Question

1,Rút gọn

\(\sqrt[3]{3+\sqrt{9+\dfrac{125}{27}}}-\sqrt[3]{-3+\sqrt{9+\dfrac{125}{27}}}\)

Answer 1

\(A=\sqrt[3]{3+\sqrt{9+\dfrac{125}{27}}}-\sqrt[3]{-3+\sqrt{9+\dfrac{125}{27}}}\)

\(\Leftrightarrow A=\sqrt[3]{3+\sqrt{9+\dfrac{125}{27}}}+\sqrt[3]{3-\sqrt{9+\dfrac{125}{27}}}\)

\(\Leftrightarrow A^3=6+3A.\sqrt[3]{3+\sqrt{9+\dfrac{125}{27}}}.\sqrt[3]{3-\sqrt{9+\dfrac{125}{27}}}\)

\(\Leftrightarrow A^3=6+3A.\left(-\dfrac{5}{3}\right)\)

\(\Leftrightarrow A^3+5A-6=0\)

\(\Leftrightarrow\left(A-1\right)\left(A^2+A+6\right)=0\)

\(\Leftrightarrow A=1\)