Question

Answer 1

a: \(\sqrt[3]{8\sqrt{5}+16}\cdot\sqrt[3]{8\sqrt{5}-16}\)

\(=\sqrt[3]{\left(8\sqrt{5}+16\right)\left(8\sqrt{5}-16\right)}\)

\(=\sqrt[3]{8\cdot8\cdot\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}=\sqrt[3]{64}=4\)

b: \(\dfrac{\sqrt[3]{8-3\sqrt{5}}+\sqrt[3]{64-12\sqrt{20}}}{\sqrt[3]{57}}\cdot\sqrt[3]{8+3\sqrt{5}}\)

\(=\dfrac{\sqrt[3]{\left(8-3\sqrt{5}\right)\left(8+3\sqrt{5}\right)}+\sqrt[3]{\left(64-12\sqrt{20}\right)\left(8+3\sqrt{5}\right)}}{\sqrt[3]{57}}\)

\(=\dfrac{\sqrt[3]{64-45}+\sqrt[3]{\left(64-24\sqrt{5}\right)\left(8+3\sqrt{5}\right)}}{\sqrt[3]{57}}\)

\(=\dfrac{\sqrt[3]{19}+\sqrt[3]{8\cdot\left(8-3\sqrt{5}\right)\left(8+3\sqrt{5}\right)}}{\sqrt[3]{57}}\)

\(=\dfrac{\sqrt[3]{19}+2\cdot\sqrt[3]{19}}{\sqrt[3]{57}}=3\sqrt[3]{\dfrac{19}{57}}=3\cdot\sqrt[3]{\dfrac{1}{3}}=\sqrt[3]{9}\)