Prove that the product of 2pi and 3/4 is an irrational number. Find the product and explain why the value is irrational. Explain your reasoning in 1-2 sentences. Simplify it.

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 \(\pi\).