Prove that the product of 2pie and 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:

\[
2\pi \times \frac{3}{4} = \frac{3 \times 2\pi}{4} = \frac{3\pi}{2}.
\]

Now, to determine whether \(\frac{3\pi}{2}\) is rational or irrational, we note that \(\pi\) is an irrational number, and the product or quotient of a non-zero rational number (in this case, \(\frac{3}{2}\)) and an irrational number is always irrational. Since \(\frac{3\pi}{2}\) is the product of the rational number \(\frac{3}{2}\) and the irrational number \(\pi\), it follows that \(\frac{3\pi}{2}\) is also irrational.

Thus, the product \(2\pi \times \frac{3}{4} = \frac{3\pi}{2}\) is an irrational number.