Prove that \(7+6\sqrt{7}\) is irrational.
Proof:
Let \(7+6\sqrt{7} =\frac{p}{q}\) be a number, where \(p\) and \(q\) are and \(q\neq 0\)
\(6\sqrt{7}=\frac{p}{q}-7\)
\(\sqrt{7}=\frac{\frac{p}{q}-7}{6}\)
Since, , \(\frac{\frac{p}{q}-7}{6}\) will also be .
Therefore, \(\sqrt{7}\) is rational.
This the fact that \(\sqrt{7}\) is .
Hence, \(7+6\sqrt{7}\) is irrational.