81n=34n; n1=n; 34ⁿ; 243=35; 92ⁿ=(32)2ⁿ=32ⁿ⁺¹; (92ⁿ)(33)=32ⁿ⁺¹+3.
So the expression becomes:
(34n3n-34ⁿ+5)/32ⁿ⁺¹+3=
(35n-34ⁿ+5)/32ⁿ⁺¹+3.
Exponents calculation: 4n+5-2n+1-3=22n+2-2n+1=(2n-1)2+1.
(35n-34ⁿ+5)/32ⁿ⁺¹+3
Exponents calculation: 5n-2n+1-3
(35n-34ⁿ+5)/32ⁿ⁺¹+3=35n-2ⁿ⁺¹-3-3(2ⁿ-1)²+1. There are several ways to express this result.
CHECK
It's easy to make a mistake in this type of problem so let's substitute a couple of values for n and plug them in to the original and final expressions. Start with n=0:
Original: (1-3×243)/(9×27)=1/243-3.
Final: 3-5-3=1/243-3.
Let n=1:
Original: (81×3-81×243)/(81×27)=(243-19683)/2187=1/9-9.
Final: 3-2-32=1/9-9.
These two checks seem to confirm the result.