1. **State the problem:** Simplify the expression $\sqrt[6]{\sqrt{15}}$. 2. **Rewrite the expression using exponents:** The square root of 15 is $15^{\frac{1}{2}}$, so the expression becomes $\sqrt[6]{15^{\frac{1}{2}}}$. 3. **Apply the rule for roots and exponents:** $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. Here, $n=6$ and $m=\frac{1}{2}$, so $$\sqrt[6]{15^{\frac{1}{2}}} = 15^{\frac{1}{2} \times \frac{1}{6}} = 15^{\frac{1}{12}}.$$ 4. **Final simplified form:** The expression simplifies to $15^{\frac{1}{12}}$, which is the 12th root of 15. This is the simplest exact form.