To simplify the expression \( 5\sqrt{32} \cdot 2\sqrt{7} \), we can start by multiplying the coefficients and the square roots separately.
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First, multiply the coefficients: \[ 5 \cdot 2 = 10 \]
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Next, multiply the square roots: \[ \sqrt{32} \cdot \sqrt{7} = \sqrt{32 \cdot 7} \]
Now calculate \( 32 \cdot 7 \): \[ 32 \cdot 7 = 224 \] So, we have: \[ \sqrt{32} \cdot \sqrt{7} = \sqrt{224} \]
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Now we combine the results: \[ 5\sqrt{32} \cdot 2\sqrt{7} = 10\sqrt{224} \]
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Now we need to simplify \( \sqrt{224} \). We start by finding its prime factorization: \[ 224 = 2^5 \times 7 \]
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Using the properties of square roots, we can simplify \( \sqrt{224} \): \[ \sqrt{224} = \sqrt{2^5 \times 7} = \sqrt{(2^4) \cdot 2 \cdot 7} = \sqrt{16 \cdot 14} = \sqrt{16} \cdot \sqrt{14} = 4\sqrt{14} \]
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Substituting back into our expression gives: \[ 10\sqrt{224} = 10 \cdot 4\sqrt{14} = 40\sqrt{14} \]
Thus, the product of the expression \( 5\sqrt{32} \cdot 2\sqrt{7} \) in simplest radical form is:
\[ \boxed{40\sqrt{14}} \]