1. The problem asks to simplify the expression $2\sqrt{3\sqrt{x^5}}$ using rational exponents. 2. Recall that radicals can be expressed as rational exponents: $\sqrt[n]{a} = a^{\frac{1}{n}}$. 3. Rewrite the expression step-by-step: - The innermost radical is $\sqrt{x^5} = (x^5)^{\frac{1}{2}} = x^{\frac{5}{2}}$. - Next, we have $3\sqrt{x^{\frac{5}{2}}} = (x^{\frac{5}{2}})^{\frac{1}{3}} = x^{\frac{5}{2} \times \frac{1}{3}} = x^{\frac{5}{6}}$. - Finally, multiply by 2: $2 \times x^{\frac{5}{6}}$. 4. The simplified expression is $2x^{\frac{5}{6}}$. 5. Among the answer choices, the exponent on $x$ is $\frac{5}{6}$, so the correct answer is $x^{\frac{5}{6}}$. This matches the last option. Final answer: $x^{\frac{5}{6}}$