What is the following product? sqrt [3](4)cdot sqrt (3) 2(sqrt [6](3,888)) sqrt [6](12) sqrt [6](432) 2(sqrt [6](9))

1. Simplify each term<br /> $\sqrt[3]{4} \cdot \sqrt{3} = 4^{1/3} \cdot 3^{1/2}$, $2(\sqrt[6]{3888}) = 2 \cdot 3888^{1/6}$, $\sqrt[6]{12} = 12^{1/6}$, $\sqrt[6]{432} = 432^{1/6}$, $2(\sqrt[6]{9}) = 2 \cdot 9^{1/6}$.<br /><br />2. Combine exponents<br /> Combine all terms under a single radical: $(4^{1/3} \cdot 3^{1/2} \cdot 2 \cdot 3888^{1/6} \cdot 12^{1/6} \cdot 432^{1/6} \cdot 2 \cdot 9^{1/6})$.<br /><br />3. Simplify the expression<br /> Convert to a common base: $= (2^2)^{1/3} \cdot 3^{1/2} \cdot 2 \cdot (2^5 \cdot 3^5)^{1/6} \cdot (2^2 \cdot 3)^{1/6} \cdot (2^4 \cdot 3^3)^{1/6} \cdot 2 \cdot (3^2)^{1/6}$.<br /><br />4. Calculate powers<br /> Simplify each power: $= 2^{2/3} \cdot 3^{1/2} \cdot 2 \cdot 2^{5/6} \cdot 3^{5/6} \cdot 2^{1/3} \cdot 3^{1/6} \cdot 2^{2/3} \cdot 3^{1/2} \cdot 2^{4/6} \cdot 3^{3/6} \cdot 2 \cdot 3^{2/6}$.<br /><br />5. Combine like terms<br /> Combine powers of 2 and 3: $= 2^{2/3 + 1 + 5/6 + 1/3 + 2/3 + 4/6 + 1} \cdot 3^{1/2 + 5/6 + 1/6 + 1/2 + 3/6 + 2/6}$.<br /><br />6. Simplify exponents<br /> Calculate the sum of exponents: $= 2^{17/6} \cdot 3^{11/6}$.<br /><br />7. Final simplification<br /> Simplify further if possible: $= (2^{17} \cdot 3^{11})^{1/6}$.