Given that \(\sqrt{7}\) is irrational, prove that \(8 - 12\sqrt{7}\) is irrational.
Proof:
Let \(8 - 12\sqrt{7}\) is
\(8 - 12\sqrt{7} = \frac{p}{q}\) be a rational number, where \(p\) and \(q\) are and
After simplification, we get
\(\sqrt{7}=-\frac{1}{12}(\frac{p}{q}-8)\)
Since, \(p\) and \(q\) are , \(-\frac{1}{12}(\frac{p}{q}-8)\) will also be .
Therefore, \(\sqrt{7}\) is .
This contradicts the fact that \(\sqrt{7}\) is .
Hence, \(8 - 12\sqrt{7}\) is .