Let $n$ be even. Are there any constructible numbers of the form $\sqrt[n+1]{2} + \sqrt[n+3]{2}$?
Attempt
My initial hunch is no. Because the $\sqrt[n+1]{2}$ is a root of the polynomial $f(x)=x^{n+1}-2$ and applying Eiseinstein's criterion with $p=2$, $f$ is irreducible thus the minimal polynomial for $\sqrt[n+1]{2}$ over $\mathbb{Q}$. Hence $\sqrt[n+1]{2}$ is algebraic of degree $n+1$ which is odd since $n$ is even. Thus, $n+1 \neq 2^k$. Hence is not constructible. By similar argument, $\sqrt[n+3]{2}$ is not constructible. So does this imply $\sqrt[n+1]{2} + \sqrt[n+3]{2}$ is not constructible? If no, how should I approach this problem?