Solve the Nested Radical: Fourth Root of Square Root of 6

To solve $\sqrt[4]{\sqrt{6}}$, we'll simplify it using the property of roots:

According to the property $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x}$, we are to multiply the indices of the roots.

Here, the indices are 4 (for the fourth root) and 2 (for the square root), hence $n = 4$ and $m = 2$.

Therefore, we calculate:

$4 \times 2 = 8$

This indicates that the expression can be rewritten as a single root of index 8, giving us:

$\sqrt[4]{\sqrt{6}} = \sqrt[8]{6}$

The answer is therefore $\sqrt[8]{6}$, which matches answer choice 4.