To simplify the expression \( 4\sqrt{10} \times 7\sqrt{32} \), we can follow these steps:
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Multiplying the coefficients and the square roots separately: \[ 4 \times 7 = 28 \] \[ \sqrt{10} \times \sqrt{32} = \sqrt{10 \times 32} = \sqrt{320} \]
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Now, we have: \[ 28\sqrt{320} \]
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Next, we need to simplify \(\sqrt{320}\). We start by factoring \(320\): \[ 320 = 32 \times 10 = 16 \times 20 = 16 \times 4 \times 5 = (4^2) \times 4 \times 5 \]
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Recognizing \(16\) as a perfect square: \[ \sqrt{320} = \sqrt{16 \times 20} = \sqrt{16} \times \sqrt{20} = 4\sqrt{20} \]
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Now, we replace \(\sqrt{320}\) in the expression: \[ 28\sqrt{320} = 28 \times 4\sqrt{20} = 112\sqrt{20} \]
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Additionally, we can simplify \(\sqrt{20}\): \[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \]
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Substitute this back into the expression: \[ 112\sqrt{20} = 112 \times 2\sqrt{5} = 224\sqrt{5} \]
Thus, the fully simplified exact answer is: \[ \boxed{224\sqrt{5}} \]