Solution:

We are given an expression -

`((1(4ab^2c))^2/((3/(2a^2bc^2))^4`

We can simplify this expression using the following steps -

Step 1: Simplify the numerator `(1(4ab^2c))^2`

- `(1(4ab^2c))^2 = (4ab^2c)^2 = 16a^2b^4c^2`

Therefore, the numerator simplifies to `16a^2b^4c^2`.

Step 2: Simplify the denominator `((3/(2a^2bc^2))^4`

- `((3/(2a^2bc^2))^4 = (3^4)/((2a^2bc^2)^4)`
- `= 81/(16a^8b^4c^8)`

Therefore, the denominator simplifies to `81/(16a^8b^4c^8)`.

Step 3: Simplify the entire expression `((1(4ab^2c))^2/((3/(2a^2bc^2))^4` by substituting the simplified numerator and denominator from steps 1 and 2 respectively.

- `((1(4ab^2c))^2/((3/(2a^2bc^2))^4`
- `= (16a^2b^4c^2)/(81/(16a^8b^4c^8))`
- `= 16a^2b^4c^2 * 16a^8b^4c^8/81`
- `= 512a^10b^8c^10/81`

Therefore, the simplified expression is `512a^10b^8c^10/81`.

Hence, the final simplified expression is `512a^10b^8c^10/81`.