To solve this equation, we will first simplify by applying the properties of exponents.

(0.75)^(-3) * √(4/3)^(8/X) = 9/16

(1/0.75)^3 * (4/3)^(8/X) = 9/16

(1.333)^3 * (4/3)^(8/X) = 9/16

(2/√3)^3 * (4/3)^(8/X) = 9/16

(8/(√3)^3 ) * (4/3)^(8/X) = 9/16

(8/(3^(3/2)) ) * (4/3)^(8/X) = 9/16

(8/3√3) * (4/3)^(8/X) = 9/16

Now we can simplify further by multiplying the terms on the left side:

(32/(9√3)) * (4/3)^(8/X) = 9/16

To further simplify, we can rewrite (4/3)^(8/X) as (2/√3)^(8/X) = 2^(8/X) / 3^(4/X).

Our equation becomes:

(32/(9√3)) * (2^(8/X) / 3^(4/X)) = 9/16

Simplify further:

(322^(8/X)) / (9√33^(4/X)) = 9/16

322^(8/X) / (93^(4/X)) = 9/16

To solve for X, we need to find the value of X that makes this equation true. There may not be a simple algebraic solution for X.

If you have restrictions on the domain of X or more information about the equation, please provide them and we can assist further.