To simplify the expression \( 8 \cdot 8\sqrt{32} \), we first calculate the product of the coefficients:
\[ 8 \cdot 8 = 64 \]
Next, we will simplify \( \sqrt{32} \). We can factor \( 32 \) as follows:
\[ 32 = 16 \cdot 2 \]
Now, we can take the square root:
\[ \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \]
Now we can substitute back into our expression:
\[ 8 \cdot 8\sqrt{32} = 64 \cdot \sqrt{32} = 64 \cdot 4\sqrt{2} \]
Calculating this gives:
\[ 64 \cdot 4 = 256 \]
Thus, we can express it as:
\[ 256\sqrt{2} \]
In summary, the product \( 8 \cdot 8\sqrt{32} \) in simplest radical form is:
\[ \boxed{256\sqrt{2}} \]
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