To simplify the expression \( 6 \times 7\sqrt{128} \), we start by calculating \( \sqrt{128} \).
First, we can factor \( 128 \):
\[ 128 = 64 \times 2 = 8^2 \times 2 = 4^2 \times 2^3 = 2^7 \]
Now, we know:
\[ \sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2} \]
Now we substitute \( \sqrt{128} \) back into the expression:
\[ 6 \times 7\sqrt{128} = 6 \times 7 \times 8\sqrt{2} \]
Calculating the product:
\[ 6 \times 7 = 42 \]
Now, multiply by \( 8\):
\[ 42 \times 8 = 336 \]
Thus, we have:
\[ 336\sqrt{2} \]
So the expression \( 6 \times 7\sqrt{128} \) in simplest radical form is:
\[ \boxed{336\sqrt{2}} \]
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