Question

so sánh

\(M=\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}\) và\(N=\dfrac{4}{\sqrt[3]{9}}\)

Answer 1

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

\(\Leftrightarrow M^3=7+5\sqrt{2}+7-5\sqrt{2}+3\left(\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}\right)\sqrt[3]{\left(7+5\sqrt{2}\right)\left(7-5\sqrt{2}\right)}\)

\(\Leftrightarrow M^3=14-3M\)

\(\Leftrightarrow M^3-14+3M=0\)

\(\Leftrightarrow\left(M-2\right)\left(M^2+2M+7\right)=0\)

De thay: \(M^2+2M+7=\left(M+1\right)^2+6>0\forall M\)

\(\Leftrightarrow M=2>\dfrac{4}{\sqrt[3]{9}}=N\)