Simplifying the expression:

To evaluate the given expression, we need to simplify it step by step.

Step 1: Simplifying the innermost parentheses:
Inside the parentheses, we have the expression 64^(1/3) * 125^(1/3).

Calculating 64^(1/3):
64^(1/3) means finding the cube root of 64. The cube root of 64 is 4 because 4 * 4 * 4 = 64.

Calculating 125^(1/3):
125^(1/3) means finding the cube root of 125. The cube root of 125 is 5 because 5 * 5 * 5 = 125.

Therefore, the expression 64^(1/3) * 125^(1/3) simplifies to 4 * 5 = 20.

Step 2: Evaluating 81^(1/2):
Next, we need to find the square root of 81. The square root of 81 is 9 because 9 * 9 = 81.

Step 3: Simplifying the expression inside the outermost parentheses:
Now, we have the expression √81 * 20.

Calculating √81:
√81 means finding the square root of 81. We already found that the square root of 81 is 9.

Therefore, the expression √81 * 20 simplifies to 9 * 20 = 180.

Step 4: Evaluating the final expression:
The final expression is (180)^(1/4), which means we need to find the fourth root of 180.

Calculating (180)^(1/4):
To find the fourth root of 180, we need to find a number that, when raised to the power of 4, equals 180. This is not a perfect fourth root, so we can use approximation methods or a calculator to find the value.

Using a calculator, we find that the fourth root of 180 is approximately 4.326748710922224.

Therefore, the value of the expression {[81^(1/2)(64^(1/3) 125^(1/3))]^(1/4)} is approximately 4.326748710922224.