prove that the product of 2pie and 3/4 is an irrational number. Find the product and explain why the value is irrational. Explain in 1-2 sentences

The product of \( 2\pi \) and \( \frac{3}{4} \) is

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

Since \( \pi \) is an irrational number, and the product of a non-zero rational number (like \( \frac{3}{2} \)) and an irrational number is also irrational, \( \frac{3\pi}{2} \) is irrational. Thus, \( 2\pi \cdot \frac{3}{4} \) is irrational because it represents a non-repeating, non-terminating decimal.