Question

Chứng minh đẳng thức :

\(\sqrt{53+12\sqrt{10}}-\sqrt{47-6\sqrt{10}}=3\sqrt{2}\)

Answer 1

\(\sqrt{53+12\sqrt{10}}-\sqrt{47-6\sqrt{10}}=\sqrt{45+2.3\sqrt{5}.2\sqrt{2}+8}-\)\(\sqrt{45-2.3\sqrt{5}.\sqrt{2}+2}\)

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