To solve the equation, we first simplify both sides of the equation:

9 * 81^(1-2x) = 27^(2-x)

9 81^(1) 81^(-2x) = 27 * 27^(2-x)

9 81 (1/81)^(2x) = 27 * (1/27)^(x)

729 (1/81)^(2x) = 27 (1/27)^x

Now we can rewrite the equation using exponential properties:

729 (1/81)^(2x) = 27 (1/27)^x

729 (1/81)^(2x) = 27 (1/27)^x

3^6 (3^-4)^(2x) = 3^3 (3^-3)^x

3^(6 - 8x) = 3^(3 - 3x)

Now, since the bases are the same, we can set the exponents equal to each other:

6 - 8x = 3 - 3x

Now we solve for x:

6 + 3 = 8x - 3x

9 = 5x

x = 9/5

Therefore, the solution to the equation is x = 9/5.