Given $x^3$ mod $55$, find its inverse So i am wondering how i can figure out what the functional inverse  of $x^3$ mod $55$ is.
I can only assume it is $x^{1/3}$ mod $55$ but i am not sure if that is the form i should keep it in

Suppose that $x^3\equiv y\pmod{55}$, where $y$ is relatively prime to $55$. Then $x^{20}\equiv 1\pmod{55}$, since $x^4\equiv 1\pmod{5}$ and $x^{10}\equiv 1\pmod{11}$. 
It follows that $x^{21}\equiv x\pmod{55}$. Note that in fact this congruence holds for all $x$. Thus if $y\equiv x^3\pmod{55}$, then
$$x\equiv (x^3)^7\equiv y^7\pmod{55}.$$
