To find the product of \(2\pi\) and \(\frac{3}{4}\), we multiply:
\[ 2\pi \cdot \frac{3}{4} = \frac{3 \cdot 2\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2}. \]
The number \(\pi\) is known to be irrational, and the product of a non-zero rational number (\(\frac{3}{2}\)) and an irrational number (\(\pi\)) results in an irrational number. Therefore, \(\frac{3\pi}{2}\) is also irrational.
In conclusion, the product \(\frac{3\pi}{2}\) is irrational because it is a product involving the irrational number \(\pi\).