(280) \( \sqrt[3]{\frac{1}{2}} \cdot \sqrt[4]{\frac{2}{3}} \)

Let's simplify the expression \( \sqrt[3]{\frac{1}{2}} \cdot \sqrt[4]{\frac{2}{3}} \). Start with rewriting the roots in exponent form: \[ \sqrt[3]{\frac{1}{2}} = \left(\frac{1}{2}\right)^{1/3} = \frac{1^{1/3}}{2^{1/3}} = \frac{1}{2^{1/3}} \] \[ \sqrt[4]{\frac{2}{3}} = \left(\frac{2}{3}\right)^{1/4} = \frac{2^{1/4}}{3^{1/4}} \] Now combine both parts: \[ \sqrt[3]{\frac{1}{2}} \cdot \sqrt[4]{\frac{2}{3}} = \frac{1}{2^{1/3}} \cdot \frac{2^{1/4}}{3^{1/4}} = \frac{2^{1/4}}{2^{1/3} \cdot 3^{1/4}} \] Next, simplify the base of \(2\): \[ 2^{1/4} \div 2^{1/3} = 2^{1/4 - 1/3} = 2^{3/12 - 4/12} = 2^{-1/12} \] Now we can write it in a single fraction: \[ \frac{2^{-1/12}}{3^{1/4}} = \frac{1}{2^{1/12} \cdot 3^{1/4}} \] Finally, the simplified result is: \[ \frac{1}{2^{1/12} \cdot 3^{1/4}} \] Therefore, the original expression simplifies to: \[ \boxed{\frac{1}{2^{1/12} \cdot 3^{1/4}}} \]