modulus in number theory

Find all solutions of x^11 ≡ 1 (mod 23). Justify your work

If I attempt to apply power of 11 to all values from 1-23, I get too large a value to then be able to see if it can be reduced modulo 23.

Is there a simpler way? any help would be appreciated

When doing modular arithmetic you never need to deal with really big numbers because you reduce along the way. So for $2^{11} \pmod{23}$ you compute $$ \begin{align} 2^2 & = 4\\ 2^3 & = 8\\ 2^4 & = 16\\ 2^5 & = 32 \equiv 9\\ 2^6 & \equiv 2 \times 9 = 18\\ 2^7 & \equiv 2 \times 18 = 36 \equiv 13 \end{align} $$ and so on.

There are other shortcuts that help with this particular problem, but the general principle (you never need big numbers) is worth remembering all the time.