What is the degree $[\Bbb Q(\sqrt{p},\sqrt[3]{p},\sqrt[4]{p},\cdots,\sqrt[n]{p}):\Bbb Q]$?

Question

Suppose $p$ is a prime, and $\mathbb Q$ the field of rational numbers. Calculate the value of $$[\mathbb Q(\sqrt{p},\sqrt[3]{p},\sqrt[4]{p},\dots,\sqrt[n]{p}):\mathbb Q].$$

I know it is not smaller than the least common multiple of $1,2,..., n$, but I have no idea about its exact value.

Any answer will be appreciated, thank you so much.

Answer 1

Hint: this field has a generator of the form $p^{1/N}$ for some $N$. Can you find this $N$?