$$\eqalign{
{\text{Упростите:}} \hfill \\
\sqrt[3]{{38 + 17\sqrt 5 }} + \sqrt[3]{{72 - 32\sqrt 5 }}. \hfill \\
} $$
$$\eqalign{
38 + 17\sqrt 5 = {\left( {a + b\sqrt 5 } \right)^3} = {a^3} + 3{a^2}\sqrt 5 + 15a{b^2} + 5\sqrt 5 {b^3} = \hfill \\
{a^3} + 15a{b^2} + \left( {3{a^2} + 5{b^3}} \right)\sqrt 5 \hfill \\
\left\{ \begin{gathered}
{a^3} + 15a{b^2} = 38 \hfill \\
3{a^2} + 5{b^3} = 17 \hfill \\
\end{gathered} \right. \Rightarrow \left\{ \begin{gathered}
a\left( {{a^2} + 15{b^2}} \right) = 2 \cdot 19 \hfill \\
3{a^2} + 5{b^3} = 17 \hfill \\
\end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered}
a = 2 \hfill \\
b = 1 \hfill \\
\end{gathered} \right. \hfill \\
38 + 17\sqrt 5 = {\left( {2 + \sqrt 5 } \right)^3} \hfill \\
{\text{Аналогично}}{\text{, }}72 - 32\sqrt 5 = {\left( {3 - \sqrt 5 } \right)^3}. \hfill \\
} $$
\[{\text{5}}\]
