Question

tính \(\sqrt[3]{4+\sqrt{5}}-\sqrt[3]{4-\sqrt{5}}\)

Answer 1

A=\(\sqrt[3]{4+\sqrt{5}}\)-\(\sqrt[3]{4-\sqrt{5}}\)

\(^{A^3}\)= (\(\sqrt[3]{4+\sqrt{5}}-\sqrt[3]{4-\sqrt{5}}\))³

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

=4+\(\sqrt{5}\) -4+\(\sqrt{5}\)\(-3\sqrt[3]{4+\sqrt{5}}.\sqrt[3]{4-\sqrt{5}}\left(\sqrt[3]{4+\sqrt{5}}-\sqrt[3]{4-\sqrt{5}}\right)=2\sqrt{5}-3\sqrt[3]{\left(4+\sqrt{5}\right).\left(4-\sqrt{5}\right)}.A=2\sqrt{5}-3\sqrt[3]{16-5}.A=2\sqrt{5}-3\sqrt[3]{11}.A\Rightarrow A^3+3\sqrt[3]{11}.A-2\sqrt{5}=0\)