Inverting Modular Exponentiation How can I go about solving the equation $4 = y^4 \bmod{7}$?  Do I have to try all of the possible $y$'s in between $1$ and $7-2$ or is there a smarter way that can be generalized for larger numbers?

The following is -in principle-still "searching" but structures the space to be searched into simpler subspaces:
$$ \begin{array}{} &4 &= y^4 \pmod 7 \\
& y^4 - 4 &\equiv 0 \pmod 7 \\
&(y^2 - 2)(y^2+2) &\equiv 0 \pmod 7 \\
&& \text{giving two factors}\\
&y^2 - 2 &\equiv 0 \pmod 7 \\
 \text{ or } & y^2 + 2 &\equiv 0 \pmod 7  \\
& y^2 &\equiv k \cdot 7 +2 & \to k=1,y^2=9 \text{ or } k=2,y^2=16 \text{ or ...}\\
 \text{ or }  & y^2 &\equiv j \cdot 7- 2  &\to k=?? \\
\end{array}$$
