Step 1: Simplifying Each Component
- Start with \((27)^{2/3}\):
- \(27 = 3^3\), therefore \((27)^{2/3} = (3^3)^{2/3} = 3^{(3 \cdot 2/3)} = 3^2 = 9\).
- Next, simplify \(\sqrt{9}\):
- \(\sqrt{9} = 3\).
- Now for \(3\sqrt{9^3}\):
- \(9^3 = (3^2)^3 = 3^{2 \cdot 3} = 3^6\),
- Thus, \(\sqrt{9^3} = \sqrt{3^6} = 3^{6/2} = 3^3 = 27\),
- Therefore, \(3\sqrt{9^3} = 3 \cdot 27 = 81\).
- Finally, evaluate \(9^{1/2}\):
- \(9^{1/2} = \sqrt{9} = 3\).

Step 2: Combining All Components
- Combine all the simplified parts:
- \((27)^{2/3} \times \sqrt{9} \times 3\sqrt{9^3} \times 9^{1/2} = 9 \times 3 \times 81 \times 3\).

Step 3: Performing the Multiplication
- Calculate step-by-step:
- First, multiply \(9 \times 3 = 27\).
- Next, multiply \(27 \times 81 = 2187\).
- Finally, \(2187 \times 3 = 6561\).

Final Result
- The simplified value of the expression \((27)^{2/3} \times \sqrt{9} \times 3\sqrt{9^3} \times 9^{1/2}\) is **6561**.