To find the product of \( 8\sqrt{14} \) and \( 5\sqrt{12} \), we can follow these steps:
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Multiply the coefficients (numbers outside the radical): \[ 8 \times 5 = 40 \]
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Multiply the radical parts: \[ \sqrt{14} \times \sqrt{12} = \sqrt{14 \times 12} \]
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Calculate \( 14 \times 12 \): \[ 14 \times 12 = 168 \] So, we have: \[ \sqrt{14} \times \sqrt{12} = \sqrt{168} \]
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Now combine the results: \[ 8\sqrt{14} \times 5\sqrt{12} = 40\sqrt{168} \]
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Next, simplify \( \sqrt{168} \):
- Factor \( 168 \) into its prime factors: \[ 168 = 2^3 \times 3 \times 7 \]
- The perfect squares in \( 168 \) are \( 4 \) (which is \( 2^2 \)): \[ \sqrt{168} = \sqrt{4 \times 42} = \sqrt{4} \times \sqrt{42} = 2\sqrt{42} \]
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Substitute \( \sqrt{168} \) back into the expression: \[ 40\sqrt{168} = 40 \times 2\sqrt{42} = 80\sqrt{42} \]
Therefore, the product of \( 8\sqrt{14} \) and \( 5\sqrt{12} \) in simplest radical form is: \[ \boxed{80\sqrt{42}} \]
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