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The product of \(2\pi\) and \(\frac{3}{4}\) is
\[
2\pi \cdot \frac{3}{4} = \frac{3\pi}{2}.
\]
Since \(\pi\) is an irrational number, any non-zero rational multiple of \(\pi\) (such as \(\frac{3}{2}\)) is also irrational. Therefore, \(\frac{3\pi}{2}\) is irrational, confirming that the product \(2\pi \cdot \frac{3}{4}\) is indeed irrational.