Simplify the Expression: Cube Root of √25 × Cube Root of √64

To solve the problem $\sqrt[3]{\sqrt{25}} \cdot \sqrt[3]{\sqrt{64}}$, we will work through it step by step:

Step 1: Simplify the inner square roots.

• $\sqrt{25}$ simplifies to $5$ because $5 \times 5 = 25$.
• $\sqrt{64}$ simplifies to $8$ because $8 \times 8 = 64$.

Step 2: Evaluate the cube roots.

• $\sqrt[3]{5}$ remains as $5^{1/3}$.
• $\sqrt[3]{8}$ evaluates to $2$ because $2 \times 2 \times 2 = 8$.

Step 3: Multiply the results of the cube roots.

• $5^{1/3} \cdot 2 = 2 \sqrt[3]{5}$.

Thus, the simplified expression is $2 \sqrt[3]{5}$.

Therefore, the solution to the problem is $2\sqrt[3]{5}$.