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

To simplify the expression \(\sqrt[3]{3} \cdot \sqrt[3]{3} \cdot \sqrt[6]{3}\), we can use the properties of exponents and radicals.<br /><br />First, recall that \(\sqrt[3]{3}\) can be written as \(3^{1/3}\) and \(\sqrt[6]{3}\) can be written as \(3^{1/6}\).<br /><br />So, the expression becomes:<br />\[ 3^{1/3} \cdot 3^{1/3} \cdot 3^{1/6} \]<br /><br />Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents:<br />\[ 3^{1/3 + 1/3 + 1/6} \]<br /><br />Next, we need to find a common denominator for the fractions \(1/3\) and \(1/6\). The least common multiple of 3 and 6 is 6. Converting the fractions:<br />\[ \frac{1}{3} = \frac{2}{6} \]<br /><br />So, the expression becomes:<br />\[ 3^{2/6 + 2/6 + 1/6} \]<br /><br />Adding the fractions:<br />\[ \frac{2}{6} + \frac{2}{6} + \frac{1}{6} = \frac{5}{6} \]<br /><br />Thus, the expression simplifies to:<br />\[ 3^{5/6} \]<br /><br />Therefore, the simplified form of \(\sqrt[3]{3} \cdot \sqrt[3]{3} \cdot \sqrt[6]{3}\) is:<br />\[ 3^{5/6} \]