Question

[nguyen thi my chi]

rút gọn (√(2+√3)-√(3+√5))²

Answer 1

[Minh Triều]

Đặt A=\(\left(\sqrt{2+\sqrt{3}}-\sqrt{3+\sqrt{5}}\right)^2\)

\(\Rightarrow2A=2.\left(\sqrt{2+\sqrt{3}}-\sqrt{3+\sqrt{5}}\right)^2\)

\(=\left(\sqrt{2}\sqrt{2+\sqrt{3}}-\sqrt{2}\sqrt{3+\sqrt{5}}\right)^2\)

\(=\left(\sqrt{4+2\sqrt{3}}-\sqrt{6+2\sqrt{5}}\right)^2\)

\(=\left(\sqrt{3+2\sqrt{3}+1}-\sqrt{5+2\sqrt{5}+1}\right)^2\)

\(=\left(\sqrt{\left(\sqrt{3}+1\right)^2}-\sqrt{\left(\sqrt{5}+1\right)^2}\right)^2\)

\(=\left(\sqrt{3}+1-\sqrt{5}-1\right)^2\)

\(=\left(\sqrt{3}-\sqrt{5}\right)^2\)

\(=8-2\sqrt{15}\Rightarrow A=4-\sqrt{15}\)

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