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

To solve the problem \( \sqrt[4]{6} \cdot \sqrt[3]{6^{2}} \) using fractional exponents, we first express each root as a fractional exponent: \[ \sqrt[4]{6} = 6^{1/4} \quad \text{and} \quad \sqrt[3]{6^{2}} = (6^{2})^{1/3} = 6^{2/3} \] Now we can multiply the two expressions: \[ 6^{1/4} \cdot 6^{2/3} \] When multiplying with the same base, we add the exponents: \[ 6^{1/4 + 2/3} \] To add the fractions, we need a common denominator, which would be 12 in this case: \[ 1/4 = 3/12 \quad \text{and} \quad 2/3 = 8/12 \] Now we add the fractions: \[ 3/12 + 8/12 = 11/12 \] Therefore, we have: \[ 6^{1/4 + 2/3} = 6^{11/12} \] So, the final answer is: \[ \sqrt[4]{6} \cdot \sqrt[3]{6^{2}} = 6^{11/12} \]