Question

Cho \(x=\sqrt[3]{7+5\sqrt{2}}-\dfrac{1}{\sqrt[3]{7+5\sqrt{2}}}\)

Tính giá trị biểu thức: F=\(x^3+3x-14\)

Answer 1

\(\dfrac{1}{\sqrt[3]{7+5\sqrt{2}}}=\dfrac{\sqrt[3]{7-5\sqrt{2}}}{\sqrt[3]{\left(7+5\sqrt{2}\right)\left(7-5\sqrt{2}\right)}}=-\sqrt[3]{7-5\sqrt{2}}\)

\(\Rightarrow x=\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}\)

\(\Rightarrow x^3=\left(\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}\right)^3=14-3\left(\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}\right)\)

\(\Rightarrow x^3=14-3x\Rightarrow x^3+3x-14=0\)

Vậy F=0