To find the product of \( 8\sqrt{2} \) and \( 5\sqrt{10} \), we start by multiplying the two expressions together:
\[ 8\sqrt{2} \times 5\sqrt{10} \]
First, we can multiply the coefficients (the numbers in front) and then the square roots:
\[ (8 \times 5) \cdot (\sqrt{2} \times \sqrt{10}) \]
Calculating the coefficients:
\[ 8 \times 5 = 40 \]
Now we multiply the square roots. Recall that \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \):
\[ \sqrt{2} \times \sqrt{10} = \sqrt{2 \times 10} = \sqrt{20} \]
Now we can combine the results:
\[ 40 \cdot \sqrt{20} \]
Next, we simplify \( \sqrt{20} \). We can factor \( 20 \) into \( 4 \times 5 \):
\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \]
Now substituting this back into the expression gives us:
\[ 40 \cdot 2\sqrt{5} = 80\sqrt{5} \]
Thus, the product of \( 8\sqrt{2} \) and \( 5\sqrt{10} \) in simplest radical form is:
\[ \boxed{80\sqrt{5}} \]
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