Solve the Nested Radical: Cube Root of Square Root of 64

Solve the following exercise:

$\sqrt[3]{\sqrt{64}}=$

To solve this problem, we'll express the nested roots using exponents and then simplify:

We start with the inner expression:

$\sqrt{64}$ is equivalent to $64^{1/2}$.

Next, apply the cube root:

$\sqrt[3]{64^{1/2}}$ is equivalent to $(64^{1/2})^{1/3}$.

Using properties of exponents, we simplify the expression:

$(64^{1/2})^{1/3} = 64^{(1/2) \cdot (1/3)} = 64^{1/6}$.

Now, evaluate $64^{1/6}$:

Since $64 = 2^6$, we have:

$64^{1/6} = (2^6)^{1/6} = 2^{6 \cdot (1/6)} = 2^{1} = 2$.

Therefore, the solution to the exercise $\sqrt[3]{\sqrt{64}}$ is $\mathbf{2}$.