To determine if the product of \(3\pi\) and \(\frac{2}{5}\) is irrational, we begin by noting that \(\pi\) is an irrational number. The product of a non-zero rational number (like \(\frac{2}{5}\)) and an irrational number (like \(3\pi\)) is also irrational. Therefore, \(3\pi \cdot \frac{2}{5}\) is irrational.
For a product that is rational, consider \(3 \cdot \frac{2}{5}\). Both \(3\) (a rational number) and \(\frac{2}{5}\) (also a rational number) multiplied together yield \(\frac{6}{5}\), which is rational. The reasoning lies in the fact that the product of two rational numbers is always rational.
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