1. The problem is to simplify the expression $\sqrt{\sqrt{\sqrt{6,561}}}$. 2. Recall that the square root of a number $x$ is $x^{\frac{1}{2}}$. Applying this repeatedly, the expression can be written as: $$\sqrt{\sqrt{\sqrt{6,561}}} = (6,561)^{\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}} = (6,561)^{\frac{1}{8}}$$ 3. Next, we need to find the eighth root of 6,561. 4. Note that $6,561 = 9^4$ because $9^2 = 81$ and $81^2 = 6,561$. 5. Therefore: $$ (6,561)^{\frac{1}{8}} = (9^4)^{\frac{1}{8}} = 9^{\frac{4}{8}} = 9^{\frac{1}{2}} $$ 6. The square root of 9 is 3, so: $$ 9^{\frac{1}{2}} = \sqrt{9} = 3 $$ 7. Hence, the simplified value of the original expression is 3.