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

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

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

=>\(M^3=14+3M\cdot\left(-1\right)=14-3M\)

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

=>\(M^3-2M^2+2M^2-4M+7M-14=0\)

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

=>M-2=0

=>M=2

\(\Leftrightarrow M=\dfrac{4}{2}=\dfrac{4}{\sqrt[3]{8}}>\dfrac{4}{\sqrt[3]{9}}=N\)