This is the following result from Milne's Fields and Galois Theory (page 72):
Milne only showed the "only if" part but I admit that I cannot give the reason why the "if" part is true.
My attempt: In case we have $a= b^r c^n$, it is $a^{1/n}/c$ a root of $X^n - b^r$. Now I think it might be a good idea to use the fact that $r$ is coprime to $n$. However, here is the part where I don't know how to use the fact and continue.
Could you please help me elaborating on this part? Thank you!
The idea is that the stated condition is, when $r$ is relatively prime to $n$, actually symmetric in $a$ and $b$, even though it appears otherwise.
In this it is clear that under those conditions, $a^{1/n}$ is in $F(b^{1/n})$, so you only need the other direction.
There are integers $x,y$ such that $rx + ny = 1$. Then take the $x$th power of both sides: $$a^x = b^{rx} c^n.$$ Multipy both by $ny$ and combine the exponent on the RHS $b$ into a $1$: $$a^x b^{ny} = b^{rx+ny} c^n = b^{1} c^n.$$ Then divide by $c^n$ and group terms on the left: $$a^x \left(\frac{b^y}{c}\right)^n = b$$
And so $b^{1/n}$ is in $F(a^{1/n})$, giving the other inclusion.