Prove that the product of 2π

Question

Prove that the product of  2π
  and  34
  is an irrational number. Find the product and explain why the value is irrational. Explain your reasoning in 1–2 sentences.(2 points)

GPT-4o mini · Answer

To find the product of $ 2\pi $ and $ 34 $, we calculate: $$ 2\pi \times 34 = 68\pi. $$ The number $ \pi $ is known to be irrational, which means it cannot be expressed as a fraction of two integers. The product of a non-zero rational number (like $ 68 $) and an irrational number ($ \pi $) is always irrational. Therefore, $ 68\pi $ is an irrational number.