The width of a rectangle is $\sqrt[3]{8^{4}} \mathrm{ft}$ and the length is $64^{\frac{2}{3}} \mathrm{ft}$. What is the area of the rectangle?

Step 1 ：The width of the rectangle is given as \(\sqrt[3]{8^{4}}\) feet.

Step 2 ：By simplifying this expression, we find that the width is approximately 16 feet.

Step 3 ：The length of the rectangle is given as \(64^{\frac{2}{3}}\) feet.

Step 4 ：By simplifying this expression, we find that the length is also approximately 16 feet.

Step 5 ：The area of a rectangle is calculated by multiplying its length by its width.

Step 6 ：So, the area of the rectangle is \(16 \times 16 = 256\) square feet.

Step 7 ：Final Answer: The area of the rectangle is approximately \(\boxed{256}\) square feet.