To determine which product is rational, we will analyze each option:
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12⋅5.3420941...: This is a product of a rational number (12) and an irrational number (5.3420941...). The product of a rational and an irrational number is irrational.
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10⋅√8: Here, \( \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \), which is irrational. Since 10 is rational, the product \( 10 \cdot \sqrt{8} \) is irrational.
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12⋅π: π is an irrational number. Since 12 is rational, the product \( 12 \cdot π \) is irrational.
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8⋅25−−√: \( \sqrt{25} = 5 \), which is rational. Thus, \( 8 \cdot \sqrt{25} = 8 \cdot 5 = 40 \), which is rational.
Based on this analysis, the only rational product is:
8⋅25−−√ (which equals 40).