+\sqrt{2\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}\)
\(=\sqrt{10}-\sqrt{2}+\sqrt{2.1}=\sqrt{10}\)
