Question

rút gọn biểu thức A= ∛2 + √7+2√10 + ∛3∛4 - 3∛2 -1

Answer 1

Ta có: \(\left(1-\sqrt[3]{2}\right)^3\)

\(=1^3-3\cdot1^2\cdot\sqrt[3]{2}+3\cdot1\cdot\sqrt[3]{2^2}-\left(\sqrt[3]{2}\right)^3\)

\(=1-3\sqrt[3]{2}+3\cdot\sqrt[3]{4}-2=3\sqrt[3]{4}-3\sqrt[3]{2}-1\)

=>\(\sqrt[3]{3\cdot\sqrt[3]{4}-3\cdot\sqrt[3]{2}-1}=\sqrt[3]{\left(1-\sqrt[3]{2}\right)^3}=1-\sqrt[3]{2}\)

Ta có: \(A=\sqrt[3]{2}+\sqrt{7+2\sqrt{10}}+\sqrt[3]{3\cdot\sqrt[3]{4}-3\cdot\sqrt[3]{2}-1}\)

\(=\sqrt[3]{2}+1-\sqrt[3]{2}+\sqrt[2]{\left(\sqrt5+\sqrt2\right)^2}\)

\(=1+\sqrt5+\sqrt2\)