Hello again!

You can try finding a pattern in the remainders of powers of $3$ $\pmod{11}$. For instance, $$ \begin{array}{rcl} 3^1 & \equiv & 3 \pmod{11}\\ 3^2 & \equiv & 9 \pmod{11}\\ 3^3 & \equiv & 5 \pmod{11}\\ 3^4 & \equiv & 4 \pmod{11}\\ 3^5 & \equiv & 1 \pmod{11}\\ 3^6 & \equiv & 3 \pmod{11}\\ & \vdots & \end{array} $$ so, you can see that the remainders follow a pattern that repeats every $5$ powers: $$3,9,5,4,1.$$ This way, for example, you can see that $3^{3^5} \equiv 5 \pmod{11}$ since the exponent $3^5$ has remainder $3$ when dividing by $5$.