Solve the Expression: Cube Root of Square Root of 64 Times Square Root of 64

To solve the expression $\sqrt[3]{\sqrt{64}}\cdot\sqrt{64}$, we follow these steps:

• Step 1: Express $\sqrt{64}$ as a power:
  Since $\sqrt{64} = 64^{1/2}$ and $64 = 2^6$, substituting gives $(2^6)^{1/2} = 2^{6 \cdot 1/2} = 2^3 = 8$.
• Step 2: Express $\sqrt[3]{\sqrt{64}}$ as a power:
  Since from Step 1, $\sqrt{64} = 2^3 = 8$, then $\sqrt[3]{\sqrt{64}} = \sqrt[3]{8}$.
  Now, $\sqrt[3]{8} = 8^{1/3}$ and $8 = 2^3$, so $(2^3)^{1/3} = 2^{3 \cdot 1/3} = 2^1 = 2$.
• Step 3: Multiply the simplified expressions:
  We now have $\sqrt[3]{\sqrt{64}} = 2$ and $\sqrt{64} = 8$.
  Thus, $\sqrt[3]{\sqrt{64}} \cdot \sqrt{64} = 2 \cdot 8 = 16$.

Therefore, the solution to the problem is $16$.