Call $d$ the degree I want to find.
I know that
$d=[\mathbb{Q}(\sqrt[4]{3},\sqrt[5]{3}):\mathbb{Q}(\sqrt[4]{3})][\mathbb{Q}(\sqrt[4]{3}):\mathbb{Q}]=[\mathbb{Q}(\sqrt[4]{3},\sqrt[5]{3}):\mathbb{Q}(\sqrt[5]{3})][\mathbb{Q}(\sqrt[5]{3}):\mathbb{Q}]$
$[\mathbb{Q}(\sqrt[4]{3}):\mathbb{Q}]=4$
$[\mathbb{Q}(\sqrt[5]{3}):\mathbb{Q}]=5$
So $4$ and $5$ divides $d$
My guess is that $d=20$ so I'm trying to prove that $f=x^5-3$ is irreducible over $\mathbb{Q}(\sqrt[4]{3})$, but I'm stuck at it.
Any tips?