Simplify the Expression: Cube Root of Square Root of 64 × Cube Root of 64

Let's solve the problem step-by-step.

• Step 1: Simplify $\sqrt[3]{\sqrt{64}}$.
• Step 2: Simplify $\sqrt[3]{64}$.
• Step 3: Multiply the results of Step 1 and Step 2.

Step 1: Consider $\sqrt[3]{\sqrt{64}}$.

We can write $\sqrt{64}$ as $64^{1/2}$. Thus, $\sqrt[3]{\sqrt{64}} = \sqrt[3]{64^{1/2}}$.

Using the property $\sqrt[n]{a^m} = a^{m/n}$, we have $(64^{1/2})^{1/3} = 64^{1/6}$.

Step 2: Simplify $\sqrt[3]{64}$.

The cube root of a number $b$ is expressed as $b^{1/3}$. Therefore, $\sqrt[3]{64} = 64^{1/3}$.

Step 3: Multiply the two results.

We now compute $64^{1/6} \cdot 64^{1/3}$.

Using the property of exponents, $a^m \cdot a^n = a^{m+n}$, thus $64^{1/6} \cdot 64^{1/3} = 64^{(1/6 + 1/3)} = 64^{(1/6 + 2/6)} = 64^{3/6} = 64^{1/2}$.

Finally, $64^{1/2}$ is simply $\sqrt{64}$, which equals $8$.

Therefore, the solution to the problem is $8$, which corresponds to choice (3).