Solution:

Given expression is 9^y . 3^2 . (3^-y)^-1 - 27^y 3^3x . 2^3 = 1 27

Let's simplify the expression step by step:

1. Simplify the expression inside the brackets first:

(3^-y)^-1 = 3^y

2. Substitute the simplified expression inside the brackets in the main expression:

9^y . 3^2 . 3^y - 27^y . 3^3x . 2^3 = 1 27

3. Simplify the powers of 3:

9^y . 3^(2+y) - 27^y . 3^(3x) . 2^3 = 1 27

4. Simplify the powers of 9 and 27:

(3^2)^y . 3^(2+y) - (3^3)^y . 3^(3x) . 2^3 = 1 27

3^(2y) . 3^(2+y) - 3^(3y) . 3^(3x) . 2^3 = 1 27

5. Combine the powers of 3:

3^(2y+2+y) - 3^(3y+3x) . 2^3 = 1 27

3^(3y+2) - 3^(3y+3x) . 2^3 = 1 27

6. Simplify the powers of 3 and 2:

27 . 3^(y+2) - 8 . 3^(3y+3x) = 1 27

7. Simplify the left-hand side:

27 . 3^(y+2) - 2^3 . 3^(3y+3x) = 1 27

27 . 3^(y+2) - 8 . 3^(3y+3x) = 1 27

8. Rewrite the right-hand side as a power of 3:

27 . 3^(y+2) - 8 . 3^(3y+3x) = 3^3

9. Simplify the left-hand side:

27 . 3^(y+2) - 8 . 3^(3y+3x) = 27

10. Divide both sides by 27:

3^(y+2) - 8/27 . 3^(3y+3x) = 1

Therefore, the simplified expression is 3^(y+2) - 8/27 . 3^(3y+3x) = 1.