Starting equation:

(

1
9

)

x + 1

= 81

Re-write

1
9

as 3^-2, and

Also re-write the right side as 3⁴;
The above two revisions result in this equation:

(3^{- 2})^x+1 = 3⁴
or
3^{- 2x - 2} = 3⁴
If the powers are equal, and the bases (here, 3) are equal
then the exponents must also be equal:
- 2x - 2 = 4
- 2x = 6

x = -3

See Rules for exponents?