Prove that the product of 2π

Question

and 3/4
is an irrational number. Find the product and explain why the value is irrational. Explain your reasoning in 1–2 sentences.(2 points) simple!!!!!

Answer 1

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.