Powerful Finale

Answer: $7$

To find the next power of $3$, multiply by $3$

To find the last digit of the next power of $3$, multiply the last digit by $3$

$3^1 = 3$                                       $3^5 = ...3$, since $1\times3=3$

$3^2=9$                                       $3^6=...9$, since $3\times3=9$

$3^3=...7$                                    $3^7=...7$                                   $3^{2011}= ...7$

$3^4=...1$, since $7 \times 3 = 21$         $3^8=...1$                                   $3^{2012}= ...1$