Solve the Nested Root Equation: Fifth Root of Square Root of x^10 = √√81

To solve this problem, we'll begin by simplifying both sides of the equation:

• Simplifying the left-hand side:

$\sqrt[5]{\sqrt{x^{10}}}$ can be rewritten using properties of exponents and roots. The inner square root is $(x^{10})^{1/2} = x^{10 \cdot \frac{1}{2}} = x^{5}$.

Then, take the fifth root: $(x^{5})^{\frac{1}{5}} = x^{5 \cdot \frac{1}{5}} = x^{1} = x$.
Thus, the left-hand side simplifies to $x$.

• Simplifying the right-hand side:

$\sqrt{\sqrt{81}}$ simplifies as follows: First, find $\sqrt{81} = 9$.
Then, compute $\sqrt{9} = 3$.

So, the equation reduces to $x = 3$.

Therefore, the solution to the problem is $\boxed{x = 3}$.