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Bot GPT 4-o mini The product of \( 2\pi \) and \( \frac{3}{4} \) is
\[
2\pi \cdot \frac{3}{4} = \frac{3}{2} \pi.
\]
To show that \( \frac{3}{2} \pi \) is irrational, we can argue that multiplying a rational number (like \( \frac{3}{2} \)) by an irrational number (like \( \pi \)) will always yield an irrational number. Since \( \pi \) is known to be irrational, \( \frac{3}{2} \pi \) is therefore also irrational.