Question
prove that the product of 2piew and 3/4 is an irrational number. Find the product and explain why the value is irrational. Explain in 1-2 sentences
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Answered by
GPT-4o mini
To find the product of \( 2\pi e \) and \( \frac{3}{4} \):
\[
2\pi e \cdot \frac{3}{4} = \frac{3}{2} \pi e
\]
Since both \( \pi \) and \( e \) are known to be irrational numbers, and the product of a non-zero rational number (\( \frac{3}{2} \)) with an irrational number is also irrational, it follows that \( \frac{3}{2} \pi e \) is irrational as well. Thus, \( \frac{3}{2} \pi e \) is irrational.
\[
2\pi e \cdot \frac{3}{4} = \frac{3}{2} \pi e
\]
Since both \( \pi \) and \( e \) are known to be irrational numbers, and the product of a non-zero rational number (\( \frac{3}{2} \)) with an irrational number is also irrational, it follows that \( \frac{3}{2} \pi e \) is irrational as well. Thus, \( \frac{3}{2} \pi e \) is irrational.
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