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)

Question

GPT-4o mini · Answer

To evaluate the product of $2\pi$ and $34$, we calculate: $$ 2\pi \times 34 = 68\pi $$ Now, we need to determine if $68\pi$ is irrational. Since $\pi$ is known to be an irrational number, and the product of a non-zero rational number (68) and an irrational number ($\pi$) is also irrational, we conclude that $68\pi$ is irrational. Thus, the product $68\pi$ is an irrational number because it cannot be expressed as a fraction of two integers.