Question

Tính: \(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}-\sqrt{2}\)

Answer 1

\(Q=\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}-\sqrt{2}\\ \Leftrightarrow Q^2=4+\sqrt{7}+4-\sqrt{7}+2-2\sqrt{\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)}+2\sqrt{2.\left(4-\sqrt{7}\right)}-2\sqrt{2.\left(4+\sqrt{7}\right)}\\ \Leftrightarrow Q^2=10-2\sqrt{16-7}+\sqrt{32-8\sqrt{7}}-\sqrt{32+8\sqrt{7}}\\ \Leftrightarrow Q^2=10-6+\sqrt{\left(2\sqrt{7}\right)^2-2.2.2\sqrt{7}+2^2}-\sqrt{\left(2\sqrt{7}\right)^2+2.2.2\sqrt{7}+2^2}\\ \Leftrightarrow Q^2=4+2\sqrt{7}-2-2\sqrt{7}-2=0\\ \Rightarrow Q=\sqrt{0}=0\)