To determine whether the product of \(2\pi\) and \(\frac{5}{6}\) is rational or irrational, we can analyze each component.
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\(2\pi\) is an irrational number because \(\pi\) itself is irrational, and multiplying an irrational number by a non-zero rational number (in this case, \(2\), which is rational) results in an irrational number.
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\(\frac{5}{6}\) is a rational number because it is the ratio of two integers (5 and 6).
Now, when we multiply \(2\pi\) (irrational) by \(\frac{5}{6}\) (rational), the product will remain irrational. Thus, the product:
\[ 2\pi \cdot \frac{5}{6} \]
is irrational.
In conclusion, the product of \(2\pi\) and \(\frac{5}{6}\) is irrational.