Given that \(\sqrt{11}\) is irrational, prove that \(6 + 13\sqrt{11}\) is irrational.
Proof:
Let \(6 + 13\sqrt{11}\) is
\(6 + 13\sqrt{11} = \frac{p}{q}\) be a rational number, where \(p\) and \(q\) are and
After simplification, we get
\(\sqrt{11}=\frac{1}{13}(\frac{p}{q}-6)\)
Since, \(p\) and \(q\) are , \(\frac{1}{13}(\frac{p}{q}-6)\) will also be .
Therefore, \(\sqrt{11}\) is .
This contradicts the fact that \(\sqrt{11}\) is .
Hence, \(6 + 13\sqrt{11}\) is .