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

To find the product of \(2\pi\) and \(\frac{3}{4}\), we calculate:

\[
\text{Product} = 2\pi \cdot \frac{3}{4} = \frac{3 \cdot 2\pi}{4} = \frac{3\pi}{2}
\]

Now, we need to determine whether \(\frac{3\pi}{2}\) is irrational. Since \(\pi\) itself is an irrational number, and multiplying an irrational number by a non-zero rational number (like \(\frac{3}{2}\)) results in an irrational number, \(\frac{3\pi}{2}\) is also irrational. Thus, the product \(\frac{3\pi}{2}\) is irrational.