HINT $\rm\ \ mod\ 23\::\ \ 2\ \equiv\ 5^{\:2}\ \ \Rightarrow\ \ 2^{\:11}\ \equiv\ 5^{\:22}\ \equiv\ 1\ $ by Fermat's little Theorem.
See Euler's Criterion and quadratic reciprocity to understand what happens generally.
Regarding square-roots, $\rm\ x^2 = a^2\ \iff\ (x-a)\ (x+a) = 0\ \iff\ x = \pm\: a\ \ $ holds true in any integral domain, i.e. it's true in any ring without zero-divisors. More concretely, in $\rm\ \mathbb Z/p\:,\: $ we have prime $\rm\ p\ |\ (x-a)\ (x+a)\ \Rightarrow\ p\ |\ x-a\ $ or $\rm\ p\ |\ x+a\:,\: $ so $\rm\ x \equiv \pm\: a\ \ (mod\ p)\:.$