Multiply \( \sqrt[4]{6} \cdot \sqrt[3]{6^{2}} \) using fractional exponents.

To solve \( \sqrt[4]{6} \cdot \sqrt[3]{6^{2}} \) using fractional exponents, we first express each term in fractional exponent form: \[ \sqrt[4]{6} = 6^{1/4} \] \[ \sqrt[3]{6^{2}} = (6^{2})^{1/3} = 6^{2/3} \] Now, we can multiply these two expressions: \[ 6^{1/4} \cdot 6^{2/3} \] According to the properties of exponents, when multiplying two expressions with the same base, we add the exponents: \[ 6^{1/4 + 2/3} \] Next, we need to find a common denominator to add the fractions \(1/4\) and \(2/3\). The least common multiple of 4 and 3 is 12. Converting \(1/4\) to have a denominator of 12: \[ \frac{1}{4} = \frac{3}{12} \] And converting \(2/3\): \[ \frac{2}{3} = \frac{8}{12} \] Now we can add these two fractions: \[ \frac{3}{12} + \frac{8}{12} = \frac{11}{12} \] Thus, we can rewrite the expression: \[ 6^{1/4 + 2/3} = 6^{11/12} \] Finally, the answer is: \[ \sqrt[4]{6} \cdot \sqrt[3]{6^{2}} = 6^{11/12} \]