Solve the Nested Radical: Cube Root of Square Root of 729

Solve the following exercise:

$\sqrt[3]{\sqrt{729}}=$

To solve this problem, we need to evaluate $\sqrt[3]{\sqrt{729}}$.

Step 1: Convert the expression into exponent form. We know $\sqrt{729} = 729^{\frac{1}{2}}$. Thus, $\sqrt[3]{\sqrt{729}} = (729^{\frac{1}{2}})^{\frac{1}{3}}$.

Step 2: Apply the formula for exponents $(a^m)^n = a^{m \cdot n}$. Thus, $(729^{\frac{1}{2}})^{\frac{1}{3}} = 729^{\frac{1}{2 \cdot 1/3}} = 729^{\frac{1}{6}}$.

Step 3: Find the base of 729 as a power of an integer. Observing, 729 = $3^6$ because $3^6 = 729$.

Step 4: Substitute the power of the base: $729^{\frac{1}{6}} = (3^6)^{\frac{1}{6}} = 3^{6 \cdot \frac{1}{6}} = 3^1 = 3$.

Therefore, the solution to the problem is $\sqrt[3]{\sqrt{729}} = 3$.