To determine which product is rational, we need to analyze each option:
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\(8 \cdot \sqrt{25}\):
- The square root of 25 is 5, so this simplifies to \(8 \cdot 5 = 40\), which is a rational number.
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\(10 \cdot \sqrt{8}\):
- The square root of 8 is \(\sqrt{4 \cdot 2} = 2\sqrt{2}\), making it \(10 \cdot 2\sqrt{2} = 20\sqrt{2}\). Since \(\sqrt{2}\) is irrational, this product is irrational.
-
\(12 \cdot \pi\):
- Since \(\pi\) is an irrational number, this product (\(12 \pi\)) is also irrational.
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\(12 \cdot 5.3420941...\):
- The number \(5.3420941...\) seems to be a decimal representation of a rational number, but we don't know if it terminates or is a repeating decimal. However, if it's just a non-terminating decimal with no specific repeating pattern indicated, we can't assume it's rational based on the number given.
The only option which we can confidently say is rational is:
- \(8 \cdot \sqrt{25}\), which equals 40.
Thus, the rational product is \(8 \cdot \sqrt{25}\).