Let $k \subseteq F$ be a radical extension, and let $k \subseteq K$ be any extension; assume F and K are contained in a larger field.
Prove that $K \subseteq FK$ is radical.
I know that I need to use meets and joins for show that $F$ and $K$ are in $FK$ which would be the larger field; however, I know know how to go about proving that $K \subseteq FK$ is radical
Since $F/k$ is radical it is contained in a larger extension $E/k$ such that $E$ is the top of a tower
$k=E_0\subset E_1\subset\cdots\subset E_n=E$
Where $E_{i+1}=E_{i}(\sqrt[n_{i}]{\alpha_{i}})$ for some $n_i$ and $\alpha_{i}\in E_{i}$ such that $\sqrt[n_{i}]{\alpha_{i}}\not\in E_{i}$. (Sorry for the lengthy notation).
The idea is that we can just adjoin the $\sqrt[n_i]{\alpha_i}$ to $K$ to get a field $E'/K$ which is on the top of the same sort of tower of field extensions. Then we can show that $FK\subset E'$, proving that $FK/K$ is a radical extension.