Find rational numbers $\alpha $ and $\beta$ in $\sqrt[3]{7+5\sqrt{2}}=\alpha+\beta\sqrt{2}$ How should we find two rational numbers $\alpha$ and $\beta$ such that
$$\sqrt[3]{7+5\sqrt{2}}=\alpha+\beta\sqrt{2}$$
The answer I got $\alpha = 1 $ and $\beta = 1$. If I'm wrong, please correct me. Thank you

By Gauss' lemma, if $\alpha$ and $\beta$ are rational, then they are integers. Simply cube both sides to get
$$7+5\sqrt{2}=(\alpha+\beta\sqrt{2})^3.$$
Expanding the right hand side and comparing coefficients shows that
\begin{eqnarray*}
7&=&\alpha^3+6\alpha\beta^2&=&\alpha(\alpha^2+6\beta^2),\\
5&=&3\alpha^2\beta+2\beta^3&=&\beta(3\alpha^2+2\beta^2),
\end{eqnarray*}
so $\alpha$ divides $7$ and $\beta$ divides $5$. This leaves very few options to check.
