Question

\(\sqrt[3]{200+126\sqrt[3]{2}+\dfrac{54}{1+\sqrt[3]{2}}}\)+\(\sqrt[3]{\dfrac{18}{1+\sqrt[3]{2}}-6\sqrt[3]{2}}\)

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Answer 1

\(\sqrt[3]{200+126\sqrt[3]{2}+\dfrac{54}{1+\sqrt[3]{2}}}+\sqrt[3]{\dfrac{18}{1+\sqrt[3]{2}}-6\sqrt[3]{2}}\)

\(=\sqrt[3]{200+126\sqrt[3]{2}+\dfrac{54\left(1-\sqrt[3]{2}+\sqrt[3]{4}\right)}{3}}+\sqrt[3]{\dfrac{18\left(1-\sqrt[3]{2}+\sqrt[3]{4}\right)}{3}-6\sqrt[3]{2}}\)

\(=\sqrt[3]{218+108\sqrt[3]{2}+18\sqrt[3]{4}}+\sqrt[3]{6-12\sqrt[3]{2}+6\sqrt[2]{4}}\)

\(=\sqrt[3]{216+3.6^2\sqrt[3]{2}+3.6\sqrt[3]{4}+2}+\sqrt[3]{8-3.2^2\sqrt[3]{2}+3.2\sqrt[2]{4}-2}\)

\(=\sqrt[3]{\left(6+\sqrt[3]{2}\right)^3}+\sqrt[3]{\left(2-\sqrt[3]{2}\right)^3}\)

\(=6+\sqrt[3]{2}+2-\sqrt[3]{2}=8\)