KEY:

Topics for:

 1 x 1 = 1

 2 x 2 = 4

 4 x 4 = 6

 5 x 5 = 5

7 x 7 = 9

Reality:
 1 x 1 = 1
 2 x 2 = 4
 4 x 4 = 16 (remove number 1 and keep number 6)
 5 x 5 = 25 (remove number 2 and keep number 5)
 7 x 7 = 49 (remove 4 and keep 9)
=> The rule of this problem is that the final result will be the last number to the right of the correct answer.

So 9 x 9 = 1.