Question

\(A=\left(4+\sqrt{15}\right)\left(\sqrt{10}-\sqrt{6}\right)\sqrt{4-\sqrt{15}}\)

Số trên là số hữu tỉ hay vô tỉ

Answer 1

\(A=\left(4+\sqrt{15}\right)\left(\sqrt{10}-\sqrt{6}\right)\sqrt{4-\sqrt{15}}\)

\(=\left(\sqrt{10}-\sqrt{6}\right).\sqrt{4+\sqrt{5}}.\)\(\sqrt{4+\sqrt{5}}.\sqrt{4-\sqrt{5}}\)

\(=\left(\sqrt{10}-\sqrt{6}\right).\sqrt{4+\sqrt{5}}.\)\(\sqrt{\left(4-\sqrt{5}\right)\left(4+\sqrt{5}\right)}\)

\(=\left(\sqrt{10}-\sqrt{6}\right)\)\(\sqrt{4+\sqrt{5}}.\sqrt{16-15}\)

\(=\left(\sqrt{5}-\sqrt{3}\right)\sqrt{8+2\sqrt{5}}\)

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

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

\(=5-3=2\)

\(\Rightarrow A\)là số hữu tỉ