sqrt[3](3) cdot sqrt(3) cdot sqrt[6](3)

To simplify the expression \(\sqrt[3]{3} \cdot \sqrt{6} \cdot \sqrt[6]{3}\), we can rewrite each term using fractional exponents:<br /><br />\[<br />\sqrt[3]{3} = 3^{1/3}, \quad \sqrt{6} = 6^{1/2}, \quad \sqrt[6]{3} = 3^{1/6}<br />\]<br /><br />Now, multiply these together:<br /><br />\[<br />3^{1/3} \cdot 6^{1/2} \cdot 3^{1/6}<br />\]<br /><br />Combine the exponents of 3:<br /><br />\[<br />3^{1/3 + 1/6} = 3^{2/6 + 1/6} = 3^{3/6} = 3^{1/2}<br />\]<br /><br />So the expression simplifies to:<br /><br />\[<br />3^{1/2} \cdot 6^{1/2}<br />\]<br /><br />This can be written as:<br /><br />\[<br />(3 \cdot 6)^{1/2} = 18^{1/2} = \sqrt{18}<br />\]<br /><br />Therefore, the simplified form of the given expression is:<br /><br />\[<br />\boxed{\sqrt{18}}<br />\]