Question

is the number $\pi \cdot -6$ rational or irrational?
rational irrational
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Explanation:

Step1: Recall definitions

A rational number can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. An irrational number cannot be expressed in this form. $\pi$ is irrational (it has a non - repeating, non - terminating decimal expansion: $3.1415926535\cdots$).

Step2: Analyze the product $- 6\pi$

When we multiply a non - zero rational number ($-6$ is rational as $-6=\frac{-6}{1}$) with an irrational number ($\pi$), the result is irrational. Suppose, for the sake of contradiction, that $-6\pi$ is rational. Then $-6\pi=\frac{a}{b}$ for some integers $a$ and $b$ with $b
eq0$. But then $\pi=\frac{-a}{6b}$, which would imply that $\pi$ is rational (since $-a$ and $6b$ are integers and $6b
eq0$), contradicting the fact that $\pi$ is irrational. So $-6\pi$ (which is the same as $\pi\cdot(-6)$) is irrational.

Answer:

irrational